Relaxation in Optimization Theory and Variational Calculus [2nd Revised and Extended Edition] 9783110590852, 9783110589627

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Tomáš Roubíček Relaxation in Optimization Theory and Variational Calculus

De Gruyter Series in Nonlinear Analysis and Applications

Editor in Chief Jürgen Appell, Würzburg, Germany Editors Catherine Bandle, Basel, Switzerland Alain Bensoussan, Richardson, Texas, USA Avner Friedman, Columbus, Ohio, USA Mikio Kato, Tokyo, Japan Wojciech Kryszewski, Torun, Poland Umberto Mosco, Worcester, Massachusetts, USA Simeon Reich, Haifa, Israel Vicenţiu D. Rădulescu, Krakow, Poland

Volume 4

Tomáš Roubíček

Relaxation in Optimization Theory and Variational Calculus 2nd Edition

Mathematics Subject Classification 2020 Primary: 49-02; 49J, 49K, 54D35, 90C, 91A. Secondary: 34H05, 34H05, 35Q93, 37N40, 46A55, 46N10, 65K10, 74B20, 74N, 78A30.

Author Prof. Ing. Tomáš Roubíček, DrSc. Mathematical Institute Faculty of Mathematics & Physics Charles University Sokolovská 83, CZ-186 75 Praha 8 and Institute of Thermomechanics Czech Academy of Sciences Dolejškova 5, CZ-182 00 Praha 8 Czech Republic

ISBN 978-3-11-058962-7 e-ISBN (PDF) 978-3-11-059085-2 e-ISBN (EPUB) 978-3-11-058974-0 ISSN 0941-813X

Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2020 Walter de Gruyter GmbH, Berlin/Boston Typesetting: Tomáš Roubíček Printing and binding: CPI books GmbH, Leck www.degruyter.com

Ê

To the memory of Marie and Dr. Ervin Robits hek, vi tims of the Holo aust.

Contents Prefa e Ê XI Prefa e to the se ond edition Ê XVIII 1

Ê1 Ê1

Ba kground Generalities

1.1

Order and topology

1.2

Linear, nonlinear, and onvex analysis

Ê 10

Ê9

1.2.a

Linear fun tional analysis

1.2.b

Convex sets

1.2.

Means of ontinuous fun tions

1.2.d

Solving abstra t nonlinear equations

1.3

Ê 14

Fun tion and measure spa es

Ê 18 Ê 22

Ê 25 Ê 26

1.3.a

Bo hner and Lebesgue spa es

1.3.b

Spa es of measures

1.3.

Spa es of smooth fun tions and Sobolev spa es

1.4 1.4.a 1.4.b 1.4. 1.4.d 1.5 1.5.a 1.5.b 1.5. 1.5.d

2 2.1

Ê 30

Some dierential and integral equations

Ê 38

Ordinary dierential and dierential-algebrai equations

Ê 45 Partial dierential equations of paraboli type Ê 50 Integral equations of Hammerstein type Ê 54 Basi s from optimization theory Ê 55 Existen e, stability, approximation Ê 55 Optimality onditions of the 1st order Ê 61 Multi riteria optimization Ê 70 Non- ooperative game theory Ê 72 Partial dierential equations of ellipti type

Theory of Convex Compa ti ations Ê 81 Convex ompa ti ations Ê 82

Ê 84

2.2

Canoni al form of onvex ompa ti ations

2.3

Convex

2.4

Approximation of onvex ompa ti ations

2.5

Extension of mappings

2.6

Inverse systems of onvex ompa ti ations

3 3.1 3.1.a 3.1.b 3.1.

Ê 33

- ompa ti ations Ê 93

Ê 106

Young Measures and Their Generalizations Classi al Young measures Ê 118 Basi s enario and results Ê 118 Some illustrations Ê 131 Some more results Ê 133

Ê 103 Ê 111

Ê 117

Ê 38

VIII 3.2

Ë

Contents

Various generalizations

Ê 135 Ê 136

3.2.a

Generalization by Fattorini

3.2.b

Generalization by S honbek, Ball, Kinderlehrer and Pedregal

3.2. 3.2.d 3.3 3.3.a 3.3.b 3.3. 3.3.d 3.3.e 3.4 3.5 3.5.a 3.5.b 3.5. 3.5.d 3.6 3.6.a 3.6.b 3.6.

4 4.1

Ê 146 L1 -spa es Ê 163 p A lass of onvex ompa ti ations of balls in L -spa es Ê 166 p Generalized Young fun tionals YH % ; S Ê 166 The omposition h DZ  Ê 174 Some on rete examples Ê 176 Coarse polynomial ompa ti ation by algebrai moments Ê 186 p Ê 188 Compatible systems of Young fun tionals on B I ; L p A lass of onvex  - ompa ti ations of L -spa es Ê 191 Approximation theory Ê 204 A general onstru tion Ê 205 An approximation over Ê 211 An approximation over S Ê 215 Higher-order onstru tions by quasi-interpolation Ê 219 Extensions of Nemytski mappings Ê 222 One-argument mappings: ane extensions Ê 223 Two-argument mappings: semi-ane extensions Ê 226 Two-argument mappings: bi-ane extensions Ê 236 Generalization by DiPerna and Majda Fonse a's extension of

;

(

)

(

Ê 243 Abstra t optimization problems Ê 244

)

Relaxation in Optimization Theory

Ê 263

4.2

Optimization problems on Lebesgue spa es

4.3

Optimal ontrol of nite-dimensional dynami al systems

Ê 277

4.3.a

Original problem

4.3.b

Relaxation s heme, orre tness, well-posedness

4.3.

Optimality onditions

4.3.d

Ê 295 Approximation theory Ê 305

Illustrative omputational simulations: os illations

4.3.f

Illustrative omputational simulations: os illations and on entrations

4.3.g

Optimal ontrol of dierential-algebrai systems

4.4

4.4.b 4.4. 4.4.d 4.5 4.5.a 4.5.b

Ellipti optimal ontrol problems

Ê 325

Ê 277

Ê 288

4.3.e

4.4.a

Ê 138

Ê 310

Ê 315 Ê 318

Ê 325 Ê 333 Optimal ontrol of Navier-Stokes' equations Ê 339 Optimal material design of some stratied media Ê 342 Paraboli optimal ontrol problems Ê 346 Innite-dimensional dynami al-system approa h Ê 348 The original problem and its relaxation

Optimality onditions in semilinear ase

An approa h through paraboli partial dierential equations

Ê 355

Contents

Optimal ontrol of Navier-Stokes equations

4.5. 4.6

5 5.1 5.2 5.3 5.4 5.5 5.6

6 6.1 6.2 6.3 6.4 6.5 6.6 6.7

7

Optimal ontrol of integral equations

Ê 373

Ê 382 Ê 383 Relaxation of variational problems; p ¡ 1. Ê 394 Optimality onditions for relaxed problems Ê 401 Relaxation of variational problems; p # 1. Ê 410 Convex approximations of relaxed problems Ê 416 Example: Mi rostru ture in ferromagneti materials Ê 429 Convex ompa ti ations of Sobolev spa es

Relaxation in Variational Cal ulus: Ve torial Case Ê 435 Prerequisites around quasi onvexity Ê 436 Gradient generalized Young fun tionals Ê 441 Variational problems and their relaxation Ê 453 FEM-approximation Ê 458 Further approximation: an inner ase Ê 461 Further approximation: an outer ase Ê 464 Multiwell problems: illustrative al ulations Ê 469 Relaxation in Game Theory

Ê 483

Ê 484

Abstra t game-theoreti al problems

7.2

Games on Lebesgue spa es

7.3

Example: Games with dynami al systems

7.4

Example: Ellipti games

8.1

8.2.b 8.2. 8.3

Ê 507

Ê 493

Ê 513

Evolution on abstra t onvex ompa ti ations Rate-independent evolution

Ê 515

Quasistati rate-dependent evolution

8.1.b

8.2.a

Ê 490

Relaxation in evolutionary problems

8.1.a

8.2

Ê 365

Relaxation in Variational Cal ulus: S alar Case

7.1

8

Ë IX

Ê 513

Ê 521

Appli ations of relaxation in rate-independent evolution Perfe t plasti ity at small strains

Ê 524

Ê 523

Ê 526 Ê 528 Notes about measure-valued solutions to paraboli equations Ê 532 Evolution of mi rostru ture in ferromagneti materials

Evolution of mi rostru ture in shape-memory materials

Bibliography Ê 539 List of Symbols Ê 571 Index Ê 577

Prefa e Has not every ... variational problem a solution, provided ... if need be that the notion of a solution shall be suitably extended? [386, p.470℄

David Hilbert This

1

senten e

had

an

immense

(1862-1943)

inuen e

on

analysis: it led ultimately to `weak' or generalised solutions, to my generalised urves and to S hwartz distributions, ... , and then e to hattering ontrols, to mixed strategies in Game Theory, all things that we nd essential today. [809, p.241℄

Lauren e Chisholm Young

(1905-2000)

Let us begin with a pie e of history. In his 20th problem, David Hilbert started a fas inating development going through the whole 20th entury, namely an eort to generalize more and more the notion of solutions to various applied problems that mathemati s met. This in ludes in parti ular the al ulus of variations, ordinary and partial dierential equations, problems from optimization theory and game theory. The generalization is always based on a natural (i.e. ontinuous2) extension of the original problems; su h ontinuous extension is often addressed as relaxation.3 The rst su

ess was a generalization of the lassi al solution to dierential equations to the weak solution4, whi h admits less smoothness than ne essary to evaluate the original dierential equation in the usual sense. This is basi ally related with the theory of distributions invented later by L. S hwartz [723℄. The weak formulation of boundary and/or initial value problems for dierential equations is nowadays a

epted so generally that the original lassi al solution is often re koned as less natural. Later it was found that in some nonlinear problems one must handle beside the loss of

1

It refers to a senten e by D. Hilbert with a similar meaning as that one ited above.

2

The ontinuity always refers to some topologies and, in fa t, may mean also lower semi ontinuity,

f. Remark 2.38.

3

A tually, the word relaxation is used also in quite very many dierent o

asions with ompletely

dierent meanings, ranging from time-evolving pro esses in various rheologi al materials or in nu lear physi s or hemistry aiming towards some stress-free states or rest states or, in a ertain parallel of de reasing some stress whatever stress may mean, to relaxation in physiotherapy or re reology or psy hology (e.g. before visiting dentist or so). Even in mathemati s itself, the word relaxation is also used as iterative te hniques for solving systems of equations or as spe i handling of onstraints in mathemati al programming, or for relaxation os illations generated by nonlinear os illators.

4

First appearan e of this philosophy is probably in the work by J. Leray [488℄, where essentially the

spa es later named as Sobolev had been introdu ed. The huge development took pla e sin e fties sin e the works by S.L. Sobolev, J.-L. Lions, and many other.

https://doi.org/10.1515/9783110590852-201

XII

Ë

Prefa e

smoothness also two other phenomena: os illations and on entrations of the solutions. The former phenomenon was for the rst time treated in the pioneering works by L.C. Young [805807℄, followed also by E.J. M Shane [526, 527℄. The latter one was investigated typi ally in onne tion with the famous Plateau's minimal surfa e problem by a lot of authors5 . The renaissan e of Young's idea was in onne tion with generalization of solutions to some optimal ontrol problems6 and mu h later to some nonlinear partial dierential equations and non onvex variational problems arising in ontinuum me hani s7. Just re ently, DiPerna and Majda in their pioneering work [266℄ made the rst attempt to oup with on entration and os illations ee ts simultaneously. The ommon feature of this ever going generalization is to solve more and more general problems and to ensure existen e of their solutions (in a reasonable sense) in larger lasses than the original ones where the existen e an a tually fail. Typi al property that makes it possible is ompa tness. The enlarged sets, where the solutions are sought, represent thus ertain ompa ti ations of the original sets where the problems are  lassi ally formulated, and the ner the ompa ti ations, the more generalized the solutions we thus obtain. This an, in prin iple, yield eventually very ne and abstra t ompa ti ations whi h are not easy to imagine, nor to use for more detailed investigations. A natural restri tion of a generality is to require the existen e of some auxiliary algebrai stru ture whi h ould be also used for a more detailed analysis. It appears useful to require the investigated ompa ti ation to be a onvex subset of some linear topologi al spa e; then we will speak about a onvex ompa ti ation.8 Typi al usage of the onvex stru ture is for optimality onditions. The development of the resear h of optimality onditions has been also en ouraged by David Hilbert [386℄, namely in his 23th problem in onne tion with the al ulus of varia-

5

See Giusti [356℄ for detailed referen es.

6

The rst ontributions appeared from sixties by Gamkrelidze [343, 344℄, Ghouila-Houri [352℄, M -

Shane [528℄, Medhin [531℄, Rishel [644℄ and Warga [786790℄. Re ently many other authors dealt with these so- alled relaxed ontrols, e.g. Ahmed [5, 6, 8℄, Balder [5053℄, Berlio

hi and Lasry [114℄, Carlson [170℄, Chryssoverghi [209, 210℄, Goh and Teo [358, 758℄, Halanay [373℄, Fattorini [294299℄, Papageorgiou [588, 589, 591℄, Rosenblueth [651653℄, S hwarzkopf [724℄, et . Cf. also M Shane [529℄ for a survey. Another relaxation approa h has been invented by J.E. Rubio, see Remark 4.49.

7

This was initialized by Tartar [747℄, followed by a lot of other authors, espe ially by Ball [61℄, Ball

and James [63, 64℄, Chipot and Kinderlehrer [204℄, Da orogna [241, 242℄, DiPerna [265℄, Evans [287℄, Kinderlehrer and Pedregal [424, 426℄, Murat [564℄, S honbek [721℄, et .

8

In fa t, my rst attempts in late 80tieth were to deal with problems formulated on Bana h spa e

(as is indeed a typi al ase) and then rather to extent as mu h as possible (in an as mu h ontinuous way as possible) the linear-spa e algebrai stru ture on the ompa ti ation, see [656℄. Later, another attempt of su h a sort has been done by J. Perán [607℄. Su h onstru tions o

ur however very umbersome and I thus developed the theory of onvex ompa ti ations not relying on the linear stru ture of original spa es, if any.

Prefa e

Ë XIII

tions. This led to the weak formulation of the Euler-Lagrange equation, and later also to an appropriate optimality ondition for problems involving os illation phenomena, namely the so- alled Euler-Weierstrass ondition for variational problems9 and the Pontryagin maximum prin iple for optimal ontrol problems.10 The essential advantage of the onvex- ompa ti ation method is that the onvex geometry is used to derive the optimality onditions by a quite onventional dierential al ulus in various problems, not only in optimal ontrol.11 During the whole 20th entury, we an also observe a parallel, intensive development of the supporting bran hes of mathemati s, in parti ular of general topology, abstra t fun tional analysis, and later also nonlinear analysis and optimization theory. The purpose of this book is to ree t these a hievements and give a fairly abstra tanalysis viewpoint to the on rete problems mentioned above. Also it an be said that the presented viewpoint represents a properly nonlinear approa h be ause it forgets (to more or less extent) the original linear stru ture (if any) and imposes a new one for the relaxed problem. I believe this ree ts genuinely the fa t that original nonlinear problems themselves violate (to more or less extent) the linear stru ture of the original spa es. Let us now go briey through the ontent of the book. After Chapter 1, whi h only summarizes very briey and mostly without proofs some more or less standard needed mathemati al ba kground, the general theory of onvex ompa ti ation is introdu ed in Chapter 2. This represents an abstra t framework of our relaxation method. Then, in Chapter 3, this general onvex-



ompa ti ation theory is applied to get ( - or also lo ally) ompa t onvex envelopes of Lebesgue spa es, whi h represents a basi tool for relaxation of on rete problems appearing in variational al ulus and optimization of systems des ribed by dierential equations. In parti ular ases, su h envelopes an be sequentially lo ally ompa t onvex subsets, imitating thus basi attributes of nite-dimensional (Eu lidean) spa es and fa ilitating usage of onventional analyti al methods.

9

For one-dimensional relaxed problems it was rst formulated by Young [806℄ and M Shane [527℄,

and generalized for spe ial two-dimensional ases in [807℄.

10

First it was formulated only for ordinary ontrols for system governed by ordinary dierential

equations by Russian s hool around L.S. Pontryagin, involving also V.G. Boltyanski , R.V. Gamkrelidze, and E.F. Mish henko in [125, 127, 344, 616, 617℄, following some earlier ideas of D.W. Bushaw [158℄ and M. Hestenes [382℄. Origin of these ideas an be found, however, already in works by C. Carathéodory [168℄, F.A Valentine [775℄ and K. Weierstraÿ [796℄; f. also Pes h at al. [608611℄ for histori al surveys. The extension of maximum prin iple for relaxed ontrols is due to Gamkrelidze [345℄ and Rishel [644℄ and, in a more general form by Avakov and Magaril-Il'yaev [39, 40℄, Fattorini [294, 298℄, Halanay [373℄, Kaskosz [420℄, S hwarzkopf [724℄, Warga [791℄, and many others.

11

This generalizes R.V. Gamkrelidze's proof whi h used hattering ontrols in ontrast to the origi-

nal derivation of maximum prin iples whi h exploited quite te hni al, so- alled needle (sometimes

alled also spike) variations invented by V.G. Boltyanski  and E.J. M Shane, f. [610℄ for a histori al reminis en e. Cf. [509℄ for a ertain simpli ation of su h arguments.

XIV

Ë

Prefa e

Having the tools prepared, we will be able to treat on rete problems. In Chapters 46 they will have typi ally the abstra t stru ture

(P )

J : U,Y Ù

where

J(u ; y) (u ; y) # 0 ; B(u ; y) ¢ 0 ; u ò U; y ò Y;

Minimize . 6

subje t to > 6 F

R is a ost fun tion, Y a Bana h spa e of states, U is a set of

 : Y , U Ù  and B : U , Y Ù 1 are mappings forming respe tively the state equation and state onstraints,  and  1 are Bana h spa es, admissible  ontrols, and

the latter one being ordered. The extended (relaxed) problems will then have the stru ture: (RP)

Minimize . 6

subje t to > 6 F

J (z ; y)  (z ; y) # 0 ; B (z ; y) ¢ 0 ; z ò K ; z ò Z ; y ò Y;





J : Z , Y Ù R, K is a onvex set in a lo ally onvex linear topologi al spa e Z ,  : Z , Y Ù  and B : Z , Y Ù 1 are ontinuous mappings. The original set U is to be onsidered as densely embedded into K and J ,  , and B as extensions of J ,  , and B , respe tively.

where and













The following questions will be pursued both on an abstra t level and in parti ular

ases:12 

Relation between (P ) and (RP); in parti ular a so- alled orre tness of the relaxation s heme.



Relations between various relaxation s hemes (RP) for a given (P ).



Existen e and stability of solutions to (RP); well-posedness of (RP).



First-order optimality onditions for (RP); Pontryagin's or Weierstrass' maximum prin iples.



Impa ts of results for (RP) to the original problem (P ).



Approximation theory for the relaxed problem (RP).



Numeri al implementation of approximate relaxed problems.

Going from simpler tasks to more ompli ated ones, we begin in Chapter 4 with the optimal ontrol problems, whi h ertainly represents the simplest variant of (P ), at least if the state onstraints have a reasonable stru ture. A typi al example is an optimal ontrol problem for a nonlinear dynami al system (

t ò (0; T)

represents a

time variable):

12

As the emphasis is put to relaxation method itself, a lot of other aspe ts will remain untou hed;

this in ludes higher-order optimality onditions, problems yielding a nonsmooth relaxed problems, sensitivity analysis, et .

Ë XV

Prefa e

T . 6 Minimize X ' ( t ; y; u ) d t 6 6 6 0 6 6 6 6 d y 6 6 # f(t ; y; u) for a.a. subje t to 6 6 dt

(P1 )

( ost fun tional)

t ò (0; T);

(state equation)

y(0) # y0 ; (initial ondition) u(t) ò S(t) for a.a. t ò (0; T); ( ontrol onstraints) ( t ; y ( t )) ¢ 0 for all t ò [0 ; T ℄ ; (state onstraints) y ò W 1 q (0; T; Rn ); u ò L p (0; T; Rm );

> 6 6 6 6 6 6 6 6 6 6 6 6 F

;

' : (0; T) , Rn , Rm Ù R, f : (0; T) , Rn , Rm Ù Rn , : [0; T℄ , Rn Ù y0 ò Rn , and S(t) ’ Rm are subje ted to ertain data quali ation, W 1 q and

where

R

k,

;

L p denote respe tively the Sobolev and the Lebesgue spa es. Quite equally, the state

equation an (and will) be a nonlinear partial dierential equation, say of an ellipti or a paraboli type, or a nonlinear integral equation. In every ase, the resulted relaxed

 (z ; y) # 0 admits, for any z ò K , pre isely one solution y #  ( u ) and the mapping  : K Ù Y , alled an problem (RP) has the property that the extended state equation





(extended) state operator, is ontinuous and even smooth. Chapter 5 is devoted to s alar-variational- al ulus problems of the type

. Minimize

(P )

> F subje t to

2

X ' ( x ; y ( x ) ; › y ( x ))d x

y ò W 1; p ( ) ;

Rn is a bounded Lips hitz domain and the energy density ' : , R , R Ù R satises ertain data quali ation but ' x; r; : Rn Ù R is allowed to be

where

n

’

(

-)

non onvex. Then the relaxed variational problem has the form (RP) with

onvex

- ompa t but non ompa t, and B # 0.





linear,

K



The ve torial-variational- al ulus problems of the type

. Minimize

(P )

> F subje t to

3

X ' ( x ; y ( x ) ; › y ( x ))d x

y ò W 1; p ( ; m ) ;

R

' : ,R R Ù R are handled in Chapter 6; the adje tive ve torial refers m to that y is R -valued, m ¡ 1. Although at the rst sight (P3 ) has the same form as (P ), (P ) is mu h more di ult than (P ) when simultaneously n £ 2 and m £ 2 and 2 3 2 m,

with

m,n

there are several essential dieren es between Chapters 5 and 6. One of them is that either

K

is non onvex or



is nonlinear and also

B #Ö 0 in general. Contrary to (P2 )

whose understanding is fairly omplete, there are still many open essential questions as far as the relaxation of (P3 ) on erns. This is basi ally onne ted with our poor understanding of quasi onvexity and related questions. In Chapters 46, two fundamental on epts (i.e. ompa tness and onvexity) have been used rather separately  the former one ensured existen e and stability of solutions while the latter one enabled us to make a more detailed analysis, e.g. to pose optimality onditions.

XVI

Ë

Prefa e

However, there are appli ations with mu h more intimate onne tions between

onvexity and ompa tness. We have in mind the non ooperative game theory, or more generally the underlying S hauder-type xed-point theory, where typi ally ompa tness and onvexity are required simultaneously to ensure mere existen e of solutions. This will be the topi of Chapter 7, though it represents rather a sample of the wide area of potential appli ations. Let me onje ture here that every abstra t problem where ompa tness and onvexity plays a ertain role an reasonably be interpreted as a relaxed problem to a ertain original problem. Another usage of ompa tness together with onvexity is for evolutionary relaxed problems where hosen onvexity serves for denition of some time derivative. On top of it, the mentioned S haudertype xed point te hnique an be used here for existen e of time-periodi solutions. This is exposed in Chapter 8.13 It should be emphasized that the relaxation method has, beside its purely mathe-

inf (P ) ¡ min(RP) must be in some situations prevented while in other situations it is wel ome.

mati al aspe ts, also an essential interpretation aspe t. For example, the fa t

The former ase is typi al for variational problems where this fa t means that optimal relaxed solution annot be attained by a minimizing sequen e for the original problem. This is related with a ne essity to hold the variational onstraint

u # ›y exa tly.14

The latter ase appears typi ally in state- onstrained optimal ontrol problems where only the state equation is to be held exa tly while the state onstraints may be held only approximately, with a ertain toleran e. Then the gap

inf (P ) ¡ min(RP) means

that relaxed ontrols an a hieve a lower  ost than the original ones, whi h is naturally wel ome.15 The reader is asked for tolerating o

asionally a bit unusual notation, reated as a ompromise by unifying the standard notation from various fairly diverse elds.16 Sometimes, the very standard notation appearing in fairly dierent o

asions was kept hopefully without any onfusion.17 Bibliographi al notes are mostly mentioned as footnotes; anyhow, be ause of the wideness of the subje t, only basi referen es are provided either from the histori al purposes or just as a sour e of other referen es for a more detailed study.

13

The evolution on onvex ompa ti ations and in parti ular periodi -solution existen e theory (i.e.

the whole Chapter 8) has expanded only the 2nd edition in 2019.

14

Cf. also Remark 5.38.

15

This aspe t an be ree ted by introdu ing suitable toleran es in optimization problems; f. [658℄

for a systemati pursuit of the toleran e approa h.

16

A typi al dilemma was, e.g., that 

u normally stands for the ontrol variable in optimal ontrol p denotes the polyno-

theory while in the variational al ulus it denotes usually the state variable, 

L p -spa es while in optimal ontrol it stands for the adjoint state, et . Æ stands, beside a small positive real number, also for the Dira distribution as well as for the

mial growth in

17

E.g.,

indi ator fun tion.

Prefa e

Ë XVII

Finally, I would like to mention a plenty of olleagues with whom I had a lot of fruitful dis ussions, among them espe ially Professors K. Bhatta harya, N.D. Botkin, B. Da orogna, H.O. Fattorini, J. Haslinger, J. Malý, S. Müller, I. Netuka, J.V. Outrata, W.H. S hmidt, J. Sou£ek, and V. ’verák. Within writing this monograph, I also beneted from the ourses I hold during the a ademi al years 1993/94 and 1995/96 at Charles University in Prague for under- and graduate students. Moreover, M. Kruºík and M. Mátlová ontributed, beside areful reading and ommenting the whole manus ript, by omputer implementation of proposed algorithms and by al ulation of the examples. It is my duty and pleasure to express my deep thanks to all of them. Last but not least, spe ial gratitude is to Professor Karl-Heinz Homann, Professor Jind°i h Ne£as and Dr. Ji°í Jaru²ek, who inuen ed very essentially both my intelle tual live and professional areer and thus, dire tly or not, the theme of this book. I also warmly a knowledge the hospitality of Institut für Angewandte Mathematik und Statistik (TU Mün hen), where a great deal of the book has been a

omplished. Besides, shorter stays at ENS (Lyon), IMA (Minneapolis), EPF (Lausanne), and ErnstMoritz-Arndt-Universität (Greifswald) were inspiring.

Praha / Mün hen, January 1996 18

18

Tomá² Roubí£ek

A tually, in 2020, few updates and ompletions were added also into this 1996-prefa e to make it

more relevant regarding this 2nd edition.

Prefa e to the se ond edition The relaxation method in optimization and variational al ulus has remained an a tive topi after the rst edition was published more than two de ades ago, as it is do umented by a dozen of new spe ialized monographs.19 This se ond, revised and substantially expanded edition ree ts in parti ular few dozens of new papers of mine (and my oauthors)20 together with hundreds of other relevant referen es, without

laiming ompleteness. The main enhan ement an be spe ied as follows: First, the ba kground generalities (Chapter 1) has been rewritten or extended at several spots. The next two hapters have been enhan ed in parti ular by inverse systems of onvex ompa ti ations (Se t. 2.6 and 3.3e-f), oarse polynomial onvex ompa ti ations (Se t. 3.3.d), and by a higher-order approximation using quasiinterpolation (Se t. 3.5.d). Many hanges have been done in Chapter 4: The Filippov-Roxin existen e theory has been applied both to optimal ontrol of system governed by ordinary-dierential equations (for what it was originally devised) and also for other optimization problems. In addition to optimal ontrol of systems governed by ordinary-dierential equations where now also some omputational illustration has been added (Se t. 4.3e,f), also optimal ontrol of dierential-algebrai systems has been presented. An illustration of usage of the maximum prin iple is there for adaptive numeri al approximation. The ellipti and paraboli equations are now more general, not relying on potentiality, and systems of su h equations being onsidered, using also simpler arguments in the proofs. As a ertain illustration, the Navier-Stokes system des ribing in ompressible uid ows is taken. In the ellipti equations, optimal ontrol in oe ients (optimal material design) in ertain stratied materials has been in luded in Chapter 4, too. An in remental formula is illustrated in some situations, leading to onvexity of the relaxed problems. The state onstraints are treated also in the qualied way. Beside the onventional single- riteria optimization, the multi riteria Pareto/Slater on ept is applied at some spots, allowing also an interpretation as ooperative games. Variational problems in Chapters 5 and 6 have enjoyed parti ularly vast number of new results emerged in relaxation during past de ades, and only sele ted ones are re orded in this new edition. On top of it, s alar variational problems (Chap. 5) have been illustrated also by relaxation in ferromagneti materials. A newly added Chapter 8 addresses the evolution on onvex ompa ti ations, exploiting both ompa tness and onvex stru ture governed by gradient ow involving

19

Sin e the rst edition in 1997, the relaxation theme has been published in the monographs [171, 187,

192, 268, 314, 601, 603, 605, 622, 623℄, or also [33, Chap. 4,11℄, [113, Chap. 36℄, [320, Chap. 8℄, and [643, Chap. 4,7℄.

20

Spe i ally, it on erns the arti les [89, 104, 106, 141, 179, 453, 459461, 520, 536, 543, 545, 619, 675

678, 681684, 686, 688, 689, 693702, 784℄.

https://doi.org/10.1515/9783110590852-202

Prefa e to the se ond edition

Ë XIX

also a dissipation potential, as devised in an older arti le [668℄ for quadrati potentials. This is addressed both on the abstra t level and on spe i appli ations for evolution of os illations (mi rostru ture) in ferroi materials or on entrations (slip bands) in perfe tly plasti materials. Simultaneous usage of onvexity and ompa tness of the relaxed problems is illustrated, beside non- ooperative game theory, now also on a feasibility/ ontrollability via xed-point theorems and on existen e of periodi solutions in evolution problems. Moreover, the notation has been o

asionally slightly modied to be more onventional or to gain a better logi . And the (rather too mu h) abstra t topologi al on ept of nets has been suppressed in favor to the more onventional on ept of sequen es, relying on an assumption of separability of spa es of test fun tions in parti ular in Chapters 4-8. Also, beside the mentioned enhan ements of the 1st edition, some redu tion of the presentation has o

asionally been applied. Some dieren es likely also have o

urred be ause all the nal les from the 1st edition whi h in luded all galley proofs and language orre tions, kept ex lusively by the Publisher/typesetting ompany, were unfortunately unprofessionally annihilated in the meantime and no further language orre tions have been exe uted in the old and the new text in this se ond edition. This new, extended edition also benets from the lasses on Sele ted parts from optimization theory whi h I had during 1993-2010 at Charles University in Prague and ree ts also some re ent resear h21. Spe ial thanks are to Miroslav Hu²ek for advising me the old general-topologi al onstru tions of inverse systems and threads, and to my oauthors Sören Bartels, Barbora Bene²ová, and Martin Kruºík for providing the omputational al ulations and gures added to this new edition. Parti ular thanks are to Dr. Apostolos Damialis, the former A quisitions Editor for Mathemati s in W. de Gruyter in Berlin, who initiated this 2nd edition in 2017.

Praha, July 2020

21

Tomá² Roubí£ek

In this ontext, the grant proje ts 19-04956S and 19-29646L of the Cze h S ien e Foundation are

a knowledged.

1 Ba kground Generalities If a fun tion

Fx is ontinuous from x

# a to x # b

in lusive, then among all the values whi h it takes, ... there is always a greatest and also a smallest ...

Bernard Bolzano

(17811848)

The theory of operators ... has penetrated several highly important areas of mathemati s in an essential way. ... The theory often makes possible altogether unforeseen interpretations of the theorems of set theory or topology.

Stefan Bana h

(18921945)

This hapter is to remind sele ted fundamental on epts and results on erning general topology, fun tional analysis, and optimization and game theory. Besides this abstra t topi s, some results from theory of fun tion spa es, theory of means on spa es (or rings) of ontinuous fun tions, and from dierential and integral equations will be summarized too. By no means this hapter is intended as an survey of these elds be ause only items needed frequently throughout the book are in luded here. Also the generality is rather restri ted to the level whi h is a tually needed in what follows. Moreover, some notions needed only lo ally have not been in luded into this

hapter, and will be reminded as footnotes at relevant pla es in the further hapters. As the reader is supposed to have a basi knowledge from general topology, fun tional analysis, and fun tion spa es, most of the results in this hapter are presented without any proofs. Some others, though being more or less also quite standard, are a

ompanied by (at least sket hed) proofs. As a result, this hapter is intended rather for a onsultation via Index within reading the further hapters but not for a thorough systemati study. Moreover, basi set-theoreti al notions like relations, mappings, inverse mappings, Cartesian produ ts, et ., are supposed well known and will not be spe i ally dened here at all.

1.1

Order and topology

In this se tion we will briey summarize fundamental ideas and results on erning ordered sets and general topology.1

1

For more details the reader is referred, e.g., to the monographs by Bourbaki [144℄, ƒe h [190℄, Csaszar

[240℄, Engelking [284℄, Köthe [436℄, and Kuratowski [471℄.

https://doi.org/10.1515/9783110590852-001

2

Ë

1 Ba kground Generalities

¢, on a set X will be alled ordering if it is reexive x ¢ x for any x ò X ), transitive (i.e. x1 ¢ x2 & x2 ¢ x3 imply x1 ¢ x3 for any x1 ; x2 ; x3 ò X ) and antisymmetri (i.e. x1 ¢ x2 & x2 ¢ x1 imply x1 # x2 ). The set A binary relation, denoted by

(i.e.

equipped with the ordering will be alled ordered. The ordering

¢ is alled linear if

x1 ¢ x2 or x2 ¢ x1 always hold for any x1 ; x2 ò X . An ordered set X is alled dire ted if for any x 1 ; x 2 ò X there is x 3 ò X su h that both x 1 ¢ x 3 and x 2 ¢ x 3 . Instead of x 1 ¢ x 2 , we will also write x 2 £ x 1 . By x 1   x 2 we will understand2 that x1 ¢ x2 but x1 #Ö x2 . Having two ordered sets X1 and X2 and a mapping f : X1 Ù X2 , we say that f is non-de reasing (resp. non-in reasing) if x 1 ¢ x 2 implies f ( x 1 ) ¢ f ( x 2 ) (resp. f ( x 1 ) £ f ( x 2 )). We say that x 1 ò X is the greatest element of the ordered set X if x 2 ¢ x 1 for any x2 ò X . Similarly, x1 ò X is the least element of X if x1 ¢ x2 for any x2 ò X . We say that x 1 ò X is maximal in the ordered set X if there is no x 2 ò X su h that x1   x2 . Let us note that the greatest element, if it exists, is always maximal but not

onversely. Similarly, x 1 ò X is minimal in X if there is no x 2 ò X su h that x 1 ¡ x 2 . The ordering ¢ on X indu es also the ordering on a subset A of X , given just by the restri tion of the relation ¢. We say that x 1 ò X is an upper bound of A ’ X if x 2 ¢ x 1 for any x 2 ò A . Analogously, x 1 ò X is alled an lower bound of A if x 1 ¢ x 2 for any x2 ò A. If every two elements x1 ; x2 ò X possesses both the least upper bound and the greatest lower bound, denoted respe tively by sup( x 1 ; x 2 ) and inf( x 1 ; x 2 ) and alled the supremum and the inmum of { x 1 ; x 2 }, then the ordered set ( X ; ¢) is alled a latti e. Then the supremum and the inmum exist for any nite subset and is determined uniquely be ause the ordering is antisymmetri . If they exist for an arbitrary subsets,

X ; ¢) is alled a omplete latti e. A subset A of a dire ted set X is alled onal if for any x 1 ò X there is x 2 ò A su h that x 1 ¢ x 2 . The following assertion, though being highly non onstru tive unless X ’ N, plays (

a fundamental role in many further onsiderations.

Lemma 1.1 (K. Kuratowski [470℄ and M. Zorn [821℄).3 If every linearly ordered subset of X has an upper bound in X , then X has at least one maximal element.

 and another set X , we say that {x  } ò is a net in X if there  Ù X :  ÜÙ x  . Having another net { x  } ò in X , we say that this net

Having a dire ted set is a mapping



j :  Ù  su h that, for any  ò  , it and moreover, for any  ò  there is  ò  large enough so that j (  1 ) £  x

is ner than the net {  }  ò  if there is a mapping holds

2

x  # x j 

(

 )













However, in a linear topologi al spa e order by a one with a non-empty interior, the relation

  will

have a bit stronger meaning, f. p. 55.

3 

This assertion is equivalent to the axiom of hoi e: for every set

X

and every olle tion {

A x }xòX ,

#Ö A x ’ X , there is a mapping f : X Ù UxòX A x su h that f(x) ò A x for any x ò X ; f., e.g., Engelking

[284, Se t. 1.4℄.

1.1 Order and topology

Ë 3

 1 £  . For example, every non-de reasing mapping j :  Ù  su h that j(  ) is onal in  produ es a ner net by putting x  # x j  . The reader should realize

whenever









(



)

that a ner net may have the index set of stri tly greater ardinality than the original net. Having in mind a ertain property of the nets (e.g. boundedness, onvergen e,

x

et .), we will say that this property holds eventually for a net {  }  ò  in question if there is

0 ò  su h that the net {x  } ò  £0 ;

has this property.

Example 1.2 (Con ept of sequen es). The set of all natural numbers N ordered by the standard ordering ¢ is a dire ted set. The nets having N (dire ted by this standard ordering) as the index set are alled sequen es. Any subsequen e of a given sequen e

an be simultaneously understood as a ner net.4 A olle tion

F

of subsets of

X will be alled a lter on X if A ; B ò F

implies

A  Bò

“ B ò F implies A ò F , and if  ò Ö F . Furthermore, a olle tion B of subsets X will be alled a lter base on X if A1 ; A2 ò B implies B ’ A1  A2 for some B ò B and if  ò Ö B . For B a lter base, the olle tion { A ’ X ; ; B ò B : A “ B } is a lter on X ; F , if A

of

we will say that this lter is generated by the lter base Furthermore, we will introdu e a topology of subsets of

X su h that T

le tion of sets from

T

T

B.

X , whi h will be a olle tion X itself, and with every nite ol-

of a set

ontains empty set and

it ontains also their interse tion, and also with every arbitrary

olle tion of sets from

T

also their union. The elements of

T

are alled open sets (or

T -open, if we want to indi ate expli itly the topology in question), while their ompleX endowed with a topology T will be alled a topologi al spa e; sometimes we will denote it by ( X ; T ) to refer to T expli itly. Having a subset A ’ X , T A :# {A  B; B ò T } is a topology on A; we will address it as a relativized

ments are alled losed. A set

topology.

T0 of subsets of X is alled a base (resp. a pre-base) of a topology T T -open set is a union of elements of T0 (resp. a union of nite interse tions of elements of T0 ). A olle tion

if every

x ò N ’ X , we say that N is a neighbourhood of x if there is an open set A x ò A ’ N . It is easy to see that the olle tion of all neighbourhoods of a given point x , denoted by N ( x ), is lter on X ; we will alled it a neighbourhood lter of x . Besides, we dene the interior, the losure, and the boundary of a set A respe tively by Having

su h that

int(A) :# {x ò X; ;N ò N (x) : N ’ A} ;

l(A) :# {x ò X; :N ò N (x) : N  A #Ö } ; bd(A) :# l(A) \ int(A) :

4



Indeed, having a sequen e { x k }kòN and its subsequen e { x k }kòN with some N ’ N, one an put # (N; ¢),  # (N; ¢), and j :  Ù  the in lusion N ’ N; note that j is nonde reasing and, sin e N

is innite,

j(N) is onal in

N

, as required.

Ë

4

Having

1 Ba kground Generalities

A ’ B ’ X , we say that A

is dense in

B

if l(

A) “ B. A topologi al

spa e is

alled separable if it ontains a ountable subset whi h is dense in it.

x

X , we say that it onverges to a point is  0 ò  large enough so that x  ò N

Having a net {  }  ò  in the topologi al spa e

xòX

N of x, there  £ 0 ; then we say also that x is the limit point of the net in question, and write lim ò  x  # x or simply x  Ù x . This on ept of onvergen e is alled the MooreSmith onvergen e [547℄. Let us note that x ò l( A ) if and only if there is a net in A

onverging to x ; in this ase we also say that x is attainable by a net from A . A point x ò X is alled a luster point of the net {x  } ò if, for any neighbourhood N of x and for any  0 ò  , there is  £  0 su h that x  ò N . Obviously, every limit point is a luster point as well, but not onversely. Nevertheless, for any luster point x of a net { x  }  ò  there exists5 a ner net { x  }  onverging to x . If  is ri h enough, we an even onsider   ò if, for any neighbourhood

whenever



 ’ ; f. Example 1.4 below. 

X1 ; T1 ) and (X2 ; T2 ) and a mapping f : X1 Ù X2 , "1 (A) # {x ò we say that f is ontinuous (or, more pre isely, (T1 ; T2 )- ontinuous) if f 1 X1 ; f(x1 ) ò A} ò T1 whenever A ò T2 . Alternatively, f is ontinuous if it maps every T1 onvergent net onto a T2 - onvergent one. The set of all ontinuous mappings X 1 Ù X 2 will be denoted by C ( X 1 ; X 2 ); if X 2 # R endowed with the standard topology, then we n will write briey C ( X 1 ) instead of C ( X 1 ; X 2 ). For an open domain ’ R , C ( ) an be identied with the subspa e of C ( ) onsisting from fun tions that possess ontinuous extension to the losure :# l( ). If f is a one-to-one mapping and both f and "1 are ontinuous, then f is alled a homeomorphism. Also we the inverse mapping f say that f : X 1 Ù X 2 realizes a homeomorphi al embedding of X 1 to X 2 if f is a homeomorphism between X 1 and f ( X 1 ) (equipped with the relativized topology oming from X2 ). The inje tive mapping f : X1 Ù X2 is alled ontinuous (resp. dense) embedding if f is ontinuous (resp. f ( X 1 ) is dense in X 2 ). The set of all topologies on a given set X is ordered naturally by the in lusion: Having two topologies T1 and T2 on a set X , we say6 that T1 is ner than T2 or T2 is oarser than T1 if T1 “ T2 (or, equivalently, if the identity on X is (T1 ; T2 )- ontinuous). The X

oarsest and the nest topologies, namely { X ; } and 2 , are alled indis rete and disHaving two topologi al spa es (





rete, respe tively.

A fun tion d : X , X Ù R is alled a metri on X if, for all x 1 ; x 2 ; x 3 ò X , d(x1 ; x2 ) £ 0, d(x1 ; x2 ) # 0 is equivalent to x1 # x2 , d(x1 ; x2 ) # d(x2 ; x1 ), and d(x1 ; x2 ) ¢ d(x1 ; x3 ) % d(x3 ; x2 ). Every metri d indu es a topology T by a base {{ x ò X ; d ( x ; x 1 )   " }; x 1 ò X ; " ¡ 0}. Conversely, a topology is alled metrizable

5

6

 #  , N ( x ) dire ted by the ordering ¢ , “ and to take, for any  ,   # ( ; N) ò  # x  ò N with  £  ; see, e.g., Engelking [284, Proposition 1.6.1℄ for details.

It su es to put

some

 x 

In the literature, the notions of stronger and weaker are sometimes used in pla e of ner and

 oarser, respe tively.

1.1 Order and topology

Ë 5

if there exists a metri that indu es it. However, it should be emphasized that there exist nonmetrizable topologies.7 Topologies of a given set

X may have various important properties. One of them is T

a separation property. We say that

is a T0 -topology, resp. T1 -topology, (sometimes

x1 ; x2 ò X there is N1 ò N (x1 ) su h that N2 ò N (x2 ) su h that x1 ò Ö N2 . If, for any x1 ; x2 ò X , N2 ò N (x2 ) su h that N1  N2 # , then T is alled a T2 -

alled Kolmogorov, resp. Fré het) if for any

x2 ò Ö N1

or (resp. and) there is

there are

N1 ò N (x1 ) and

topology, or also a Hausdor topology. Every net in a Hausdor spa e may have at most

X ; T ) is alled ompletely regular8 if, x ò X and any N ò N (x), there is a ontinuous fun tion f : X Ù R su h that f(x) # 0 and f(X \ N) # 1. Eventually, a Hausdor topology T is T4 , also alled normal, if for every losed mutually disjoint subsets M ; N ’ X there is a ontinuous fun tion f : X Ù [0; 1℄ su h that f(N) # 0 while f(M) # 1.

one limit point. A Hausdor topologi al spa e ( for any

Example 1.3 (Topologies on R).

The standard topology on

pre-base omposed from all open intervals, i.e. {(

;

R whi h is indu ed by the

a ; b); a ; b ò R € {"; %}}, is Haus-

dor, i.e. T2 and even normal, i.e. T4 -topology. If nothing is said, always we will un-

R equipped with this Hausdor topology. A olle tion a ; % ; a ò R is another topology on R whi h is T but not T , f. Remark 2.38 for its usage. An example of a T -topology whi h is not T is the olle tion R’ F ; F nite .

derstand

{(

0

1

)

}

1

{

2

Example 1.4 (Universal index set).

}

X is a ompletely regular topologi al spa e, then there is a lter base U on X,X su h that, for any x ò X , U ( x ) # {{ x ò X; ( x ; x) ò B}; B ò U } is a base of the neighbourhood lter N (x). For many investigaIt is known9 that, if





tions in ompletely regular spa es, a universal su iently ri h index set is

 # U ordered by the in lusion whi h makes it dire ted. Then, for example, every

x ò l(A)

x ’ A.10 Moreover, if a net {x  } ò has a luster point x ò X , then we an laim that there exists a ner net { x  } ò (using the same index set) whi h onverges to x .11

an be attained by some net {  }  ò 



7

An example of a nonmetrizable topology is the produ t topology on

ber of Hausdor topologi al spa es

8

Completely

regular Hausdor

Xj

AjòJ

X j on an un ountable num-

having at least two elements.

spa es

are

denoted

as T

3

1 -spa es 2

and

sometimes

also

alled

Tikhonov spa es in the literature.

9

In fa t, it su es to take a base of any uniformity stru ture on

X ; f., e.g., Bourbaki [144, Chap. II℄

or Engelking [284, Chap. 8℄.

 # B ò  # U , some x  su h that (x  ; x) ò B. B ò U there is some  B ò  su h that  B £ B and (x B ; x) ò B. Then it su es to put  x B # x B for any B ò U .

10

Indeed, it su es to take, for

11

Indeed, we know that, for any

6

Ë

1 Ba kground Generalities

Having a topologi al spa e ( X ; T ) and taking the T0 -topology from Example 1.3, a T ; T0 )- ontinuous fun tion f : X Ù R will be alled also lower semi ontinuous (with respe t to the topology T ). A fun tion f : X Ù R is alled upper semi ontinuous if (

f : X Ù R is alled ontinuous (with respe t X ) if it is both lower and upper semi ontinuous. The reader an easily verify that this ontinuity is equivalent to the (T ; T1 )- ontinuity where T1 denotes the standard topology on R indu ed by the metri d ( a 1 ; a 2 ) #  a 1 " a 2 . Besides, the lower (resp. upper) semi ontinuity of f is equivalent to liminf xÙ x f ( x ) £ f ( x ) (resp. limsup xÙ x f ( x ) ¢ f ( x )), where the limit inferior and limit superior are dened respe -

"f

is lower semi ontinuous. A fun tion

to the topology

T

on





tively by

lim inf f( x ) :# sup inf f( x ) xÙ x



N òN ( x ) xò N

and



lim sup f( x ) :# inf sup f( x ) : 

xÙ x

N òN ( x ) xò N



The entral topologi al notion we will rely on is the ompa tness. A topology

T

X is alled ompa t if every over of X by open subsets ontains a nite sub- over. Equally12 we an dene T ompa t if every net in X has a luster point in X . Con-

on

tinuous mappings map ompa t sets onto ompa t ones. On a given set, ompa t topologies are minimal in the lass of all Hausdor topologies. Every lower (resp. upper) semi ontinuous fun tion

X Ù

R on a ompa t topologi al spa e

(

X ; T ) attains

its minimum (resp. maximum), whi h is known as (a generalization of) the Bolzano-

Weierstrass theorem.13 A topologi al spa e ( X ; T ) is alled sequentially ompa t if every sequen e in

X admits a subsequen e that onverges in X . A metrizable topology is

ompa t if and only if it is sequentially ompa t,14 while for non-metrizable topologies these notions are not omparable.15 A subset

A of a topologi al spa e (X ; T ) is alled A is (sequentially) ompa t in X . A

relatively (sequentially) ompa t if the losure of topologi al spa e is alled

- ompa t if it is a union of a ountable number of ompa t

subsets, and it is alled lo ally (sequentially) ompa t if every point of its possesses a (sequentially) ompa t neighbourhood.

X j ; Tj )}jòJ of topologi al spa es, we dene the topology T on the produ t X # A j ò J X j anoni ally as the oarsest topology on X that makes (T ; T j )- ontinuous all the proje tion X Ù X j ; this topology has16 a base Having an arbitrary olle tion {(

12

See, e.g., Engelking [284, Thm. 3.1.23℄.

13

More pre isely, B. Bolzano [130℄ showed that a real ontinuous fun tion of a bounded losed in-

terval is bounded. A tually, that time, the trans edental numbers were not dis overed so the lo al

ompa tness of the reals was only intuitively understood, and the on ept of what later was alled Cau hy sequen es had to be invented.

14

We refer, e.g., to Engelking [284, Thm. 4.1.17℄.

15

For examples of a non ompa t sequentially ompa t and a ompa t but not sequentially ompa t

spa es we refer to Köthe [436, Se t. 3.4℄.

16

See, e.g., Bourbaki [144, Se t. I.8.1℄ or Engelking [284, Proposition 2.3.1℄.

1.1 Order and topology

{A

jòJ

A j ; :j ò J : A j ò Tj ; & A j # X j

for all but a nite number of indi es

Ë 7

j ò J}. The fol-

lowing assertion, based on the Kuratowski-Zorn lemma 1.1,17 is of a vital importan e:

Theorem 1.5 (A.N. Tikhonov).18 The produ t spa e (X ; T ) is ompa t if and only if all (

X j ; Tj ) are ompa t.

X ; T ), we say that a pair ( X ; i) is a ompa ti ation X into X and if X is ompa t. If, in addition, the embedding i is homeomorphi al, then X will be alled T Having a topologi al spa e (

of

X

if

i : X Ù X

is a ontinuous19 dense embedding of

onsistent. As homeomorphi al topologi al spa es are equivalent to ea h other from

X and i(X) in the T - onsistent ase. The dieren e X \ X will be addressed as a remainder. If T is a ompletely regular topology, then ( X ; T ) admits a T - onsistent ompa ti athe general-topology viewpoint, we will sometimes not distinguish between

tion.20 In general, a ( ompletely regular) topologi al spa e may admit a large amount of (T - onsistent) ompa ti ations, so it is worth introdu ing a natural ordering of them. Having two ompa ti ations (

1

X ; i1 ) and (

2

X ; i2 ) of X , we say that the for-

mer one is a ner ompa ti ation than the latter one (or, equivalently, the latter one is oarser than the former one) and write ( if there is a ontinuous mapping

:

1

XÙ

1

X ; i1 ) ³ ( 2 X ; i2 ) (or briey 1 X ³ 2 X ) Ž i1 # i2 . 2 X xing X in the sense that

This mapping is inevitably surje tive, and we will refer to it as a anoni al surje tion.

X³

1

2

X and

X²

X , then these ompa ti ations will be alled equiv1X Ê 2 X . If 1X ³ 2 X but 1 X ÊÖ 2 X, then we will write 1 X ± 2 X , saying that 1 X is stri tly ner than 2 X (or 2 X is : 1X Ù 2X stri tly oarser than 1 X ). If 1 X ± 2 X , then the anoni al surje tion inevitably glues at least two points of the remainder of 1 X together, i.e. there are x1 ; x2 ò 1 X \ i1 (X) su h that (x1 ) # (x2 ). If a ompa ti ation is T - onsistent, then

If both

1

2

alent to ea h other and then we will write

any ner ompa ti ation is

T - onsistent, too.

Any olle tion of ompa ti ations {( j

X ; i j )}jòJ

of a given ompletely regular

spa e X admits its supremum ( X ; i ), whi h an be onstru ted21 by putting X #

l(i(X)) where i : X Ù AjòJ j X : x ÜÙ i j (x) jòJ and AjòJ j X is endowed with the standard produ t topology whi h makes it ompa t by the Tikhonov theorem. In parti ular, there exists the supremum of all ompa ti ations of

17

Noteworthy, if

J

is ountable and all (

X j ; Tj )

lemma is rather trivial so that the ompa tness of

18

X , denoted by X . This is simul-

are metrizable, the usage of the Kuratowski-Zorn

AjòJ

Xj

has a onstru tive hara ter.

A.N. Ty hono (19061993) formulated this theorem in [772℄. Beside German spelling in this origi-

nal arti le, often it is referred with dierent spelling, transliterating his name in Cyrilli , as Ty honov, Tikhonov, Tihonov, or Ti honov. See also, e.g., Bourbaki [144, Setion I.10.5℄, Dunford and S hwartz [275, Thm. I.8.5℄, Engelking [284, Thm. 3.2.4℄, Köthe [436, Se t. 3.3℄.

19

Mostly, a narrower on ept of ompa ti ations, requiring the embedding to be homeomorphi al,

is adopted in general topology; see, e.g., [190, 240, 284, 471℄. For our purposes it appears useful to a

ept su h wider on ept, the narrower on ept being spe ied by the adje tive  onsistent.

20 21

See, e.g., Engelking [284, Thm. 3.5.1℄. We refer to Engelking [284, Thm. 3.5.9℄ for more details.

8

Ë

1 Ba kground Generalities

taneously the nest ompa ti ation of

tion [189, 736℄. It is always

X , being alled the ƒe h-Stone ompa ti a-

T - onsistent. Sometimes, (X ; T

) admits also the oarsest

T - onsistent ompa ti ation, denoted by X , and alled the Alexandro ompa ti ation. A non- ompa t ompletely regular spa e admits the Alexandro ompa ti a-

X is either void (if X itself X is only lo ally ompa t but non- ompa t).

tion if and only if it is lo ally ompa t.22 The remainder of is ompa t) or a singleton (if

Example 1.6 (Compa ti ations of R). point ompa ti ation

If X # R, the real line, the Alexandro oneR just adds  to R, gluing thus both free ends of R to-

R is then homeomorphi with a ir le. Then the standard two-point

ompa ti ation R # [" ; %℄ is stri tly ner. The nest, ƒe h-Stone ompa ti ation R is still stri tly ner23 than R. Thus we get the situation gether so that

R ° Let us still remark that

R ° R :

R are here metrizable, while R is not.

R and

Another useful on ept generalizes the usual (= single-valued) mapping: having

X2 , a mapping S : X1 ٠2X2 , with 2X2 denoting the set of all subsets of X 2 , will be also alled a multivalued mapping from X 1 to X 2 , denoted by S : X 1 ± X2 . Having (X1 ; T1 ) and (X2 ; T2 ) two topologi al spa es, it is worth generalizing the two sets

X1

and

on ept of ontinuity. We dene

Limsup S( x ) :# Žx ò X ; x1 Ù x 1 ; a net (x  ; x  ) Ù (x ; x 1



2

2

1

Liminf S( x ) :# Žx ò X ; x1 Ù x 1 : net x  Ù x 

1

2

1

2

2)

in

T1 , T2

: x  ò S(x  )DZ ;

(1.1a)



: x Ù x

(1.1b)

2

1

2

1

1

in



T1 ; a net x2 ò S(x1 )

2

2

in

T2 DZ;

The above introdu ed upper and lower limits are alled the Kuratowski limits.24

S is alled upper (resp. lower) semi ontinuous at x1 if S( x 1 ) ’ S(x1 ) (resp. Liminfx1 Ùx1 S( x 1 ) “ S(x1 )). Of ourse, S is alled upsemi ontinuous25 if it is upper (lower) semi ontinuous at every x 1 ò X 1 .

The multivalued mapping Limsup x1 Ù x 1 per (lower)





22

See, e.g., Engelking [284, Theorems 3.5.11-12℄.

23

In fa t, the remainder of



R

is very large, ontaining at least

2

N



2

points; f. Engelking [284,

Thm. 3.6.11℄.

24

X # %}, being denoted by the symbols Ls and Li. Let us also mention that sometimes the sym-

They were invented in Kuratowski [471, Se tions 29.I and 29.III℄ for the ase of sequen es, i.e.

N€

1

{

bols Limsup and Liminf  may have another meaning, being dened without referring to any topology on

   X2 , namely LimsupnÙ S(x n ) # V n#0 Uk#0 S ( x n%k ) and LiminfnÙ S ( x n ) # Un#0 Vk#0 S ( x n%k ),

f. [471, Se t. I.V℄.

S is alled upper (lower) semi ontinux1 ò X1 ; S(x1 ) ’ A} is T1 -open (T1 - losed) for any A whi h is T2 -open (T2 - losed). However, both denitions oin ide with ea h other provided X 2 is ompa t, see Kuratowski [471, Se t. 43.II℄ 25

Sometimes, these notions are dened by other ways: namely

ous if the set {

or Deimling [256, Se tions 1 and 2℄.

Ë 9

1.2 Linear, nonlinear, and onvex analysis

If Limsup x1 Ù x 1 Lim x1 Ù x 1

S( x 1 ) # 

S( x 1 ).

Liminf x1 Ù x 1

S( x 1 ), 

we will denote this ommon limit set as



S is single-valued (i.e. S(x) # {f(x)} for a mapping f : X1 Ù X2 ) well as the lower) semi ontinuity of S is equivalent to the usual

Let us note that if then the upper (as

ontinuity of Another

f. elegant

onstru tion

has

S. Lefs hetz:26 onsidering a dire ted set spa es

X

been

,

invented

by

a olle tion (

P. Alexandro

X  ) ò

is alled an inverse system if

:  £  ;  21 : X 2 Ù X 1 ontinuous : :  #  :  21 # identity and 2

1

1

  31

: ¢  ¢  : 1

2

The mappings

3



 21

inverse system as for any

and

of topologi al

#

  21

Ž

  32

(1.2a)

2

:

(1.2b)

from (1.2) are alled bonding mappings. Shortly, we write this



S # (X  ;   ) ;  ò ;  £ . Ea h x ò A  ò X 

 £  . The set

is alled a thread if

S :# Žx ò I X  ; x is a threadDZ lim ØÚ



 x # x (1.3)

 ò

proje tions



S # (X  ;   ) ;  ò ;  £ . We further dene the of the limit of the inverse system Pr : lim S Ù X  as the restri tion on

is alled the limit of the inverse system

ØÚ

 lim S of the proje tions A ò X  Ù X  . For all  ¢  , it holds Pr #   Ž Pr . ØÚ 

X  in (X  ;   )  ò  £ are Hausdor topologi al spa es, then lim S is losed ØÚ in A  ò  X  equipped with the anoni al produ t topology; f. [284, Prop. 2.5.1℄. If all X  are ompa t, this topology is ompa t by the Tikhonov theorem, so that we then have immediately the ompa tness of the inverse limit S. If all

1.2

;

;

Linear, nonlinear, and onvex analysis

In this se tion we will briey summarize fundamental ideas and results on erning linear topologi al spa es, their duals, and onvex subsets, as well as linear or nonlinear mappings or onvex fun tionals on them.27 Throughout the whole book we will

onne ourselves to topologi al ve tor spa es over the eld of reals

26

R.

The on ept of limits of the inverse systems has been invented in early 30ties of the last entury

in [486℄ exploiting a bit modied denition in [12℄ and then developed in [282℄; see e.g. [284, Se t.2.5℄ for a omprehensive exposition.

27

For more details the reader is referred, e.g., to the monographs by Choquet [208℄, Day [252℄, Dunford

and S hwartz [275℄, Edwards [278℄, Holmes [392℄, Kolmogorov and Fomin [434℄, Köthe [436℄, Taylor [751℄, Valentine [776℄ and Yosida [804℄.

10

Ë

1.2.a

1 Ba kground Generalities

Linear fun tional analysis

X ; T ) is alled a (real) linear topologi al spa e if it is equipped x1 ; x2 ) ÜÙ x1 % x2 : X , X Ù X whi h makes it a ommutative topologi al group28 and with a jointly ontinuous multipli ation by s alars ( a ; x ) ÜÙ ax : R , X Ù X satisfying (a1 %a2 )x # a1 x % a2 x, a(x1 %x2 ) # ax1 % ax2 , (a1 a2 )x # a1 (a2 x), and 1x # x. The point 0 ò X is also alled the origin. A subset K ’ X is alled

onvex if ax 1 % (1" a ) x 2 ò K whenever x 1 ; x 2 ò K and 0 ¢ a ¢ 1, and it is alled a

one (with the vertex at the origin) if ax ò K whenever x ò K and a £ 0. A topologi al linear spa e X is alled ordered by a relation £ if this relation is an ordering and, in

A topologi al spa e (

with the binary operation (

addition, it is ompatible with the linear and topologi al stru ture in the sense that

ax £ 0 if x £ 0 and a £ 0, that x1 % x2 £ 0 if both x1 £ 0 and x2 £ 0, that x1 £ x2 implies x 1 % x 3 £ x 2 % x 3 for any x 3 , and that x  £ 0 and x  Ù x implies x £ 0. It is easy to see that D # { x ò X ; x £ 0} is a losed onvex one whi h does not ontain a line. Conversely, having a losed onvex one D ’ X whi h does not ontain a line, the relation £ dened by x 1 £ x 2 provided x 1 " x 2 ò D makes X an ordered linear topologi al spa e. For a subset

A ’ X , we dene a so- alled indi ator fun tion Æ A (x) : X Ù {0; %}

by

0 %

Æ A (x) :# ® Let us note that

for

x ò A;

otherwise

:

A is onvex (resp. losed) if and only if Æ A is onvex (resp. lower semi-

ontinuous).

X1 and X2 and a mapping A : X1 Ù X2 , A is a ontinuous linear operator if it is ontinuous and satises A(a1 x1 % a2 x2 ) # a1 A(x1 ) % a2 A(x2 ) for any a1 ; a2 ò R and x1 ; x2 ò X . Often we will write briey Ax instead of A(x). If X1 # X2 , a linear ontinuous operator A : X1 Ù X2 is alled a proje tor if A Ž A # A . The set of all linear ontinuous operators X 1 Ù X 2 will be denoted by L( X 1 ; X 2 ), being itself a linear spa e when equipped with the addition and multipli ation by s alars dened respe tively by ( A 1 % A 2 ) x # A 1 x % A 2 x and ( aA ) x # a ( Ax ). As R is itself a linear topologi al spa e, we an onsider the linear spa e L( X ; R), being also denoted by X and alled the dual spa e to X . Having two topologi al linear spa es

we say that

*

The topology of a topologi al linear spa e is fully determined by a base lter of neighbourhoods of the origin

N

0}

forms a base of the topology of

N

N

0

of the

0) be ause the olle tion {x % A; x ò X ; A ò

(

X . An important lass of topologi al linear spa es

onsists of lo ally onvex spa es, having a base of

N

0) omposed from onvex sets.

(

x ÜÙ x : X Ù R is alled a seminorm if x £ 0, ax # ax, and x1 % x2  ¢  x 1  %  x 2 . A fun tion x ÜÙ x : X Ù R is alled a norm if it is a seminorm and if x # 0

A fun tion

28 (

It means that this mapping is jointly ontinuous and satises

x1 % x2 ) % x3 , ;0 ò X : x % 0 # x, and :x1 ò X ;x2 : x1 % x2

# 0.

x1 % x2

# x % x , x % (x % x ) # 2

1

1

2

3

1.2 Linear, nonlinear, and onvex analysis

implies

Ë 11

x # 0. Having a olle tion of seminorms { -  } ò on a linear spa e X , we an :# {{x ò X; max ò x ¢ "}; " ¡ 0;  ’  nite} is a lter base

see that the olle tion B



0

and, taking the lter generated by it as the neighbourhood lter of the origin N ( ), we obtain a lo ally onvex spa e. Conversely, every lo ally onvex spa e an be obtained by this manner if taking the olle tion of seminorms appropriately. A lo ally onvex spa e equipped with a norm and with the topology generated by this norm is alled a

normed linear spa e, its topology being also addressed as strong. Nets onverging in strong topology are alled strongly onvergent. If the olle tion of seminorms { -   }  ò  generating the topology of a Hausdor lo ally onvex spa e is ountable (i.e. we may suppose

 :# N and write { - k }kòN ) then d(x1 ; x2 ) :#



"k

H2 k #1

x " x k 1 % x "x k 1

2

1

(1.4)

2

denes a translation-invariant29 metri whi h indu es the topology of Having two normed linear spa es

L(X1 ; X2 ) by

A L X 1 (

;

X2 )

X1

and

X2 ,

X.

we an introdu e a norm on

:# sup Ax X2 ;

(1.5)

x X1 ¢1



X1 ; X2 ) a normed linear spa e. An operator A ò L(X1 ; X2 ) is alled omX1 onto relatively

ompa t subsets of X 2 . A net { x  }  ò  in a topologi al linear spa e is alled Cau hy30 if, for any N ò N (0), there is  N ò  su h that x  1 " x  2 ò N for any  1 £  N and  2 £  N . If every Cau hy net

onverges in X , then X is alled omplete. A Hausdor omplete lo ally onvex spa e

whi h makes L(

pa t if it maps bounded (with respe t to the norm - X 1 ) subsets of

0

with N ( ) having a ountable base is alled a Fré het spa e, while a omplete normed linear spa e is alled a Bana h spa e. Having a lo ally onvex spa e

X and its dual X

*

, we an see that { -  x * } x * ò X * with

x ; x> is a olle tion of seminorms; the bilinear form : X , X Ù R dened by < x ; x > :# x ( x ) is alled the anoni al bilinear pairing. The topology generated on X by this olle tion of seminorms is alled the weak topology and it an be seen that this topology is always oarser than the original topology and X equipped with 

xx :# *


for some x ò X ; in other words, the dual *

*

*

spa e to ( X If

X

*

; weak*) is again X .

is a normed linear spa e equipped with the norm - , then

sup x ¢ is a norm on X *





*

*

1

, whi h makes

norm as the dual norm. Considering

X

its dual, denoted by

:# (X

**

*

X

*

X

*

x

ÜÙ

*

the Bana h spa e. We will refer to this

normed by the dual norm, we an think about

*

) , equipped again with the dual norm. This spa e is

X . The mapping i : X Ù X dened by :# is alled the anoni al embedding of X into its bidual, and it realizes a (weak,weak*)**

alled the bi-dual spa e to

*

*

as well as (strong,strong)-homeomorphi al embedding.33 Moreover, the Goldstine the-

B% X

orem34 says that, for dense in the ball in

ive if i ( X )

#X

**

the ball of the radius

**

% ¡ 0 in X , the image i(B % ) is weakly* X is alled reex-

of the same radius. A normed linear spa e

. The reader ertainly noti ed that we have dened three lo ally on-

vex topologies on

X

*

, namely the strong, the weak, and the weak* topologies. The

weak topology is always oarser35 than the strong topology, and the weak* topology is

oarser36 than the weak one. If

X

is a Bana h spa e, the Bana h-Steinhaus prin iple [73℄ (often also alled

uniform-boundedness prin iple, or the resonan e theorem) an be applied to ing37 that a olle tion { in

x  ;  ò } is bounded in X *

*

*

, say-

x  ; x>;  ò } is bounded sequen e in X must be

provided {
£ 0 for any f £ 0, f ò F .

Proposition 1.16 (Averaging positive fun tionals 

*

, i.e.

 £ 0 means just

means).53 Let

F

that

be a linear sub-

C0 (U) ontaining onstants. Then: The set M(F ) of all means on F an be alternatively expressed as:

spa e of (i)

 Moreover, if

: F Ù R linear;  £ 0 &  ; 1 # 1 #:

F) :

M(

(1.22)

inf f ¡ 0, then even (f) ¡ 0 for any  ò M(F ). e : U Ù M(F ) is weakly* ontinuous.

(ii) The evaluation mapping (iii)

M(F ) is weakly* ompa t and onvex subset of F * .

(iv)

M(F ) is the weak* losure of the set of all nite means.

51

For more details, the reader is referred to the monographs by, e.g., Berglund et al. [108℄, ƒe h [190℄,

Edwards [278℄, Engelking [284℄, Gilmann and Jerison [355℄, and Yosida [804℄. The means an be dened even a bit more generally on a linear subspa e F of bounded fun tions U not ne essarily ontinuous and not ne essarily ontaining onstants. Namely, a mean  is by the denition a linear fun tional F Ù su h that inf uòU f ( u ) ¢  ( f ) ¢ supuòU f ( u ); f. Edwards [278, 0 Se t. 3.5℄. This oin ides with our denition provided F ’ C ( U ) and 1 ò F .

52

on

53

R

We refer to Berglund, Junghenn, Milnes [108, Se t. I.3℄; however, the presented assertion here is a

bit modied, e.g.

C0 (U) is not a omplex but a real algebra and F

need not be losed.

Ë

20

1 Ba kground Generalities

 ò M(F ) and f £ 0, f ò F . Put fmax :# sup f(U) and fmin :# inf f(U). Obvifmin £ 0. Sin e  ò M(F ), it holds

Proof. Let ously,

!! !!  ; f " !!

!! 1 ( f max % f min ) ! !! 2 !

#

!! !! ¼ ; f !!

!

¢

" " "f  F " " "

" 21 (fmax % fmin )½ !!!! !

*

"

" 1 (f % fmin )"""""C0 2 max (U)

1 (f "f ): 2 max min

#

0 ¢ fmin ¢ ¢ fmax . This proves  £ 0 and even (f) ¡ 0 if inf f ¡ 0.  : F Ù R linear su h that  £ 0 and # 1. Furthermore, take f ò F and put f # " f % f C 0 U . Obviously, f £ 0 and therefore <  ; f > # "<  ; f > % f C 0 U £ 0. This yields <  ; f > ¢ f C 0 U for any f ò F . Therefore  is ontinuous, i.e.  ò F , and even  F # 1. Thus the point (i) has been proved. The weak* ontinuity of e : U Ù F means pre isely that u ÜÙ < e ( u ) ; f > # f ( u ) : U Ù R is ontinuous for any f ò F , whi h follows dire tly from the ontinuity of ea h f ò F ’ C (U). This shows (ii).

Therefore,

Conversely, let us take

(



(



)



)

(

*

)

*

*

0

M(F ) is onvex and losed. By (1.20), M(F ) is ontained in the F * , and therefore, by Alaoglu-Bourbaki theorem 1.8, it must be weakly*

In view of (1.22), unit ball of

ompa t, as laimed in the point (iii).

 ò M(F ) and put M(F ) :# w*- l( o(e(U))) the weak*

losure of nite means. If  were not belong to M(F ), then by the Hahn-Bana h theorem 1.11 there would exist f ò F su h that <  ; f > ¡ sup  r  òM F <  ; f >; realize also Let us go on to (iv). Take





(

)



F*

that, by Theorem 1.7, every weakly* ontinuous linear fun tional on

has the form

f ò F . However, sup  òM r F <  ; f > £ sup u ò U < e ( u ) ; f > # f C 0 and we obtain a ontradi tion <  ; f > ¡ f C 0 U . This shows that  ò M(F ).  ÜÙ

for some

(

(

)



)

(

U) ,



Remark 1.17 (Conne tion between means and probability measures). From (1.22), one an easily see that in the spe ial ase F # C ( U ), the set of all means is pre isely 0

the set of all probability measures on

U.

theorem 1.32 below, M(F )

(resp. M(F )

U

Ê r a% (U) 1

is ompa t (resp. normal). If

F

Thus, in view of the Riesz representation

is smaller than

Ê rba% (U)) 1

C (U), 0

if

F

# C (U) 0

and

the means an alterna-

tively be understood as lasses of probability measures with respe t to a suitable equivalen e.54 Let us now turn our attention to multipli ative means on rings of ontinuous

’ C (U) is alled a ring if f ; f ò R implies f f ò R , where f f denotes the pointwise multipli ation dened naturally by [ f f ℄( u ) :# f (u)f (u). Obviously, f f # f f , and thus we are talking about ommutative rings. bounded fun tions. A subspa e

R

0

1

2

1 2

1 2

1

2

1 2

1 2

2 1

C0 (U), ontains onstants and separates points from losed sets in the sense that, for every A ’ U losed and u ò U \ A , R

ontains f being equal 1 on A and vanishing at u . As the onstant 1 represents a unit

A ring

54 if

R

is alled omplete if it is losed in

Namely, for

P U

f d1

# PU

1 ; 2 ò r a%1 (U) (resp. rba%1 (U)) we have in mind the equivalen e: 1 f d2 for any f ò F .

È

2 if and only

1.2 Linear, nonlinear, and onvex analysis

in the sense that

Ë 21

1f # f1 # f , su h omplete subrings are simultaneously so- alled

ommutative unital algebras55 . Also, if (

U; T ) is ompletely regular, then C0 (U) itself

is a omplete ring. The aim is to onstru t for every ompa ti ation of of multipli ative means on a suitable ring.56 If

R

U its representation in terms C0 (U), let us denote

is a subring of

the set of averaging positive multipli ative fun tionals by

R ) :#   ò M(R );

:f ; f ò R :  ; f f

Mmult (

1

2

1 2

#  ; f

1

 ; f 2  :

The elements of Mmult (R ) are also alled multipli ative means, being pre isely the

R whi h dier R ) will be endowed with the (relativized) weak* topology.

ontinuous, linear, multipli ative fun tionals on

from zero. Again,

Mmult (

Proposition 1.18 (Multipli ative means).

Let

R

be a subring of

C0 (U)

ontaining on-

stants. Then: (i)

R ) is the weak* losure of e(U), and the pair (Mmult (R ); e) is a ompa ti ation of ( U; T ). Moreover, Mmult (R ) Ê Mmult ( R) with R :# lC 0 ( U ) R . If R is a omplete ring, then the ompa ti ation (Mmult (R ) ; e ) is T - onsistent. Mmult (



(ii)



(iii) Identifying equivalent ompa ti ations, there is a one-to-one order-preserving or-

²

responden e between (T - onsistent) ompa ti ations of ( U; T ) ordered by   and ( omplete) losed rings of ontinuous bounded fun tions ontaining onstants

’

ordered by the in lusion  .

M(R ),

Sket h of the proof. Clearly, Mmult (R ) is weakly* losed subset of Proposition 1.16(iii) ompa t. By the Gelfand representation,57 morphi with

f () :# 


. Then obviously f (e(u)) e(U) must be identi ally 

is isometri ally iso-

f ÜÙ f with f ò C(Mmult (R )) dened by # f(u), so that any fun tion in C(Mmult (R )) zero. Then, by Urysohn's lemma, e ( U ) must

via the mapping

vanishing on

R

hen e by





be dense in Mmult (R ).

Q : R Ù R the in lusion so that the adjoint operator Q , realizing the restri tion of linear ontinuous fun tionals from R to R , maps R onto R and is (weak*,weak*)- ontinuous. Sin e two dierent fun tionals on R remain dierent after restri tion on a dense subspa e R , Q is also inje tive. Yet ontinuous one-to*

Let us denote by



*



*





*

one mapping between ompa t sets must be a homeomorphism. The point (i) has been thus proved.

55

Let us note that a subring of

C0 (U) ontaining onstants is simultaneously an algebra under the

multipli ation by s alars.

56

Let us only remark that there are several other equivalent onstru tions: the set of all maximal

ideals on su h a ring in

C0 (U), see Gelfand and at al. [348℄, or the set of all lters on U with a ertain

spe ial properties, e.g. the set of all maximal round (or alternatively ompressed round) lters with respe t to a given proximity stru ture, see e.g. Csaszar [240℄ for denitions and other details.

57

See Gelfand at al. [348℄ (where maximal ideals are used in pla e of multipli ative linear fun tionals)

or also, e.g., Yosida [804, Se t. XI.1℄.

22

Ë

1 Ba kground Generalities

lim ò e(u  ) # e(u) implies T -lim ò u  # u T -neighbourhood N of u and f ò R su h that f ( u ) # 0 and f ( U \ N ) # 1. Then, for every  ò  large enough one has < e ( u  ) " e(u); f>   1, whi h means that u  ò N be ause obviously # . and < J k u ; J k As A # A 1 % A 2 is oer ive, for % su iently large we have *

In other words, we seek

*

*

*

*

*

*



u V k # % âá

A ( u )

Suppose, for a moment, that

*

" f; u £ A(u); u " f u ¡ 0:

I k A ( u ) #Ö I k f *

*

mapping

u ÜÙ %

(1.30)

*

"1

for any

J k I k f " * " " "I k (f " *

u ò Vk

u

with V k

¢ %. Then the

" A(u)

A(u))""""V

(1.31)

*

k

u ò V k ; u ¢ %} into itself be ause J "k 1 # 1; note that V k # f V k . Also, the mapping u ÜÙ < A ( u ) ; v > : V k Ù R is ontinuous for k any v so that also u ÜÙ I A ( u ) : V k Ù V is ontinuous. By the Brouwer xed-point k k Theorem 1.19, the mapping (1.31) has a xed point u , this means maps the onvex ompa t set {

"1 f J

*

*

*

u#%

J "k 1 I k f " A(u) : " " " " " " I k ( f " A ( u ))" "V *

(1.32)

*

*

k

As

J "k 1 f V k # f V k , (1.32) implies u V k # %. Testing (1.32) by J k u I k (f " A(u)) V k , one

gets

*

*

%2 """"I k (f " A(u))""""V # J k u ; u """" I k (f " A(u))""""V *

*

*

*

k

*

k

# % J k u ; J "k I k (f " A(u)) # % I k (f " A(u)); u # % f " A ( u ) ; I k u # % f " A ( u ) ; u 1

*

*

(1.33)

A(u) " f; u> # "% I k (A(u) " f) V k ¢ 0, a ontradi tion with (1.30). Moreover, putting v :# u k into (1.29), we an estimate62   u k  u k ¢ A ( u k ) ; u k # < f; u k > ¢ f u k with a suitable in reasing fun tion  : R% Ù R% su h that lim  Ù () # % whi h exists due to the assumed the oer ivity (1.27) of *

whi h yields
# < f " f; u 1 " u 2 > # 0, a ontradi tion. Thus,

this limit, the identity (1.29) holds even for any

1.3

Fun tion and measure spa es

For the brevity of this se tion, we must onne ourselves only to a brief summary of basi denitions and results.63

will be a measurable subset of Rn endowed with a Lebesgue measure and 1 ¢ p ¢ %; by  A  we will denote the Lebesgue measure64 of a measurable subset A ’ . We will use the standard notation for the  onjugate If not said otherwise,

exponent

p/(p"1) . 6 p :# > %

if



6 F

if

1

if

1   p   %; p # 1; p # % :

(1.35)

S will be a separable Bana h spa e; often S will be nite-dimensional.

, we say that a property holds almost everywhere on

(in abbreviation a.e. on ) if this property holds everywhere on with the possible Besides,

Having a measurable set

63

More details an be found in the monographs by Adams [4℄, Dunford, S hwartz [275℄, Gajewski,

Gröger, Za harias [342℄, Halmos [374℄ or Kufner, Fu£ík, John [467℄. For Lebesgue spa es, see also Bourbaki [145, Chap. IV℄.

64

Let us re all that the

n-dimensional Lebesgue measure  -  is the restri tion of the n-dimensional

outer Lebesgue measure  -  on 



Rn

is dened as

n

A :# inf Ž H I b ki " a ki : A ’ k#1 i#1

-algebra of Lebesgue measurable subsets of A # A  S % A \ S for any S ’ n .

on the





k k k k ℄ [ a 1 ; b 1 ℄, - - - ,[ a n ; b n ℄ ; k#1

R

Rn

. We all a set

A’

a ki ¢b ki DZ

Rn

Lebesgue measurable if

26

Ë

1 Ba kground Generalities

ex eption of a set of Lebesgue-measure zero; referring to those holds, we will also say that it holds at almost all

x ò

x where this property x ò ).

(in abbreviation a.a.

:a.a.  will mean that something holds for almost all elements.

The notation 

1.3.a

Bo hner and Lebesgue spa es

L p ( ; S) we will denote the set of all Bo hner measurable65 fun tions66 u : Ù S su h that u L p ;S   %, where

By

(



)

u L p

(

;S)

:#

p . 6 6 ”X u ( x ) S d x •

> 6 6

1/

p

ess sup u(x) S xò

F

for

1 ¢ p   % ;

for

p # % :

(1.36)

S is separable, Bo hner measurability is the same as strong meau : Ù S is alled " 1 strongly measurable if u ( A ) :# { x ò ; u ( x ) ò A } is Lebesgue measurable for any A ò S open with respe t to the strong topology. The set L p ( ; S), endowed with a pointLet us remark that, if

surability. Strong measurability has here the usual meaning:

wise addition and s alar multipli ation, is a linear spa e. Besides, - L p ( ;S) is a norm

L p ( ; S) whi h makes it a Bana h spa e, alled Bo hner spa e or, if S is innitep dimensional, a Lebesgue spa e. If S is separable, for 1 ¢ p   %, L ( ; S) is separable too.67 Let us agree on the usual onvention that S will be omitted when equal to R. on

An important question is how to hara terize on retely the dual spa es. The natural duality pairing onsidered throughout this se tion will always ome from the

L2 -spa es, whi h means :# P u1 (x) - u2 (x) dx, where u1 - u2 will often abbreviate the duality pairing between S and S . If 1   p   % and S is p a reexive Bana h spa e, then L ( ; S) is reexive. Using the Hölder inequality,68 it p

an be shown that the dual spa e is isometri ally isomorphi with L ( ; S ). This

hara terization of the dual spa e holds true also for p # 1.69 If S is not reexive, then p L p ( ; S) Ê Lw ( ; S ), whi h is the spa e of weakly measurable70 fun tions Ù S s alar produ t in

*





*

*

*

*

*

*

with the indi ated integrability;71 this is sometimes reers as a Dunford-Pettis theorem.

65

Bo hner's measurability means that

u is a.e. the limit of a sequen e of nitely-valued measurable

fun tions.

66

As usual, we will not distinguished between fun tions that equal to ea h other a.e., so that, stri tly

speaking,

L p ( ; S) ontains lasses of equivalen e of su h fun tions.

67

See, e.g., Warga [791, Thm. I.5.18℄.

68

This is

P



u1 (x)

-

u2 (x)dx

¢

(P





p

p

u1 (x) S dx)1/p (P u2 (x) S dx)1"1/p ; *

f. e.g. Bourbaki [144,

Se t. IV.6.4℄ or Köthe [436, Se t. 14.10℄. Originally, Hölder [391℄ states it in a less symmetri al form for sums in pla e of integrals.

69

We refer, e.g., to Edwards [278, Thm. 8.20.5℄ or Gajewski et al. [342, Se t. IV.1.3℄.

70

A mapping

u* : Ù S* s ò S.

*

is alled weakly

measurable if

x

ÜÙ : Ù

measurable for any

71

We refer, e.g., to Edwards [278, Thm. 8.20.3℄ or [299, Thm. 12.2.4℄.

*

R

is Lebesgue

1.3 Fun tion and measure spa es

Moreover, for

s ; : : : sm ) #

( 1

1   p   %

m 2  i #1 s i , the spa e

and

S #

Rm

Ë 27

equipped with the standard norm

L p ( ; S) is uniformly onvex.72

An important lass of nonlinear mappings from one Lebesgue (or Bo hner) spa e into another one onsists of the

x-dependent superposition mappings:

N' (u) : L p ( ; S1 ) Ù L q ( ; S2 ) : u ÜÙ „ x ÜÙ '(x ; u(x)) … ;

(1.37)

S1 and S2 are separable Bana h spa es and ' : , S1 Ù S2 is a Carathéodory '(-; s1 ) : Ù S2 is measurable for all s1 ò S1 and '(x ; -) : S1 Ù S2 is (strong,strong)- ontinuous for a.a. x ò . The nonlinear mappings (1.37)

where

mapping, whi h means that

are alled Nemytski mappings.

Theorem 1.24 (Nemytski mappings).73 Let S ; S be separable Bana h spa es, ' : , S Ù S be a Carathéodory mapping, and 1 ¢ p   %, 1 ¢ q ¢ %. Then the following 1

1

2

2

statements are equivalent to ea h other:

L p ( ; S1 ) into L q ( ; S2 ).

(i)

N'

maps

(ii)

N'

maps bounded subsets of

L p ( ; S1 ) onto bounded subsets of L q ( ; S2 ).

; a ò L q ( ) ;b ò R: '(x ; s ) S2 ¢ a(x) % b s Sp 1q . Moreover, if q #Ö %, then the above statements are also equivalent to p q (iv) N ' maps L ( ; S ) ontinuously into L ( ; S ). /

(iii)

1

1

1

2

In fa t, from the above general theorem, we will use only the impli ations74 (iii) (ii) and (iii)

á (iv).

á

S : ± S is alled measurable if, for any open A ’ S, S"1 (A) :# {x ò ; S(x)  A #Ö } is measurable.75 An example of measurable

A multivalued mapping the set

multivalued mapping arises from level sets:

Theorem 1.25 (Measurable level-set mapping).76 Let ' : , S Ù R be a Carathéodory fun tion and let S : ± S and : Ù R be measurable. Then the multivalued 72

This result is due to Clarkson [225℄, see also Adams [4, Corollary 2.29℄ or Kufner at al. [467, Re-

mark 2.17.8℄.

73

For the full generality we refer to Lu

hetti and Patrone [499℄. If

S1

and

S2

are nite-dimensional,

su h results an also be found, e.g., in Krasnoselski  [440℄.

74

The former impli ation is just by Hölder's inequality and also the latter one has a relatively sim-

Ù u in L q ( ; Rn ), then take subsequen es onverging a.e. on . Then, by ontinuity of  ( x ; -) for a.a. x ò , N  ( u k ) Ù N  ( u ) a.e., and thus in measure, too. Due to the obviq ¢ 2q" (2 a q ( x ) % b q  u ( x )q % b q  u ( x )q ) for a.a. x ò , show that ous estimate   ( x ; u k ) "  ( x ; u ) k q {  ( x ; u k ) "  ( x ; u ) }kòN is equi-absolutely ontinuous sin e strongly onvergent sequen es are. Evenq tually ombine these two fa ts to get P   ( x ; u k ) "  ( x ; u ) Ù 0 and realize that, as the limit N  ( u )

ple proof: Take

uk

1

is determined uniquely, eventually the whole sequen e onverges.

75

For this denition (possibly with

S

only omplete separable metri spa e) and further detailed

study of measurable multivalued mappings we refer to the monographs by Aubin and Frankowska [37, Chap. 8℄, Castaing and Valadier [188℄, or Deimling [256, Se t. 3℄.

76

Cf. Aubin and Frankowska [37, Theorems 8.2.9℄.

Ë

28

mapping

1 Ba kground Generalities

± S dened by x ÜÙ Lev S

(

x ); ( x ) ' ( x ; -) :#  s ò S ( x );

'(x ; s) ¢ (x)

is measurable. Having a multivalued mapping

g(x) ò S(x) for any x ò .

S : ± S, we say that g : ٠S is its sele tion if

Theorem 1.26 (Measurable sele tions).77

A multivalued mapping

S : ± S

with

nonempty losed values is measurable if and only if there exists a sequen e { g k } k òN of its measurable sele tions

gk

su h that

S(x) # lS (U kòN g k (x)) for any x ò .

S : ± S is measurable losed-valued, then also the multivalued mapping oS :

± S : x ÜÙ o(S(x)) is measurable.78 If S # Rn , one an onsider the following

If

modi ation of the Carathéodory theorem 1.2.4:

Theorem 1.27 (Carathéodory sele tions).79 Let S : ± Rn be measurable nonempty n

losed-valued and g : Ù R be a measurable sele tion of o S . Then there are measurable sele tions g k ( x ) ò S ( x ) and measurable oe ients a k : Ù [0 ; 1℄ with k # 1; :::; n%1 su h that nk#% a k (x) # 1 and nk#% a k (x)g k (x) # g(x) for any x ò . 1

1

1

1

In the rest of this se tion, we will onne ourselves to the nite-dimensional ase,

say

S :# Rm .

L p ( ; Rm ). Bounded sets in L p ( ; Rm ) are relatively weakly or weakly* ompa t provided 1   p   % or p # %, respe tively. For p # 1 the situation is far more deli ate: Let us investigate the Lebesgue spa es

Theorem 1.28 (Weak L

1

- ompa tness). Let

M ’ L1 ( ; Rm ) be bounded. Then the fol-

lowing statements are equivalent to ea h other: (i)

M is relatively weakly ompa t in L1 ( ; Rm ),

(ii) the set

M is uniformly integrable, whi h means :

:" ¡ 0 ;K ò R% : (iii) the set

M

sup X uòM

{



x ò ;  u ( x )£ K }

u(x)dx ¢ " ;

is equi- ontinuous (or, more pre isely, equi-absolutely- ontinuous) with

respe t to the Lebesgue measure, whi h means:

:" ¡ 0 ;Æ ¡ 0 :

77

sup sup X u(x) dx ¢ " ; u ò M  A ¢ Æ A

This assertion is due to Castaing [186℄; see Aubin and Frankowska [37, Thm. 8.1.4℄ also for

other hara terization of measurability or also Castaing and Valadier [188, Se t. III.2℄, Deimling [256, Se t. 3.2℄ or Warga [791, Se t. 1.7℄.

78

Cf. Aubin and Frankowska [37, Thm. 8.2.2℄.

79

Cf. Aubin and Frankowska [37, Thm. 8.2.15℄.

1.3 Fun tion and measure spa es

(iv) there is a ontinuous fun tion

:

limaÙ% (a)/a # %) su h that:

R% Ù R%

Ë 29

with a super-linear growth (i.e.



sup X (u(x)) dx   % : uòM

The points (ii) and (iii) are alled the Dunford-Pettis ompa tness riterion [274℄80 while the point (iv) is the de la Vallée-Poussin riterion [257℄.

L1 ( ) is losely related with the so- alled 1 biting onvergen e by Cha on:81 A sequen e { u k } k òN ’ L ( ) is said to onverge to B u ò L1 ( ) in the biting sense (then we will write u k Ù u), if there is a sequen e {A j }jòN su h that A j is measurable and A j ’ A j %1 ’ for any j ò N, limj Ù  A j  #  , and u k Ù u for k Ù  weakly in L1 (A j ) with j ò N arbitrary. The so- alled Cha on biting 1 lemma [154℄ says that every bounded sequen e in L ( ) admits a subsequen e on1 verging in L ( ) in the biting sense. A bit more powerful version of the biting lemma The relatively weak ompa tness in



is the following:

Lemma 1.29 (Biting Lemma).82 Having a sequen e {u k }kòN bounded in L ( ), there are measurable A k ’ su h that A k ’ A k % for any k ò N,  U k òN A k  #  , and su h that, 1

1

after taking possibly a subsequen e (denoted, for simpli ity, by the same indi es) the set

 A k u k ; k ò N} is relatively weakly ompa t in L1 ( ), where  A k : Ù {0; 1} denotes the hara teristi fun tion of the set A k .

{

Another important

L1 -weak- ompa tness

prin iple takes pla e for a.e.- on-

verging sequen es whi h have a ommon integrable majorant:

Theorem 1.30 (Lebesgue).83 Let {u k }kòN ’ L ( ) be a sequen e su h that, for a.a. x ò , the sequen e { u k ( x )} k òN ’ R onverges to some u ( x ) and  u k ( x ) ¢ u ( x ) for some u ò L ( ). Then u lives in L ( ) and limkÙ PA u k (x) dx Ù PA u(x) dx for any A ’

measurable. In parti ular, the set { u k ; k ò N} is relatively weakly ompa t84 in L ( ). 1

0

1

0

1



1

It should be emphasized that the dual spa e to than

L



(

; Rm ) is substantially larger

L1 ( ; Rm ) and its elements an be identied with ertain measures. This leads us

to a few denitions from the measure theory. For simpli ity we will onne ourselves

80

See also, e.g., Della herie and Meyer [258, Chap.II, Theorems 19,22,25℄, Dunford and S hwartz [275,

Se t. IV.8℄, or Edwards [278, Se t. 4.21℄, where the relative sequential weak ompa tness in proved but, by the Eberlain-’muljan theorem, it is equivalent to the relative weak ompa tness.

81

See Brooks and Cha on [154℄ or also Ball and Murat [67℄.

82

We refer to Valadier [774, Thm. 23℄.

83

See, e.g., Dunford and S hwartz [275, Corollary III.6.16℄ or Kolmogorov and Fomin [434, Se t. V.5.5℄.

Let us note that the linear hull of all hara teristi fun tions  A with A ’ measurable is dense in L ( ) Ê L1 ( )* , so that the sequen e {u k }, being bounded in L1 ( ), onverges weakly in L1 ( ) and,

84

as su h, it is relatively sequentially weakly ompa t, hen e by the Eberlein-’muljan theorem relatively weakly ompa t, too.

30

Ë

1 Ba kground Generalities

to the s alar ase (i.e.

m # 1), the modi ation for the ve torial ase (i.e. m ¡ 1) being

obvious (ex ept the positive and the negative variations).

1.3.b

Spa es of measures

 of subsets of an abstra t set M will be alled an algebra if  ò , A ò  á M \ A ò , and A1 ; A2 ò  á A1 € A2 ò . If also A i ò  á UiòN A i ò , then  will be alled a  -algebra. A fun tion  Ù R is alled additive if  ( A 1 € A 2 ) # (A1 ) % (A2 ) provided A1  A2 # . If (U iòN A i ) # iòN (A i ) for any mutually disjoint A i ò  , then  is alled  -additive. For  additive, we dene the variation    I of  by   ( A ) # sup A A I ò M A  i #1   ( A i ), where M ( A ) denotes the set of all nite 1

olle tions ( A 1 ; :::; A I ) of mutually disjoint A i ò  for any i # 1 ; :::; I . Besides, the 1 1 % % positive variation  is dened by  ( A ) :#   ( A )%  ( A ), while the negative variation 2 2 " is dened by " (A) :# 12 (A) " 21 (A). The obvious identity  # % " " is alled

A olle tion



(

;:::;

)

(

)

the Jordan de omposition.

 Ù R with bounded variations will be deM; ), and its subset onsisting of -additive set fun tions will be denoted

The set of all additive set fun tions noted by ba(

M; ). If M is additionally a topologi al spa e, then a set fun tion  is alled regular if :A ò  :" ¡ 0 ;A1 ; A2 ò : l(A1 ) ’ A ’ int(A2 ) and (A2 \ A1 ) ¢ ". If a set fun tion  % " is additive,  -additive, or regular, then so are also all its variations   ,  , and  . In this ase, we an dene rba( M ;  ) as the olle tion of all regular additive set fun tions  Ù R with a bounded variation, and by r a(M; ) we denote its subset onsisting of  -additive set fun tions. The smallest  -algebra ontaining all open subsets of M

onsists just of all Borel subsets of M and, as su h, it will be alled the Borel  -algebra. n Often, M # will be a domain in R endowed not only by the Eu lidean topology, but also by the Lebesgue measure. Then another natural hoi e for the  -algebra  is the set of all subsets of that are measurable with respe t to the Lebesgue measure.85 Then by vba( ;  ) we denote the set of all additive set fun tions with bounded by a(

variations that vanish on sets having the Lebesgue measure zero. All the introdu ed spa es ba(

M; ), a(M; ), rba(M; ), r a(M; ), and vba(M; )  :#

are linear ve tor spa es whi h an be normed by means of the variation, i.e. 

(M). This makes them Bana h spa es. Let us remark that  -additive set fun tions dened on a  -algebra are alled mea-

sures, while the additive set fun tions are sometimes also alled nitely additive mea-

M is a topologi al spa e and  its Borel -algebra, the measures from a(M; ) M; ) will be then addressed % % as Radon measures. If  #  with  referring to the Jordan de omposition,  is alled sures. If

will be alled Borel measures, while the elements of r a(

85

In fa t, this is the so- alled Lebesgue extension of the Borel

sets of sets having the measure zero.

-algebra, reated by adding all sub-

1.3 Fun tion and measure spa es

Ë 31

-algebra  and a positive measure is the set of #(A) and dened as number of elements of A for nite subsets of M , otherwise as %. Moreover, a positive (nitely additive) measure  will be alled a probability measure if  ( M ) # 1. The onvex subsets of positive (resp. probability) measures will be % % % % % denoted by   (resp. by  1 ), for example r a ( M ;  ) or rba ( M ;  ) (resp. r a1 ( M ;  ) % or rba1 ( M ;  )). An important example of a probability measure is the Dira measure Æ x supported at a point x ò M , whi h is dened for any subset A ò  by a positive measure. An example of the all subsets of

M

and a so- alled ounting measure, denoted by

Æ x (A) :#



1 0

if if

x ò A; x ò M \ A:

k

a i Æ x i of Dira 's measures with some x i ò M , a i £ 0, a i # 1, and k ò N is another example of a probability measure. We will all su h a measure k -atomi . Borrowing a physi al terminology, su h measures are sometimes

A onvex ombination  i #1

k  i #1

alled mole ular, f. e.g. [501, Def. A.77℄.

Theorem 1.31 (Extreme probability measures).86 (i)

The Dira measures are extreme points in the set of all probability measures.

M is ompa t, then every extreme point in r a%1 (M; ) is of the form Æ x for some x ò M .

(ii) Conversely, if

B(M) we will denote the spa e of all bounded fun tions M Ù R endowed with u :# sup u(M), whi h makes it a Bana h spa e. If M bears 0 also a topology, say T , we denote by C ( M ) :# C ( M )  B ( M ) the linear subspa e of all By

the Chebyshev norm

ontinuous bounded fun tions endowed with the same norm, whi h makes it also a

M ; T ) is ompa t, then C0 (M) # C(M). If (M ; T ) is a lo ally ompa t spa e, then C 0 ( M ) will denote a losure of the subspa e of C ( M ) of fun tions with a ompa t support; the support of a fun tion g : M Ù S, denoted by supp(g), is dened by Bana h spa e. Let us note that, if (

supp(g) :# M \ UA ò T ; g(A) # 0: Likewise, having a measure

T ; (A)

86

# 0}. Let us note that the support is always a losed subset.

For the point (i) see, e.g., Köthe [436, Se t. 25.2℄. For

Lemma V.8.6℄ and realize that, if in

 on M , we an dene its support by supp() :# M \ U{A ò

Æx

M ompa t see also Dunford and S hwartz [275, r a(M; ), it remains extreme

is an extreme point in unit ball of

r a%1 (M; ), too. The point (ii) follows from Dunford and S hwartz [275, Lemma V.8.5℄ if one realizes r a%1 (M; ) # o({Æ x ; x ò M}) and the set {Æ x ; x ò M},

that, by Proposition 1.16 and Remark 1.17, we have being a ontinuous image of the ompa t set

M , is weakly* ompa t.

Ë

32

1 Ba kground Generalities

Theorem 1.32 (Riesz-type representations).87 Let M be a set with an algebra . Then: The dual spa e to B ( M ) is isometri ally isomorphi with ba( M ;  ) provided  is the

(i)

M. M is a normal topologi al spa e and  the algebra generated by all losed subsets 0 of M , then the dual of C ( M ) is isometri ally isomorphi with rba( M ;  ). If M is ompa t and  the Borel  -algebra, C ( M ) is isometri ally isomorphi with r a(M; ). If M is lo ally ompa t and  the Borel  -algebra, then C 0 ( M ) is isometri ally isomorphi with r a( M ;  ). n If M is a measurable domain in R endowed with the Lebesgue measure and  the -algebra of all (Lebesgue) measurable subsets of M , then the dual spa e to L (M) is isometri ally isomorphi with vba( M ;  ). algebra of all subsets of

(ii) If

(iii)

(iv)

(v)

*

*



In all ases, the isometri al isomorphism

 ÜÙ g

*

, where

g

*

is the linear ontin-

uous fun tional (as an element of the dual spa e in question) and

 is the respe tive

measure, is given by the formula

g

*

; g :#

X g ( x )  (d x ) : M

The statements (iii) and (iv) are known as the Riesz representation theorems In parti ular, the Dira measure

Æx

[640℄.

an be understood as the linear ontinuous fun -

tional on ontinuous fun tions, whi h authorizes us to write Let us agree to use the shorthand notation omitting

Æ x (g) # g(x) for g ò C0 (M).  (e.g. ba( ) in pla e of

ba( ; )) be ause the algebra  will be always lear from a ontext. As vba( ) Ê L ( ) and L ( ) Ê L ( ) , we an see that vba( ) is (isomet

*

1



ri ally isomorphi with) the bidual of

*

L ( ). We saw in Se t. 1.2a that every Bana h 1

spa e an be anoni ally embedded into its bidual. A

epting this onvention, we will o

asionally not distinguish between integrable fun tions and the orresponding nitely additive measures (though sometimes the underlying integrable fun tions

L1 ( ) an be embedded also into measures understood as linear ontinuous fun tionals on C ( );

f. also Example 1.4.11 below. If a (nitely additive) measure  possesses a density d  ò L1 ( ), whi h means (A) # PA d  (x) dx for any measurable A ’ , then 

will be addressed as densities of the measures in question). Alternatively,



has a ertain spe ial property, namely it is absolutely ontinuous with respe t to the Lebesgue measure, whi h means that 

:" ¡ 0 ;Æ ¡ 0 :A ’ measurable: A ¢ Æ âá

(A) ¢ ". Also the onverse assertion is true: every absolutely ontinuous measure L1 ( ). This is known as the Radon-Nikodým theo-

possesses a density belonging to

87

For the parti ular points (i), (ii), (iii), and (v), we refer to Dunford and S hwartz [275, IV.5, Thm. 1

and Corollary 1℄, [275, IV.6, Thm. 2℄, [275, IV.6, Thm. 3℄, and [275, IV.8, Thm. 16℄, respe tively; for (v) f. also Yosida and Hewitt [803℄. For the point (iv), see, e.g., Edwards [278, Se ts. 4.3 and 4.10℄.

1.3 Fun tion and measure spa es

Ë 33

 has a density d  , we will use the notation (dx) # d  (x) dx. Every  òr a( ) admits89 a uniquely determined de omposition  # 1 % 2 where 1 is absolutely ontinuous and 2 is singular (with respe t to the Lebesgue measure) in the sense that it is supported on some subset of having the Lebesgue measure zero; the splitting  #  1 %  2 is alled the Lebesgue de omposition. n n Considering two Lebesgue measurable sets 1 ’ R 1 and 2 ’ R 2 , the identity

rem [578, 624℄.88 If measure



g(x1 ; x2 ) dx1 dx2 X

1 , 2 holds provided

#X

1

”X g ( x 1 ; x 2 ) d x 2• d x 1

2

#X

2

”X g ( x 1 ; x 2 ) d x 1• d x 2

1

g ò L1 ( 1 , 2 ) or provided one of the double-integral does exist and

is nite. This is known as the Fubini theorem90 [338℄.

1.3.

Spa es of smooth fun tions and Sobolev spa es

Ki,

Let us turn our attention to fun tions whi h enjoy some smoothness. Considering open and an in reasing sequen e of ompa t subsets

D( ) :#

k)

k

K i ’ su h that #

U i òN

C K i ( ), where C K i ( ) denotes the spa e of all fun tions

Ù R whi h are ontinuous together with all their derivatives up to the order k and k whi h have the support ontained in K i . Ea h C K i :# V k òN C K i ( ) is endowed by the l

olle tion of seminorms {- k K i } k òN with  g  k K i :# max1¢ l ¢ k ›x g C K i , whi h makes it a l Fré het spa e; here ›x is the ve tor of all partial derivatives of the order l . Then D( ) # U i òN C K is equipped with the nest topology that makes all the embeddings C K Ù i i D( ) ontinuous,91 whi h makes it a lo ally onvex spa e. More pre isely, D( ) is a Montel spa e.92 The elements of the dual spa e D( ) are alled distributions. we put

(

U i òN V k òN

( )

()

;

(

;

)

(

)

()

()

*

An important lass of fun tion spa es onsists of the Sobolev spa es [732℄, denoted by

Wk

;

p ( ;

Rm

) and dened, for

Wk

88

;

p

(

k ò N, by

; Rm ) :# u ò L p ( ; Rm );

k

›x

u ò L p ( ; Rm,n

k

)

;

See also, e.g., Dunford and S hwartz [275, Se t. 3.10℄, Edwards [278, Se t. 4.15℄ or Halmos [374,

Se t. 31℄.

89

We refer, e.g., to Dunford and S hwartz [275, Thm. 3.4.14℄ or Edwards [278, Thm. 4.15.8℄.

90

See also, e.g., Halmos [374, Se t. 36℄, Kolmogorov and Fomin [434, Se t. 5.6.4℄, or Yosida [804,

Se t. 0.3℄. In fa t, Fubini's theorem holds in more general situations than Lebesgue measures on

91

This topology is alled the indu tive limit of the topologies on

C Ki

()

Rm

.

; see, e.g., Edwards [278, Se -

tions 5.1 and 6.3℄.

92

The Montel spa e is a barrelled spa e in whi h every bounded set is relatively ompa t; re all that

a lo ally onvex spa e is alled barrelled if every losed, balan ed, onvex, and absorbing subset is a neighbourhood of 0.

Ë

34

where D

1 Ba kground Generalities

k u denotes the set of all

k-th

order partial derivatives of

Rm

u

understood in

m # ( ;R ) k u p 1/ p ) , whi h makes it a Bana h spa e. Likewise for m ( ;R ) L p ( ;Rm,n k ) 1; p m ) are separable and, Lebesgue spa es, for 1 ¢ p   % the Sobolev spa es W ( ;

Wk

the distributional sense.93 The standard norm on

(

if

p

%

u L p

;

p ( ;

›x

) is

u W k p ;

R

1   p   %, they are uniformly onvex,94 hen e also reexive. Besides, for k £ 0 non-integer we dene Wk

;

p

(

; Rm ) :# –u ò W

[

k ℄; p

(

; Rm );

k

X X



k. For k

where [ ℄ denotes the integer part of

u(x) " u(x )p dxdx   %—; n%p k" k  x " x  

(

[

(1.38)

℄)

non-integer,

Wk

;

p ( ;

Rm

) is alled the

p-power of)

Sobolev-Slobode ki spa e and the double-integral in (1.38) is alled (the Gagliardo's seminorm. They are Bana h spa es if normed by the norm



u W k p ;

We say that

(

;Rm )



the tra e operator

C( ; R

m)

:#

” u

p W k p ( ;Rm ) [

℄;

%X

X



1 u(x) " u(x )p dxdx • n % p k " k  x " x 

/



(

[

is the Lips hitz domain if its boundary

u ÜÙ u

Ù C( ; R

p

℄)

:

is Lips hitzian.95 Then

, onsidered lassi ally as a mapping

W 1 p ( ; Rm )  ;

m ), an be extended ontinuously to a linear, ontinuous, and

surje tive operator

u ÜÙ u : W 1 p ( ; Rm ) Ù W 1"1 ;

/

p; p

where the Sobolev-Slobode ki spa e on the boundary

ation of

so that

(

; Rm ) ;

(1.39)

is dened by the lo al re ti-

is overed by Lips hitzian images of (

n " 1)-dimensional domains

on whi h the former denition of Sobolev-Slobode ki spa es an be already used.96 The losed linear subspa e { Furthermore, we have

W 1"1

/

. 6 6

pŽ:# >

u ò W 1 p ( ; Rm ); u # 0}

p; p (

;

; Rm ) ’ L

pŽ

(

; Rm ) with the notation

np " p n"p

for for

u ÜÙ u : W 1 p ( ; Rm ) Ù L p ;

Ž

(

W0

1;

p

(

; Rm ).

p   n;

p # n; for p ¡ n :

an arbitrarily large real 6 6 F %

To summarize, we have

is denoted by

(1.40)

; Rm ).

 k u/x1k1 ::: x knn with k1 % ::: % k n # k and k i £ 0 for any i # 1; :::; n is dened as an distribution su h that # ("1)k m ). for any g ò D( ;

93

For example, the distributional derivative

94

See Adams [4, Thm. 3.5℄.

95

It means that

R

an be divided into a nite number of overlapping parts, ea h of them being a

graph of a s alar Lips hitz fun tion on an open subset of

96

See, e.g., Adams [4℄ or Kufner, Fu£ík and John [467℄.

Rn"

1

.

1.3 Fun tion and measure spa es

Ë 35

Relations between various fun tion and measure spa es are often in the form of in lusions. Su h in lusions are always linear operators whi h an have some additional properties: the parti ular embedding is alled ontinuous, ompa t, dense, or

homeomorphi al if the orresponding linear operator is ontinuous, ompa t, have a dense range, or the inverse operator (restri ted on the range of the original operator) is ontinuous together with the original operator, respe tively. The following embedding theorems will be often used: The embedding

C( ) ’

1 ¢ p   % it is dense but not homeomorphi al p # % it is homeomorphi al but not dense. For 1 ¢ p ¢ q ¢ %, we have q p the ontinuous dense embedding L ( ) ’ L ( ) (re all that we supposed bounded hen e     %, otherwise this embedding would not hold). Neither of the mentioned L p ( ) is always ontinuous, and for

while for

embeddings is ompa t. On the other hand, it holds97

1 p

¡

1 q

"

k n

Wk

âá

;

q

(

) ’ L p ( )

ompa tly

;

(1.41)

n is the dimension of ’ Rn . If 1/p £ 1/q " k/n, then the embedding ’ L p ( ) is generally only ontinuous provided kq   n or kq # n # 1. Also, k q ( ) is ontinuously embedded into C ( ). Introdu ing for kq ¡ n £ 2 or for n # 1, W

re all that

W k ; q ( )

;



the notation (the so- alled Sobolev exponent)

. 6 6

p :# > *

np n"p

for

an arbitrarily large real 6 6 F %

p   n;

p # n; for p ¡ n ;

(1.42)

for

W 1 p ( ) ’ L p ( ) or W 2 p ( ) ’ L p ( ) with p :# (p ) . Also, e.g., we Ž have u Ù Ü u : W 2 p ( ) Ù L p ( ) with p Ž:# (p )Ž, referring to the notation (1.40).

we an write

*

;

;

**

;

*

*

**

*

*

*

The embeddings an be transposed, resulting thus to relations between the

G1 ’ G2 and denoting I : G1 Ù G2 , the adjoint operator I : G2 Ù G1 makes just the linear ontinuous fun tionals on G 2 . Let us distinguish

respe tive dual spa es. Having two fun tion (Bana h) spa es

*

the ontinuous embedding the restri tion on

G1

of

*

*

two typi al situations for su h

ontinuous embeddings:

where

97

Ti

! . onsistent, i.e. T2 !! !G 1 # T1 > dense, i.e. l I ( G 1 ) # G 2 F

denotes the norm topology of

 type (C)

;

 type (D)

:

(1.43)

G i , i # 1; 2. These two types more in details:

Re all that throughout the book we use the onvention

1/p :# 0 for p # %.

Ë

36

(

1 Ba kground Generalities

Su h embedding

)

I

*

: G Ù G *

*

2

1

I : G1 Ù G2

is homeomorphi al and then the adjoint operator

is surje tive be ause every linear ontinuous fun tional on

remains ontinuous also with respe t to the topology indu ed from

G1

and an

G2 by the Hahn-Bana h theorem 1.11. I : G1 Ù G2 ontinuous and dense makes the adjoint

be then extended onto

d

(

G2

The embedding operator

)

I : G2 Ù G1 is inje tive (be ause two dierent linear ontinuous fun tionals on G 2 must have also dierent tra es on any dense subset, in parti ular on G 1 ). *

*

operator

*

Sometimes the above ases an appear simultaneously, whi h gives rise to the third situation when the ontinuous embedding is simultaneously onsistent and dense:

d

(

)

I : G Ù G is homeomorphi al and dense, then the adjoint : G Ù G is one-to-one. Though I need not be a (weak*,weak*)-

If the embedding operator

I

*

1

*

*

2

1

2

*

homeomorphism, it is the (weak*,weak*)-homeomorphism if restri ted on a ball in

G2

*

duals

(whi h is weakly* ompa t). If

G1

*

and

G2

*

G1

G2

and

are normed spa es (so that the

are Bana h spa es), the inverse operator (

I )"1 *

is additionally

(strong,strong)- ontinuous thanks to the open-mapping theorem. In the situation (D) and thus also (CD), it is a ommon onvention to onsider bedded via images in

I

*

into

G1 .

G1

*

and then not to distinguish between elements of

G2

*

G2

*

em-

and their

*

Example 1.33 (Intermediate subspa e G). Let :# ["1; 1℄ and let G be a linear spa e of fun tions g : Ù R whi h are ontinuous ex ept 0 where they posses unilateral limits; this means g  " ò C(["1; 0℄) and g ò C([0; 1℄). We endow G with the (

1 ; 0)

(0 ; 1)

supremum norm. We have obviously

C(["1; 1℄) ’ G ’ L



"1; 1℄) ;

([

both embeddings98 being homeomorphi al but not dense, i.e. of the type (C) but not

G an be identied with the spa e of ertain measures, namely r a(["1 ; 0℄) , r a([0 ; 1℄). The relations between the dual spa e are obviously the surje tions: vba["1 ; 1℄ Ù G Ù r a["1 ; 1℄. Neither of these surje tions is invertible. For example, for any a ò R, the mapping aÆ 0" % (1" a ) Æ 0% : G Ù R dened by

(D). Again, the dual spa e

*

*

[

a Æ0" % (1" a) Æ0% ℄ (g) # a lim g(x) % (1" a) lim g(x) x ÷0

x ÿ0

G. Obviously, if g ò C(["1; 1℄), then aÆ0" % (1" a)Æ0% ℄ (g) # Æ0 (g) # g(0). In other words, the surje tion G Ù C(["1; 1℄) ,

forms a linear ontinuous fun tional on [

*

*

C(["1; 1℄) is a spa e of fun tions while L (["1; 1℄) onsists  of equivalen e lasses of fun tions. Nevertheless, the embedding like C (["1 ; 1℄) ’ L (["1 ; 1℄) has

98

To be pre ise, one should realize that

a good sense be ause two ontinuous fun tions, that are a.e. equal to ea h other, oin ide with ea h other; f. also Lang [475, Se t. VII.4℄.

1.3 Fun tion and measure spa es

Ë 37

C(["1; 1℄), sends the fun tional aÆ0" % (1"a)Æ0% to the Æ0 . Thus we saw the situation that the Dira measure is split onto a ontinuum of mutually dierent measures when the spa e of test fun tions C (["1 ; 1℄) is enlarged for G “ C (["1 ; 1℄). On the other hand, G is still a separable Bana h spa e, so that the weak* topology on bounded subsets of G is metrizable, ontrary to L (["1 ; 1℄). However, L (["1 ; 1℄) is no longer separable, whi h auses that the re-

whi h is just the restri tion on Dira measure

*





sulting fun tionals likely annot be des ribed expli itly but merely their existen e an be laimed with help of the Hahn-Bana h theorem (and thus of the axiom of hoi e whi h is involved nontrivially in the Hahn-Bana h theorem).99

Example 1.34.

D( ), L p ( ) an be transposed for the respe tive dual spa es. We suppose bounded and 1 ¢ p   %. The relations are summarized

C( ), L





(

For an illustration, let us realize how the interrelations between

), G

from Example 1.33, and

by the following diagram. The arrows are marked either by (C) or by (D), referring thus to the above lassi ation.

(D) ✲ (C) ✲ G (C) ✲ L ( ) D( ) C( ) ❳❳❳ ✘✘ (D) (D) ✘✘✘✘ (D) ❳❳❳❳❅❅ (D) ✠✘ ✾✘ ③❘ ❳ L p ( )

The transposed diagram is the following (the des ription 

sur and inj indi ates re-

spe tively the surje tivity or inje tivity of the mapping orresponding to the parti ular arrow):

INJ. r a( ) ✛SUR. SUR. vba( ) D( )* ✛ G* ✛ ②❳❳ ❳ ✿ ✘✘ ■ INJ. ✒ INJ.✘✘✘✘ INJ. ❳❳❳❳❅ INJ. ✘ ❳❅ ✘ L p ( ) 

The relations between the involved spa es are a

omplished by the observation that the rst diagram is onne ted with the se ond one be ause we have always the em-

L p ( ) Ù vba( ). For p £ 2, we have even stronger onne tion be ause there p p is the embedding L ( ) Ù L ( ).

bedding



Remark 1.35 (Insu ien y of the on ept of sequen es). The weak* topology on vba( ) Ê L ( ) is not metrizable even if restri ted on bounded subsets, whi h is 1

**

related with the fa t that

L



(

)

is not separable. Besides, this is an example of a

situation where sequen es are not a satisfa tory tool. Namely, no element from the

vba( ) \ L ( ) an be attained (with respe t to the weak* topology) by a sequen e from L ( ), though L ( ) is dense in vba( ). Indeed, if it were possible, 1

remainder

1

1

su h a net would be weakly Cau hy in

99

L1 ( ) be ause the tra e of the weak* topology

Cf. also the example by Lang [475, Se t. VII.4℄.

Ë

38 in

1 Ba kground Generalities

vba( ) oin ides with the weak topology in L ( ). However, the limit of su h a se1

quen e must live in

L ( ) be ause L ( ) is sequentially weakly omplete.100 Anyhow, 1

1

insu ien y of sequen es (and ne essity of the on ept Moore-Smith onvergen e) is onsidered as too far-going mathemati al abstra tion whi h dramati ally looses

onstru tivity and is attempted to be avoided in appli ations.

Some dierential and integral equations

1.4

In this se tion we will briey summarize some basi fa ts about sele ted lasses of dierential and integral equations whi h we will need in the examples of Chapter 4.

Ordinary dierential and dierential-algebrai equations

1.4.a

We will start with the initial-value problem for a (system of) ordinary dierential equa-

tions (ODE)  we will also say for a nite-dimensional dynami al system:

dy # f(t ; y) with t ò I and y(0) # y dt n n n with a Carathéodory mapping f : I , R Ù R with I :# [0 ; T ℄ and y ò R .

(1.44)

0

0

The main ingredient for estimation of evolution systems in general is the so- alled

Gronwall inequality,101 whi h we will also often use. In the general form, this inequality says that, for all

t £ 0, it holds y(t) ¢

whenever we know that

œC

t



% X b()e" P0 a # (

)

d#

0

t

d eP0 a  (

)

d

(1.45)

t

y(t) ¢ C % P0 a()y() % b() d for some a ; b £ 0 integrable.

Classi al existen e and uniqueness results are the following:

Proposition 1.36 (Ordinary dierential equation).102 Let 1 ¢ p ¢ % and the Caran n p théodory mapping f : I ,R Ù R satisfy  f ( t ; r ) ¢ a p ( t )(1 %  r ) with some a p ò L ( I ). p n Then (1.44) possesses a solution y ò W ( I ; R ). This solution is unique provided  f ( t ; r ) " f ( t ; r ) ¢ a ( t ) r " r  for some a ò L ( I ). 1;

1

2

1

1

2

1

1

 ¡ 0 with T/ ò N, let us dene the approximate solution y  ò W 1 (I; Rn ) su h that, for any k # 1 ; :::; T /  , the restri tion y   k "1  k is ane and, denoting ;

Proof. For

[(

100

)

;

℄

For this nontrivial fa t the reader is referred, e.g., to Dunford and S hwartz [275, Se t. IV.8℄ or

Edwards [278, Thm. 4.21.4℄.

101

In the general form presented here, whi h an be found, e.g., in Mordukhovi h [550, Se t. B1℄, it

is also alled the Bellman-Gronwall inequality. 1 In fa t, for p # 1, it su es to assume f (- ; 0) ò L ( I ; f(t ; -), we an see f(t ; r) ¢ a1 (t)(1 % r), too.

102 of

Rn

) be ause then, by the Lips hitz ontinuity

1.4 Some dierential and integral equations

Ë 39

y k # y  (k), given by the re ursive formula holds103 k

y k # y k"1 % X

(

k "1) 

f(t ; y k"1 ) dt ;

k # 1; :::; T/, starting for k # 1 with y0 # y0 . Due to the estimate y k " y k"1 / ¢ k "1 ) P k (1% y  a (t) dt with a0 ò L1 (I), we an see that y  is bounded in L (I; Rn ) k "1  0

for



(

)

. Sin e even a0 ò L p (I), we an also see that ddt y  is bounded in L p (I; Rn ) uniformly with respe t to . Sin e p ¡ 1, we an take a subsequen e and 1 p n some y ò W ( I ; R ) su h that, for  Ù 0, it holds

independently of

;

y Ù y

weakly* in

W 1 p (I; Rn ) ; ;

1   p   % it is the weak onvergen e while for p # 1 this onvergen e BV(I; Rn ) but, as we later show that ddt y # Nf (y) ò L (I; Rn ), the limit y n k " for t ò [( k "1)  ; k ) and n ( I ; R ) by y  ( t ) # y  belongs to W ( I ; R ). Dening y  ò L k # 1; :::; T/, we an see that

in fa t, for

1

is in rather in

1;1

y Ù y be ause

strongly in

L q (I; Rn )

for any

q   %

y  Ù y strongly in L p (I; Rn ) and be ause of the al ulus

p

y  "y  L p I;Rn # (

)

#

T/

k !! k t !!( y " y k "1 ) HX  !!  ( k "1)  k #1  T/ !!! k k"1 !!!p H !y "y ! # p%1 k#1 !!   !!

" (k"1) !!!p 

!! !

dt

 p """ dy  """ p # O ( p ) " " p%1 "" dt "" L p I;Rn (

)

I; Rn ). Passing to the limit in the obvious identity # Nf (y  ) and using the ontinuity of the Nemytski mapping Nf : L p (I; Rn ) Ù

and be ause of the bound of

d dt y 

1



y  in L



(

L p (I; Rn ), we get ddt y # Nf (y), whi h just means that y solves (1.44); note that y(0) # y0 be ause y  (0) # y 0 and be ause of the weak ontinuity of the tra e operator y ÜÙ y (0) : W 1 p (I; Rn ) Ù Rn . Supposing now that (1.44) admits two solutions y 1 and y 2 , we get by subtra tion and multipli ation by y 1 " y 2 the estimate ;

1d y " y 2 dt 1

2

2

¢ Nf (y ) " Nf (y

y1 # y2 a # a1 , and y # y1 "y2 2 .

from whi h we get

103

1

2 )

y " y2  ¢ a1 y1 " y2 2 ;

-  1

by the Gronwall inequality (1.45) used for

C # 0, b # 0,

In other words, we use the so- alled (expli it) Euler formula with an equi-distant partition of the

time interval

I.

40

Ë

1 Ba kground Generalities

A useful generalization of the initial-value problem for ordinary dierential equations (1.44) is towards dierential-algebrai 104 equations (DAE) in the so- alled semi-

expli it form.105 Conning ourselves again to nite-dimensional ases, it reads as

dy # f(t ; y; w) ; y(0) # y dt 0 # g(t ; y; w) with Carathéodory mappings

y0 ò R

and with

n . Now

f : I,

y(t) ò R

m and

r

(1.46a)

0

(1.46b)

Rn , Rm Ù Rn and g : I , Rn , Rm Ù Rm , w t ò Rm are unknown ve tors of slow and ( )

y, we will use v a pla eholder for values of w. Saying that (1.46) has a (dierential) index k means that we need to dierentiate the algebrai part (1.46b) in time ( k "1)-times to obtain

fast variables, respe tively. Like

being a pla eholder for values of

the underlying system of ordinary dierential equations (ODE) like (1.44). The simplest DAEs with index 1 arises when the algebrai part (1.46b) admits an impli it fun tion

w in the sense:

;w : I , Rn Ù Rm :

g(t ; r; v) # 0 ã v # w(t ; r) :

(1.47)

This assumption is to be veried in ea h parti ular ase in on rete appli ations. Then the so- alled underlying ODE (1.44) takes solution

y

f

as

of this underlying ODE, the pair (

f Žw : (t ; r) ÜÙ f(t ; r; w (t ; r)). Having a y; w) with w # w(t ; y) solves the DAE

(1.46). The ondition (1.47) often annot be fullled be ause the DAEs in question have an index higher than 1. For index-2 DAEs, we will assume satisfying

g smooth (of the C1 - lass),

gv # 0 ; 

this means that

(1.48)

g depends only on t and y. By dierentiation of the algebrai equation

(1.46b) on e in time and using also the dierential equation (1.46a), one gets

0#

d dy g(t ; y) # g t (t ; y) % g r (t ; y) # g t (t ; y) % g r (t ; y)f(t ; y; w) : dt dt 







(1.49)

The analog of the assumption (1.47) now reads as

;w : I , Rn Ù Rm : Then, using this repla ing

104

f

g t % g r f
(t ; r; v) # 0 



ã

v # w(t ; r) :

w, the underlying ODE is to be onstru ted and used as before when f Žw. Now, the initial ondition y is to be ompatible with the

in (1.44) by

0

The adje tive algebrai  in the ontext of DAEs does not refer to any algebra, just wants to high-

light that

w-variable is derivative free. Sometimes they are also alled singular systems, referring that E ddt y # f(t ; y) with a matrix

(1.46) an alternatively be understood as a generalization of (1.44) towards

E whi h an be singular. 105

(1.50)

The adje tive semi-expli it refers to that

d dt y o

urs expli itly in the dierential part.

1.4 Some dierential and integral equations

Ë 41

algebrai onstraint, namely

g(0; y0 ) # 0 :

(1.51)

Higher-index DAEs be ome more umbersome. Let us still present index-3 DAEs, whi h has importan e in some appli ations, f. Remark 4.73. Then of

g is to be assumed

C2 - lass satisfying, in addition to (1.48), also g r (t ; r)f v (t ; r; v) # 0 : 



(1.52)

By dierentiating the algebrai equation (1.46b) twi e in time and using also the differential equation (1.46a) and the stru tural restri tions (1.48) and (1.52) in order to eliminate expli it dependen e of

0#

w on ddwt , one obtains

d d g(t ; y) # g t (t ; y) % g r (t ; y)f(t ; y; w) dt dt dy dy # g tt (t ; y) % g tr (t ; y) % Œg tr (t ; y) % g rr (t ; y) f(t ; y; w) dt dt dy dw % g r (t ; y)Œ f t (t ; y; w) % f r (t ; y; w) % f v (t ; y; w)  dt dt # g tt (t ; y) % g rr (t ; r)f (t ; r; v) % g r (t ; y)f r (t ; y; w)f(t ; y; w) % 2g tr (t ; y)f(t ; y; w) % g r (t ; y)f t (t ; y; w) : (1.53) 2





2

















2













Let us note that (1.48) now implies also



g yv # 0. Instead of (1.50), we now assume 

;w : I , Rn Ù Rm : g tt % g rr f 2 % g r f r f % 2g tr f % g r f t
(t ; r; v) # 0 



Again, using this











ã

v # w(t ; r) :

(1.54)

w, the underlying ODE is to be onstru ted and used as before. More-

over, it is also natural (and to some extent ne essary) to assume the initial velo ity

d dt y(0) ompatible with the algebrai onstraint (1.46b), i.e. .

g t (0; y0 ) # "g r (0; y0 )y0 ; 



where

y0 :# f(0; y0 ; v) ; v ò Rm : .

independent of v . The imporg r (0; y0 )y 0 in (1.55) does not depend on r and v . be ause of the orthogonality (1.52), though y 0 itself may depend on v as indi ated in

Here we used the assumption (1.48) implying tant fa t is that the right-hand side



.

gt

(1.55)



and

gr 

(1.55). Let us summarize the above manipulations towards using Proposition 1.36:

Proposition 1.37 (Dierential-algebrai systems).106 Let (1.47), or (1.48) with (1.50) or

with (1.52) and (1.54) hold. Moreover, in the latter two ases, let y 0 be ompatible with the

f Žw ts with the assumptions y ò W 1;1 (I; m ) to ddt y #  m ) sin e w ( t ) # w( t ; y ( t )). f(t ; y; w(t ; y)) with y(0) # y0 , from whi h we then get w ò L (I;

106

Note that the assumptions on

of Proposition 1.36 with

p

# 1,

f

and

w

are just devised so that

whi h then yields a unique solution

R

R

42

Ë

1 Ba kground Generalities

algebrai part in the sense (1.51) or (1.55), respe tively. Let also

w : I ,Rn Ù Rm from

(1.47), or (1.50), or (1.54) be a Carathéodory mapping uniformly Lips hitz ontinuous in the sense 

w(t ; r)"w(t ; r ) ¢ C(1 % r" r ) for some C ò R with w(-; 0) ò L (I; Rn ) 

f : I ,Rn ,Rm Ù Rn satisfy f(-; 0; 0) ò L1 (I; Rn ) and f(t ; r; v)" f(t ; r ; v )) ¢ a1 (t)(1 % r" r  % w" v ) with some a1 ò L1 (I). Then the initial-value problem (1.46) has a 1 1 m m unique solution ( y; w ) ò W (I; R ) , L ( I ; R ). 



and and let









;



Of ourse, if (1.46b) ontains

m ¡ 1 equations, the index may be dierent in dier-

ent equations and the above al ulations should then be ombined. The presen e of the algebrai onstraint (1.46b) may bring di ulties in numeri al solutions of DAEs in omparison with ODEs, and may exhibit some hidden onstraints in parti ular in the ontext of optimal ontrol of systems governed by DAEs, f. Se tion 4.3.g. Another useful generalization of the initial-value problem for ordinary dierential equations (1.44) is towards innite-dimensional dynami al systems, i.e. (1.44) with a

f : I , V Ù V and y0 ò H with V being a separable reexive Bana h spa e and H “ V a separable Hilbert spa e; i.e. here f ( t ; -) : V Ù V strongly

ontinuous for a.a. t ò I and f (- ; v ) : I Ù V Bo hner measurable for all v ò V . To be a bit more spe i , instead of f ( t ; v ) we will onsider f ( t ; v ) " A ( v ) with A : V Ù V , so *

Carathéodory mapping

*

*

*

that (1.44) will take the form

dy % A(y) # f(t ; y) dt Moreover, we assume that

with

tòI

and

y(0) # y0 :

(1.56)

H is identied with its own dual and the embedding V ’ H

is ontinuous and dense; i.e. the embedding of type (D) so that the adjoint mapping

V ’ H ’ V . Importantly, the restri tion of the V on H is the s alar produ t (-; -) on H and we have *

is inje tive. Therefore between

V

*

and

;

duality the abstra t

by-part integration formula

t dy ; y½ X ¼ 0

dt

so that in parti ular for



%¼

dy ; y½ dt # (y(t); y (t)) " (y(0); y (0)) ; dt 



t

y # y we have P0 < ddyt ; y > dt # 



yt

1

2

( )

2

H

"

1 2

y 0) 2H . To devise

(

the abstra t s heme optimally, we expe t to have some Bana h spa e

L V;H ; p ontain1"  -

I; H) for whi h the interpolation - LV H p ¢ C - L p holds for some C ò R and 0      1.

ing

L p (I; V)  L

(1.57)





(

;

;

I V)

( ;

L



I H)

( ;

There are several te hniques to handle this evolution problem in its various generality. For simpli ity, having in mind appli ation to paraboli partial dierential equations, we onne ourselves to the monotoni ity te hnique.

Proposition 1.38 (Solutions to abstra t dynami al system). Let the embedding V ’ H V ’ V for some Bana h spa e V , A : V Ù V be ontinuous and f : I ,V Ù V be a Carathéodory mapping bounded in the

be dense and ompa t as well as the embedding *

*





1.4 Some dierential and integral equations

Ë 43

sense107

p "1

A(v)V ¢ C( v H )1% v V *

for some

 and

p "1

f t ; v) V ¢  p (t) % C( v H ) v V

(



*

R Ù R ontinuous, and let further A " f t ;

1   p   %, C :

(

-)

(1.58a)

:VÙV

*

be

semi- oer ive in the sense

A ( v )

" f(t ; v); v £ vpV "  p (t)vV "  (t) v H 

2

1

(1.58b)

 q ò L q (I), and A(u) # A(u ; u) with A(-; v) : V Ù V ontinuous with some Bana h spa e V into whi h V is embedded ompa tly and A( u ; -) : V Ù V ontinuous *

with some



*



and uniformly semi-monotone in the sense

A( u ; v )

" A(u ; v ); v" v £ v" v pV " v" v H / 2







(1.58 )



 ¡ 0 for some seminorm  - V on V satisfying v V ¢ C(vV % v H ) for some C   %, N f maps bounded sets in L p (I; V)  L (I; H) into bounded sets in Lp V H p, p 1 p and y 0 ò H. Then (1.56) has a solution y ò L ( I ; V )  W ( I ; V ) in the sense that d y % A(y) # f(y) holds a.e. on I in V and y(0) # y holds108 in H . Moreover, if also 0 dt with some



;

;



;

*

*

;  ¡ 0  ò L (I) :v; v ò V : 1

1

A ( v )



" f(t ; v) " A( v ) % f(t ; v ); v" v %  (t)% v pV % v pV  v" v H / £ 0 ; 2





1







(1.58d)

then this solution is unique and depends ontinuously on the data in the sense that, for

f(t ; y) # f0 (t ; y) % f1 (t), the mapping f1 ÜÙ y is ontinuous from L p (I; V*) Ù L p (I; V)  L (I; H). 



Sket h of the proof. We use the approximation by Faedo-Galerkin's method exploiting theory of ordinary dierential equations as in Proposition 1.36. Let us take a sequen e

V1 ’ V2 ’ V3 ’ ::: ’ V whose union is dense in V y0 k ò V k su h that y0 k Ù y0 in H . For k ò N, let us dene109 the approximate 1 p solution y k ò W ( I ; V k ) su h that y k (0 ; -) # y 0 k . 1 d 2 The test of the approximate solutions by y k is legitimate and gives y k H % 2 dt < A ( y k ) ; y k > # < f ( t ; y k ) ; y k >. Then, using (1.58b) and the growth (1.58a) of f and the of nite-dimensional subspa es

and

;

;

;

107



;

In fa t, (1.58a) an be generalized by allowing

108

f

to have also a omponent admitting the bound

¢ (t)(1 % v H ) with some ò L (I). p p ( I ; V ) is embedded into C ( I ; H) so that the initial ondition A tually, the spa e L ( I ; V )  W

f t ; v) H

(

1

1;



*

has indeed a good sense.

109

Considering a base {

tion

yk

v i }i#1;:::;k

of

Vk

and the ansatz

y k (t)

#

k i#1

is determined by a system of ordinary dierential equations

f t ; kj#1  j (t)v j ); v i > with i

< (

 i (t)v i , the approximate solu% #

d dt  i

# 1; :::; k for the oe ients  i , so that the existen e of our approximate

solutions an be laimed by Proposition 1.36 rst lo ally in time and then by ontinuation using the

L (I)-apriori estimates.

44

Ë

1 Ba kground Generalities

Young inequality, the estimate

1d y % y k pV ¢  y k H %  p y k V : 2 dt k H 2

2



1

y 0 # y0 H   %, I; H)  L p (I; V). By omparison and using (1.58a), for d any k £ l we obtain a uniform bound of dt y k in seminorms

Using the Young and the Gronwall inequalities together with k ( ) H

y k in L

we obtain the bound of

 - 



(

l :#

sup

:a.a. tòI: v(t)òV l v L I;H L p I;V ¢1 

(

)

(

T X - ; v ( t ) d t :

(1.59)

0

)

Then, by Bana h's sele tion prin iple (Theorem 1.9), we take a subsequen e

y

onverging weakly* in

L



(

I; H)  L p (I; V).

Using ompa tness of

V ’ V 

y ki Ù

and the

Aubin-Lions theorem110 about the ompa t embedding

L p (I; V)  W 1 1 (I; Vl s ) ’ L p (I; V ) ;

ompa tly



Vl s ,111 we still have y k Ù y strongly in L p (I; V ). By the interpolation with the boundedness in L ( I ; H), we have y k Ù y strongly also for any Hausdor lo ally- onvex spa e





in

Lp V ;H ; p . Furthermore, we prove strong onvergen e

strategy (1.34). More spe i ally, taking

yl

y ki Ù y

by modifying the abstra t

y

an approximation of



valued in

Vl

and

using (1.58 ), we estimate

1d 1 p y l "y k i H %   y l "y k i  V ¢ A( y k i ; y l )" A ( y k i ) ; y l " y k i % y l " y k i H 2 dt  1 # A(y k i ; y l ); y l "y k i " f(y k i ); y l "y k i % y l "y k i H ; 2



2















2



(1.60)



t-dependen e of f , y k , et . for notational simpli ity. We an onsider y l Ù y strongly in L p (I; V)  L (I; H). We then obtain the strong onvergen e y k i Ù y in L p (I; V)  L (I; H) by the Gronwall inequality and by the onvergen e p # 0

i Ù

fy

sin e { ( k i )} i òN is bounded in

110 111

Lp*

V ;H ; p



and

y l "y k i Ù 0 strongly Lp V H 

;

;

p.

See J.-P. Aubin [34℄ and J.-L. Lions [495, Chap. 1, Thm. 5.1℄. This is, in fa t, a bit te hni al generalization of the usual Aubin-Lions theorem [659℄ tted for the

d dt y k holds only in the seminorms (1.59), whi h yields the Vl s . Alternatively, one an use the Bana h spa e V * in the position of Vl s and a Hahnd Bana h extension of dt y k , f. [685, Se t.8.4℄.

Galerkin approximation, as the estimate of topology of

1.4 Some dierential and integral equations

Having now the strong onvergen e

y ki Ù y

Ë 45

proved, the limit passage in the

Galerkin approximation towards (1.56) is then easy.

y1 and y2 and using (1.58d) and the Gronwall d inequality, we an see uniqueness. More spe i ally, testing the dieren e of dt y i % A(y i ) # f(y1 ), i # 1; 2, by y1 " y2 #: y12 and using (1.58d), we obtain the estimate Eventually, omparing two solutions

1d y 2 dt from whi h we obtain

2

12

H

¢  ( t ) % y 1

1

p V

% y

2

p 2 V  y 12 H / 

y12 # 0 by the Gronwall inequality, using also y12 t#0 # 0. Moref1 ÜÙ y is by the uniform monotoni ity

over, the laimed ontinuity of the mapping

(1.58 ), just repli ating the arguments for the strong onvergen e of the Galerkin approximation above.

Remark 1.39 (Weak solutions). I

Instead of the two equations in (1.56), the former one

holding a.e. on , one an require

T X A ( y )" f ( y ) ; v 0

for any

y ò L p (I; V)  Cw (I; H) to satisfy the integral identity

" ¼y;

v ò L p (I; V)  W 1 p (I; V ;



*

dv ½ d t %  y ( T ) ; v ( T ) # y ; v (0) dt

). If also

0

d dt y

ò L p (I; V 

*

), it is equivalent to the

lassi al solution to the initial-value problem (1.56) laimed in Proposition 1.38.

Remark 1.40 (Abstra t dierential-algebrai systems).

These two generalizations of

(1.44) an be ombined and thus one gets innite-dimensional dierential-algebrai systems, f. Remark 4.114.

1.4.b

Partial dierential equations of ellipti type

Another type of (systems of rst ase

m) dierential equations ontains partial derivatives. The

is of the ellipti type.112 Su h type of equations des ribes stationary (or

steady-state) spatially-distributed-parameter systems on a spatial domain and need also an appropriate boundary onditions. We will onne ourselves to the ase of Robin-type (sometimes also alled Newton or Fourier-type) boundary-value problem for a 2nd-order systems of ellipti dierential equations in the divergen e form. So we will onsider the problem

"div a(x ; y; ›x y) % (x ; y ; ›x y) # 0 n (x) - a(x ; y; ›x y) % b(x ; y) # 0

112

; on ;

on

§

(1.61)

Here, readers are re ommended to the monographs e.g. [495, 568, 685℄ for more details and results

in more general situations.

Ë

46

where



1 Ba kground Generalities

is a bounded domain in

Rn with a Lips hitz boundary

notes the unit outward normal113 to the boundary



tion is pres ribed. The notation ›x  denotes the spatial gradient ( x 1 ;

n

: ,R

R

m,n

Ù

R

m and

b :

,R Ù m

R

de-

 x n ) while m,n Ù m,n ,

:::;

a : ,Rm , R



div is the divergen e of a ve tor, i.e.  i #1 x i (-) i . Here,

m,

n

and where

where the Robin boundary ondi-

R

m are Carathéodory mappings repre-

senting a  ondu tivity or elasti ity oe ients, distributed sour es or for es, and a boundary ux or a for e tra tion, respe tively, depending on parti ular appli ations. As the lassi al (= pointwise) understanding of the problem (1.61) is not natural from both mathemati al and physi al reasons, the standard understanding of (1.61) is in the sense of distributions, whi h leads to the notion of a so- alled weak solution. The weak formulation arises by multiplying the equation in (1.61) by some test fun tion by integration over

y, 

, by the applying Green's formula, i.e. X y

whi h holds for any

-

div A % A : ›x y dx #

y : Ù

Rm and A

X

A : (y‹ n ) dS

Rm,n smooth enough, and eventu-

: Ù

ally by substitution of the onormal derivative from the boundary ondition in (1.61), whi h eventually yields the identity

X a ( y; ›x y ) : ›x y

where 

:



% (y ; ›x y ) - y dx % X b(y) - y dS # 0 ; 

(1.62)



 and - denotes the summation over two or one indi es, respe tively. For

notational simpli ity, we omit the expli it

R

x-dependen e in a, , and b. We say that

y: Ù m is a weak solution to (1.61) if the integral identity (1.62) is fullled for any y ò W 1; p ( ; m ). Considering a suitable polynomial-like-growth exponent 1   p  

R



% of the data, we will seek a solution in the Sobolev spa e W y ò W p ( ; Rm ) in (1.62).

1;

p ( ;

Rm

) and use

1;



y; y ò b. First,

To guarantee the integral identity (1.62) to have a good sense for any

W 1 p ( ; Rm ), ;

one must impose a ertain onditions on the data

a, ,

and



all of them will be assumed Carathéodory mappings so that all terms under the integrals in (1.62) will be measurable. Moreover, the integrability of these terms will be respe tively guaranteed by the following growth onditions:

; ò L p ( ) :(x ; r ;  ) ò , Rm , Rm,n : 



Ž ; ò L p " (

; òL p (

*



)

(

)

a(x ; r ;  ) ¢

:(x ; r) ò , R :



b(x ; r) ¢

R R

") ( ) :(x ; r;  ) ò , m , m,n 

:

x ; r;  ) ¢

 (

113

x % Crp

Let us note that this normal does exist a.e. on

*

( )

/

p



"

Ž

% Cp" ; 1

x % Crp ""1 ;

( )

(1.63b)

x % Crp ""1 %Cp *

( )

be ause

(1.63a)

/(

p

*

") ; 

(1.63 )

is Lips hitzian, whi h means that

be overed by a nite number of graphs of Lips hitz fun tions on domains in

Rn"

1

.

an

1.4 Some dierential and integral equations

Ë 47

p , pŽ, and p are from (1.35), (1.40), and (1.42), respe tively, C ò R, and 0    ¢ min(p ; pŽ) " 1 is arbitrarily small. Let us a

ept the notational short ut that, for p ¡ n, the terms r% o

urring in (1.63b, ) are to be understood su h that b(x ; -), and  ( x ; - ;  ) may have an arbitrary fast growth if  r  Ù . 

where

*

*



In view of Theorem 1.24, the growth onditions (1.63) are designed so that respe tively

Na : W 1; p ( ; Rm ),L p ( ; Rn ) Ù L p ( ; Rn ) 

,

is (weak strong,strong)- ontinuous

Ž y ÜÙ Nb (y ) : W 1 p ( ; Rm ) Ù L p " ;

(



)

(

; Rm )

is ontinuous

;

(1.64a)

;

(1.64b)

N : W 1; p ( ; Rm ),L p ( ; Rm,n ) Ù L(p ") ( ; Rm ) is ontinuous : *



(1.64 )

y; v ò W 1 p ( ; Rm ), the integrands a(y; ›x y) : ›x y and (y; ›x y) - y 1 1 o

urring in (1.62) belong to L ( ) while b ( y  ) - y  belongs to L ( ). ;

In parti ular, for







Proposition 1.41 (Ellipti equations: existen e and uniqueness). Let 1   p   % and the following uniform-monotoni ity114

oer ivity onditions are valid for some  ¡ 0:

for some

:

x ò : r ò Rm ;  ;  ò Rm,n : 

a.a.

X a ( y; ›x y ) : ›x y

"a(x ; r;  ) : ( "  ) £  "  p ; 1 b(y) - y dS £  y W 1 p Rm " :

 a ( x ; r; 

% (y; ›x y) - y dx % X

(1.63) be valid and the overall-

)







1/

;

(

;



)

(1.65a) (1.65b)

Then the boundary value problem (1.61) possesses at least one weak solution

W 1 p ( ; Rm ). Moreover, if the overall stri t monotoni ity hold, i.e.

y ò

;

:y; y ò W

1;



p

(

; R m ) ; y #Ö y :

X a ( y; ›x y )



" a( y ; ›x y ) : ›x (y" y ) 





%  (y; ›x y) " ( y ; ›x y ) - (y" y ) dx % X b(y) " b( y ) - (y" y ) dS ¡ 0; 









(1.66)

then the problem (1.61) possesses just one weak solution.

Sket h of the proof. We will use Proposition 1.23. The monotone mapping

W 1 p ( ; Rm ) Ù W 1 p ( ; Rm ) ;

A( y; z ) ;

y # X a(y; ›x z) : ›x y dx

(1.67a)

with







while the lower-order ompa t part

A 2 ( y ) ;

114

:

is now determined by the inte-

*

gral identity

A1

A1 (y) # A(y; y)

;

A2 is given by

y # X (y; ›x y) - y dx % X b(y) - y dS : 







(1.67b)

When weakening (1.65a) to stri t monotoni ity, the proof is more involved, f. e.g. [495℄ or [685,

Lemma 2.32℄.

Ë

48

1 Ba kground Generalities

y; -) learly follows from (1.65a). The growth onditions A and A2 . Moreover, due to the ompa t embed1 p p "  ( ) and hen e also of L p "  ( ) ’ W 1 p ( ) , we obtain the ding W ( ) ’ L

ompa tness of A 2 . Thus, using still the oer ivity (1.65b) of A 1 % A 2 , the existen e of

The uniform monotoni ity of A(

(1.63) imply the required ontinuity of *

;

(

*



)

;

*

solution is due to Proposition 1.23(i). When (1.66) holds, we onsider two weak solutions, subtra t the respe tive integral identities (1.62), and test it by the dieren e of these solution. This yields uniqueness. Cf. also Proposition 1.23(ii). The oer ivity (1.65b) an be ensured be various ways. One parti ular ase is

; ¡ 0; d £ 0; d ò L ( ) : a(x ; r;  ) :  % (x ; r;  )- r £ p % d rq " b(x ; r)r £ b rq " b (x) ;b £ 0; b ò L ( ) :

d (x);

1

(1.68a)

1

(1.68b)

q £ 1 and d £ 0, b £ 0, min(d ; b ) ¡ 0. Then, £ p q q  ›x y L p ;Rm,n % d y L q ;Rm % b y L q ;Rm " d L1 " b L1 . In parti ular, if d # 0, the oer ivity of the mapping A # A1 %A2 follows by Poin aré's inequality.

with some (

)

(

)

(

)

(

)

(

)

Remark 1.42 (Navier-Stokes system). One an apply Proposition 1.23 on a subspa e of W p ( ; Rm ). One example is for m # n and the subspa e

a Sobolev spa e

1;

Wdiv 0 ( ; Rn ) :# y ò W 1 p ( ; Rn ); div y # 0 on ; y - n # 0 1;

p

;

;

:

on

(1.69)

p # 2, a prominent example is the Navier-Stokes system for a velo ity y and a s alar variable (pressure) p des ribing a steady ow of an in ompressible vis ous (so- alled For

Newtonian) uid:

%(y - ›x )y " 
y % ›x p # f

and

(›x y)n
t % byt # g

and

div y # 0 yn # 0

on

;

(1.70a)

on

;

(1.70b)

 ¡ 0 is the vis osity oe ient, % £ 0 mass density, and b # b(x) ¡ 0 is the sliding resistan e of the boundary wall, y # y n % y t is the de omposition of the velo ity on the boundary to the normal part y n :# ( y - n ) n and the tangential part y , and t analogously for the tra tion ve tor  (›x y ) n . We use the so- alled Navier boundary on-

where

dition, in luding nonpenetrability of the boundary wall. Let us note that the so- alled

%(y - ›x )y # %div(y‹ y) " %(div y)y # %div(y‹ y) ts with the assumpn ¢ 3.116 In the weak formulation, the pressure disappears be ause, by Green's formula, P ›x p - y d x # P p ( y - n ) d S " P p div y d x # 0 for div y # 0 and



y - n # 0. The resulted weak formulation then onsists in the integral identity

onve tive term115

tion (1.64) if



: y ò Wdiv ( ; Rn ) : 1;2



115 for

;0

X  ›x y : ›x y





% %(y-›x )y- y dx % X byt - y t dS # X f - y dx : 







More spe i ally, this onve tive term written omponentwise means [(

i

# 1; :::; n.

ÜÙ y‹y is ompa t from W ( ; Rn ) Ù L provided n ¢ 3 and 0    ¢ 2 " 4. Thus we pose a ( r; s ) # s " r ‹r .

116



Note that the mapping

y

1;2

*

*

(2

y - ›x )y℄i

")/2 ( ;

(1.71)

# nk# y k xk y i

Rn

1

)

’ L ( ; Rn ) 2

1.4 Some dierential and integral equations

The oer ivity of the underlying operator on

X %( y



- ›x )

Ë 49

1 2 n Wdiv ( ; R ) is due to the al ulus 0 ;

;

y - y dx # X %(y‹ y ) : ›x y dx 



# X %(y - y )y - n dS " X %(y‹y) : ›x y % (y - y )div y dx 







# "X %(y‹ y) : ›x y dx # "X %( y - ›x )y - y dx 

by Green's formula and by

(1.72)







div y # 0 in and by y - n # 0 on

, assuming

% ¡ 0 onstant.

Thus, all the integrals in (1.72) equal in fa t 0. For the mentioned oer ivity, (1.72) is

y # y. Let us note that the pointwise oer ivity (1.68a) does not hold, however. The uniqueness holds only for su iently small for e f and large vis osity oe ient , i.e. for small-turbulen e ows with with small (so- alled) Reynolds' numbers. More 1 2 n spe i ally, it is natural to equipped W div 0 ( ; R ) with the norm used for



;

;

y



and then, assuming

1 %

y

2

dx % X byt  dS•

1/2

2

;

(1.73)

n ¢ 3, for two weak solutions y1 and y2 , we an estimate

# X (y

2

12 ;b

 ; b :# ”X  ›x y 



x )y1

1 -›

# X (y

" (y

x ) y 2 - y 12

2 -›

x )y2

12 -›

# X (y

y12 % (y2 -›x )y12 - y12 dx

-

x )y2

12 -›

-

y12 dx ¢ ›x y2 L2

¢ "  b ›x y 2 ;

dx

;

2

;Rn,n ) y 12 L 4 ( ;Rn ) 2

(

L 2 ( ;Rn,n ) y 12  ; b 2

;

(1.74)

y12 :# y1 "y2 and  b ¡ 0 is the onstant from the Poin aré-type inequality  b ( y L 4 ;Rn % y L 4 ;Rn ) ¢ y  b . Therefore y12 # 0 prot vided ›x y 2 L 2 ;Rn,n   2  b /%. Testing (1.70a) and using (1.72), we obtain  ›x y 2L2 ;Rn,n ¢ y 2 b # f - y L1 % g- yt L1 ¢ f L4 3 ;Rn y L4 ;Rn % g L 4 3 ;Rn y L 4 ;Rn ¢ max( f L4 3 ;Rn ; g L4 3 ;Rn ) y  b /  b , from whi h we t obtain  y  b ¢ max( f L 4 3 ;Rn ; g L 4 3 ;Rn )/  b . Hen e, we obtain the unique-

where

;

;

;

(

(

(

/

(

)

;

(

(

/

/

(

;

)

;

)

;

ness if

(

;

)

)

;

)

)

(

/

)

/

)

(

(

/

Remark 1.43 (Regularity).

(

;

)

/

(

;

)

;

)

max  f L4 3 Rn ; g L4 3

(

/

)

Rn ) 

;

;

(

)

(

)

;

;

¢  b /% : 3

;

;

(1.75)

Sometimes, some additional qualitative information about

the solutions in addition to the basi quality

y ò W 1 p ( ; Rm ) is useful, in parti ular in ;

the ontext of optimal ontrol with state onstraints.

Ë

50

1.4.

1 Ba kground Generalities

Partial dierential equations of paraboli type

A further type of equations whi h we want to treat here as an example of an innitedimensional dynami al system is a system of Again we abbreviate

I :# [0; T℄

m

quasilinear paraboli equations.117

for a xed time horizon. More spe i ally, we will

onsider the Robin-type (also alled Newton-Fourier) initial-boundary-value problem for a system of

m su h equations: y " div a(y; ›x y) % (y ; ›x y) # 0 t n - a(y; ›x y) % b(y) # 0 y(0; -) # y0

on

I, ;

I, ; on : on

/ 7 7

(1.76)

? 7 7 G

The basi natural requirement we will assume through the following text is that

a : , (Rm ,Rm,n ) Ù Rm,n ; b : (I , ) , Rm Ù Rm ; and

: (I , ) , (Rm ,Rm,n ) Ù Rm

/ 7

are Carathéodory mappings : ? 7 G

(1.77)

For notational simpli ity, in (1.76) and in what follows, we did and will not write expli itly the dependen e on

x

t.

and

The further natural requirement is a ontrolled

growth, namely

; ò L p ( ); C ò R : 

; òL ; òL

#  (p ") (

p" ")

(



(



I , ); C ò R :



I , ); C ò R :

where118

p" :#

np%2p n

a(x ; r;  ) ¢

x % Crp

( )

b(t ; x ; r) ¢

(

t ; x ; r;  ) ¢

 (

p# :#

and

" /p "

t ; x) % Cr (

t ; x) % Cr

np%2p"2 n

% Cp" ;

p # "1" 

1

;

p " "1" 

provided

(1.78a)

and

(1.78b)

% C p¡

p/( p " "  )



2 n %2 : n %2

;

(1.78 )

(1.79)

 £ 0, the orresponding R , " Rm,n ) Ù L p (I , ; Rm,n ), p N : L (I , ; Rm ) , L p (I , ; Rm,n ) Ù

In parti ular, the growth onditions (1.78) ensures that, for Nemytski mappings work as

Na :

" L p (I , ;

m)

Nb : L (I , ; Rm ) Ù L (I , ; Rm ), and " L p (I , ; Rm ). Moreover, for  ¡ 0, we an rely p#

p# 



L p (I , ;



on respe tive ompa t embeddings

needed for existen e of weak solutions. Multiplying our equation by a test fun tion

y , applying the Green formula in spa e 

together with using the Robin-type boundary onditions, and making also by-part integration in time with using the initial ondition, we ome to the notion of the weak so-

lution in the spirit of Remark 1.39: a fun tion

117

y ò L p (I; W 1 p ( ; Rm ))  L ;



(

I; L2 ( ; Rm ))

For more details about su h equations we refer, e.g., to Gajewski et al. [342℄ or Lions [495℄ or also

[685℄.

118

L p (I; W 1;p ( ))  : L p (I; W 1;p ( )) 

The exponents in (1.79) are hosen in order to have the ontinuous embedding

L (I; L ( )) L (I; L2 ( )) 

2

" L p (I

’ , ) and the ontinuous tra e operator u ÜÙ uI, # Ù L p (I , ), f. [685, Se t. 8.6℄. The ondition p ¡ (2n%2)/(n%2) is needed only for

optimizing the exponent

p# and an be avoided when (1.78b) would be strengthened.

Ë 51

1.4 Some dierential and integral equations

will be alled the weak solution to (1.76) if the following integral identity is fullled

T X ”X a ( y; ›x y ) : ›x y 0



% (y; ›x y) - y " y 

y dx % X b(y) - y dS• dt t 



%X y(T) - y (T) dx # X y 



for any

y ò W1

;



(



0

y (0) dx :

-

(1.80)



I , ; Rm ). Let us note that, supposing the growth onditions (1.78),

all the integrals in (1.80) are nite and the denition has a good sele tivity in the sense that, if the solution and the data are smooth enough, one an re over all three equation in (1.76) when, after making the by-part integration on

y

hoosing suitable test fun tions

I

and Green's formula on

,

0; T) , and then

rst with ompa t support on ( 

more general to re over the boundary and the initial ondition.119

Proposition 1.44 (Paraboli equations: existen e and uniqueness). y ò L ( ; Rm ), and semi- oer ive

(1.78)(1.79) be satised,

X a ( y; ›x y ) : ›x y

Let

(1.65a)

and

2

0

% (t ; y; ›x y) - y dx % X b(t ; y) - y dS £  ›x y pLp

(

;Rm,n )

" ( t ) „ 1 % y L 2 Rm 2

;

(

)

…

(1.81)

ò L (I). Then the initial-boundary-value problem (1.76) possesses just one p p ( ; R m ))  L ( I ; L ( ; R m )) in the sense of (1.80) whi h adweak solution y ò L ( I ; W p (I; W p ( ; Rm ) ) % W m ditionally belongs also to W ( I ; L ( ; R )). If also a weak1

for some

1;

2



1;



1;

1;1

*

2

ened global monotoni ity

; ò L (I) : 1

t ò I :y; y ò W 1 p ( ; Rm ) : ;

a.a.

X  a ( y; ›x y )



" a( y ; ›x y ) : ›x (y" y ) %  (t ; y; ›x y) " (t ; y ; ›x y ) - (y" y ) dx 











% X b(t ; y) " b(t ; y ) - (y" y ) dS £ " (t) y" y L2 Rm ; 2







;

(

(1.82)

)

holds, then this solution is unique.

V # # (P ›x - p dx) p , H # L ( ; Rm ), and p " "  ( I , ; R m ) , L p # "  ( I , ; R m ) with the duality the interpolation spa e Lp VH p # L ¢ 0, whi h proves (1.105b). Putting (1.105b) into (1.107), one an write the *

resulted inequality just in the form (1.105a).

 is nite-dimensional but int(D) # , we an work, instead of , with the linear hull of D , whi h is a losed subspa e of  . Then D has nonempty interior with In ase

respe t to the relativized topology.

Convention 1.55. spa e

134

In fa t, the mappings  and R need not be dened on the whole Z but only on the onvex subset K . Then the meaning of the dierential ›R(z) ò

This type of optimality onditions was rst invented by Fritz John (19101994) in [408℄. The asser-

K has nonempty inte innite-dimensional is admitted provided ertain additional assumptions on D and › R are imposed. For ertain spe ial data  , R , and D , an innitedimensional  is also admitted in Ioe and Tikhomirov [399, Se t. 1.1.4℄. tion presented here is basi ally due to Casas [180, Thm. 5.2℄. For the ase that

rior, see also Zeidler [812, Se t. 48.3℄ where also

135

This proof is essentially due to Casas [180℄.

1.5 Basi s from optimization theory

Ë 63

L(Z ; ) of R at a point z ò K is that [›R(z)℄( z "z) # lim"ÿ0 (R(z % "( z "z)) " R(z))/" z ò K only (and not for z ò Z as usual). The modi ation for  is straightforward. 

for







This may ause the dierentials to be determined uniquely only up to a losed linear subspa e provided

K is at. From the proof of Proposition 1.54, one an also see that R(z) : Z Ù  su es to be dire tionally weakly ontinuous (i.e.

the linear operator ›

weakly ontinuous when restri ted on the segments), whi h is alled hemi ontinuity. Similarly for ›

(z) : Z Ù R. We will o

asionally use this onvention in what follows.

The rst multiplier

0

*

in the F. John onditions an sometimes degenerate to zero

 falls

ompletely out.136 Therefore, the so- alled normal ase  0 ¡ 0 (or equivalently  0 # 1) and then su h ondition be ome not mu h sele tive be ause the ost fun tion *

*

is of parti ular interests:

Proposition 1.56 (Karush-Kuhn-Tu ker onditions).137 Let K be onvex,  and R be Gâteaux dierentiable, int( D ) #Ö , and z ò Argmin(P ). Let further one of the following onstraint quali ation hold:

:  £ 0 ;  #Ö 0 ; z ò K : *

*

[› R ( z )℄

*

*



 ; z "z   0 *

(1.108)



R is D- onvex on K and ; z ò K : R( z )   0 :

or



(1.109)



 0 # 1.

Then (1.105) holds with

*

The ondition (1.109) is usually veriable quite simply, being alled the Slater

onstraint quali ation [728℄, while (1.108) is appli able to non onvex onstraint mappings

R, being alled the Mangasarian-Fromowitz onstraint quali ation138 [515℄. 0 # 0 for a moment, (1.105 ) yields  #Ö 0 £ 0. As  £ 0 but  #Ö 0, we simultaneously have # 0, f. (1.105b). 

before, and where the equality is due to the orthogonality < Thus again we obtained
  , a ontradi tion with (1.124). Here we used the fa t that, with  0 #  ( z ) "  ( z ), it holds149 Then
¢ 0. If <  ;  > # 0, then <  ; N > would be a neighbourhood of 0, a ontradi tion.

149

Indeed,

*

*

150

*

*

*

*

*

The proof of (1.126) is analogous as those of (1.125).

*

Ë

72

Thus

1 Ba kground Generalities

S1  S2 # . Then we get (1.127a)(1.127 ) by the Eidelheit theorem as in the proof

of Proposition 1.54. The point (ii) follows as in Proposition 1.56 be ause the ontradi tion step uses

0 # 0 whi h then eliminates the ve tor-valued  from the onsiderations. *

An e ient straightforward approa h to multi riteria optimization is a so- alled

s alarization, i.e. to onsider suitable s alar-valued riteria instead of the original ve tor-valued one. In general, for a fun tional

F : 0 Ù

R, we an onsider the

s alar-valued problem Minimize subje t to

FŽ (z) for z ò Z ; R(z) ¢ 0 ; z ò K

§

(1.129)

Inspired by the proof of Proposition 1.65, a worthy hoi e is a linear fun tional

F # *0

0 £ 0, 0 #Ö 0. Then any solution to (1.129) is D0 -Slater optimal for (P ). If (Dad (P )) is onvex in 0 , then this linear s alarization overs even all D0 -Slater op-

with

*

*

*

timal solutions for (P ).151 Obviously, if

 is D0 - onvex, R is D- onvex, and K is onvex, then also (Dad (P )) (Dad (P )),

is onvex and the linear s alarization is truly e ient. For a non onvex

F is parti ularly worth onsidering. For a nite number of s alar-valued n n % n fun tionals, i.e. if  # (  i ) i #1 ,  0 :# R , and D 0 :# (R ) , assuming  i ¡ 0, it is more a nonlinear

e ient to take

F (0i )ni#1  # max (*0i 0i ) i #1 ; : : : ; n

with

0i £ 0 *

and

n * H 0i i #1

# 1:

(1.130)

D0 -Slater optimal solution z, there is a suitable n-tuple (0i )ni#1 for whi h z minimizes F Ž , f. [614, Se t.2.1℄. *

Then, for any

1.5.d

Non- ooperative game theory

In this se tion we mention briey some basi on epts and results from the theory of

non- ooperative two-person games.152 Having onned ourselves to two players (distinguished by the indi es 1 and 2),

Z l and two onvex sets K l ’ Z l , as well as two ost fun tions  l : Z 1 , Z 2 Ù R, l # 1 ; 2. The rst player uses the ontrol we will now onsider two lo ally onvex spa es

D0 -Slater optimal z, put S # {(z) " ( z ); z ò Dad (P )}. Assuming S  int(D0 ) #Ö z being D0 -Slater optimal. Hen e, by the Eidelheit theorem, S * * * and int( D 0 ) an be separated by a linear fun tional, say  , i.e. <  ; int( D 0 )> ¡ 0 and <  ;  ( u ) " 0 0 0 * (Dad (P ))> ¢ 0. The former inequality gives 0 £ 0 while the latter one just says that z minimizes *0 Ž  over Dad (P ).

151

To show it, for any





, we would get a ontradi tion with

*

152

More about this topi an be found in the monographs by Aubin [35℄, Aubin and Ekeland [36℄,

Balakrishnan [49℄, or Zeidler [811, 812℄.

1.5 Basi s from optimization theory

Ë 73

z1 ò K1 with the aim to minimize the ost fun tion 1 , while the se ond player drives z2 ò K2 to minimize 2 . In ontext of game theory, the ontrols are also addressed as strategies.

1 # 2 , Z # Z1 , Z2 , K # K1 , K2 , and  # 1 in the problem (P ) there. Hen e the a tual game begins if 1 #Ö 2 , i.e. if there Let us realize that if both players have identi al goals, whi h means

then we get basi ally the situation from Se t. 1.3 if put

is (to more or less extent) a oni t of goals. As one an anti ipate, game situations are also mathemati ally mu h more ompli ated than mere minimization problems, whi h orrespond to their ability to ree t in a more proper way the reality of live whi h is so dramati just due to frequently o

urring oni ting situations. For entirely non- ooperative behaviour of two players, a suitable on ept of so-

K1 ; K2 ; 1 ; 2 ) is the Nash equilibrium: z ; z2 ) ò K1 , K2 is alled a Nash equilibrium of the game (K1 ; K2 ; 1 ; 2 ) if

lution to the game des ribed by the data ( ( 1

1 (z1 ; z2 ) # min 1 ( z 1 ; z2 ) z 1 ò K 1



2 (z1 ; z2 ) # min 2 (z1 ; z 2 ) :

and

z2 ò K 2



(1.131)

Let us denote the set of all Nash equilibria by

Nash ( ;  ) :# 1

K1 ,K2

2

† (z1 ; z2 ) ò K1

, K ; (1:131) is satised ‡ :

(1.132)

2

Sometimes, Nash equilibria are also alled non- ooperative equilibria, having the obvious meaning that ea h player follows only his or her individual prot and expe ts the same behaviour of the opponent. The existen e of the Nash equilibria often fails unless quite strong data quali ations are imposed; a tually it is not mu h surprising sin e everybody knows well from own every-day experien e that, willing to be in an equilibrium state, one should better avoid oni ting purely non- ooperative situations. The following existen e theorem is, in fa t, equivalent with Brouwer's xed-point Theorem 1.19 and is thus highly non onstru tive.153

Theorem 1.67 (Nash equilibria).154 Let the following assumptions be satised: are separately ontinuous ;

1 and 2 1 % 2

is jointly ontinuous on

(1.133a)

K1 , K2 ;

(1.133b)

:z ò K ; z ò K :  (-; z ) and  (z ; -) are onvex ; ; ¡ 0 ;K onvex ompa t :z ò K :  #Ö LevK1  (-; z ) ’ K 1

1

2

2

1

2

1;

153

2

2

(1.133 )

1

2

;

1

2

1;

;

(1.133d)

Inspe ting the proof of the Nash Theorem 1.67, we found even a series of non- onstru tive argu-

ments: a ontradi tion argument, a sele tion of nite overing relying on ompa tness, and the mentioned Brouwer xed-point theorem.

154

John Nash, a 1994 Nobel prize winner for e onomy, formulated this theorem for a spe ial ase

where the set of admissible strategies

K1 and K2 are mixed strategies for a nite game, see [567℄.

74

Ë

1 Ba kground Generalities

; ¡ 0 ;K Then

2;

onvex ompa t

:z ò K : 1

1



#Ö LevK2  (z ; -) ’ K 2

;

1

2;

:

(1.133e)

NashK1 ,K2 ( ;  ) #Ö . 1

Proof. 155 If

K1

or

K2

2

K1 ,K2 and 1 ; 2 ) # NashK1 ,K2 (1 ; 2 ) thanks to the uniform- oer ivity

is not ompa t, we an nd the Nash equilibrium on

realize that Nash K 1 , K 2 (

assumptions (1.133d,e). Let us abbreviate

u :# (u1 ; u2 ) ò K1 ,K2 and similarly z :# (z1 ; z2 ), and dene (z ; u) :# 1 (z1 ; u2 ) % 2 (u1 ; z2 ) :

We will show that minimum at

(1.134)

u $ (u1 ; u2 ) is a Nash equilibrium if and only if (-; u) attains its

u, i.e.

:z ò K ,K : 1

(u ; u) ¢ (z ; u) :

2

(1.135)

1 (u1 ; u2 ) ¢ 1 (z1 ; u2 ) and 2 (u1 ; u2 ) ¢ 2 (u1 ; z2 ), f. (1.131). Conversely, if (1.135) holds, then for z1 :# u1 one gets

For the only if  part, it su es to sum

1 (u1 ; u2 ) % 2 (u1 ; u2 ) # (u1 ; u2 ; u1 ; u2 )

¢ (u ; z ; u ; u ) #  (u ; u ) %  (u ; z 1

2

1

2

1

1

2

2 (u1 ; u2 ) ¢ 2 (u1 ; z2 ). Similarly, by putting z2 :# u2 1 (z1 ; u2 ).

so that

2

one gets

1

2)

1 (u1 ; u2 ) ¢

Suppose that there is no Nash equilibrium, whi h by (1.135) would mean that

:u ò K ,K ;z ò K ,K : 1

2

1

2

(u ; u) ¡ (z ; u) :

(1.136)

G z :# {u ò K1 ,K2 : (u ; u) ¡ (z ; u)}. By (1.133a-b), all G z are open. Then G z }zòK1 ,K2 forms an open overing of K1 ,K2 . By ompa tness of K1 ,K2 there is a nite sub overing, i.e. there is {z i }i#1 n ’ K1 ,K2 with some n ò N n su h that U i #1 G z i # K 1 , K 2 . This means pre isely

Denote

(1.136) just says that {

;:::;

:u ò K ,K ;j : (u ; u) ¡ (z j ; u) : 1

f i (u) :# max( (u ; u)" (z i ; u) ; 0). By (1.133a-b), ea h f i f i £ 0 and, by (1.137), i f i ¡ 0. Furthermore, put

Put

'(u) :#

155

For general

(1.137)

2

n i H  i (u)z i #1

and

 i (u) :#

is ontinuous. Moreover

f i (u) : n  j #1 f j ( u )

(1.138)

K1 and K2 onvex ompa t, this theorem has been proved by Nikaid and Isoda [577℄ n-player generalization). It was further

by using the Brouwer xed-point Theorem 1.19 (even for an

shown by Kindler [427℄ that, onversely, Brouwer's theorem follows from the Nikaid-Isoda theorem.

1.5 Basi s from optimization theory

As

'(K1 ,K2 ) ’ o({z i }i#1

;:::;

n)

Ë 75

#: S, we have in parti ular '(S) ’ S. As S is a ompa t

onvex nite-dimensional subset, by Brower's xed-point Theorem 1.19, there is some

u ò S su h that '(u) # u. Yet, by (1.137), (u ; u) ¡ (z j ; u) for a suitable j. By (1.133 ), (-; u) is onvex. Thus be ause

due to

'(u) # u

by on-

vexity n n # ('(u); u) # Œ H  i (u)z i ; u ¢ H  i (u) (z i ; u) i #1 i #1 #  j (u)  (z j ; u) % H   i (u) (z i ; u)    (u ; u) i #Ö j # 0 if ¢ (u ; u) if

(u ; u)

(1.138)

f i (u) # 0

be ause

 j (u) ¡ 0

f i (u) ¡ 0

   j (u) (u ; u) % H  i (u) (u ; u) # (u ; u) ; i#Ö j

whi h gives a ontradi tion.

Remark 1.68 (A spe ial ase:  2

1

and

2 ontinuous).

Let us still note that, if

1

and

themselves are jointly ontinuous, then the Nash theorem is an immediate onse-

quen e of Kakutani's xed-point Theorem 1.21 applied to the upper semi ontinuous

onvex-valued mapping

u#”

K1 ,K2 ± K1 ,K2 dened by

u1 Argmin1 (-; u2 ) • ÜÙ ” • # Argmin (- ; u ) ; u2 Argmin2 (u1 ; -)

the upper semi ontinuity follows essentially by Proposition 1.49. Cf. also Aubin and Ekeland [36, Se t. 6.3, Thm. 13℄ or [35, Thm. 12.2℄. In pra ti al omputer implementation, one is mostly for ed to approximate the

K2 by some (usually nite-dimensional) sets K1d d d d and K 2 as well as the ost fun tions  1 and  2 by some  1 and  2 with d ¡ 0 being

set of admissible strategies

K1

and

an abstra t dis retisation parameter. Then immediately one asks whether the approximate problems onverge somehow to the original problem:

Proposition 1.69 (Convergen e of approximate games). (

K1d ; K2d ; 1d ; 2d )

satisfy (1.133a- ) with

Let

K1d , K2d , 1d , 2d

(1.133)

be

satised,

K1 , K2 , 1 , 2 ,

in pla e of

respe tively, and let the following assumptions be satised:

:d £ d ¡ 0 : K d ’ K d ’ K ; K d onvex losed, lZ1 



1

1

1

1

:d £ d ¡ 0 : K d ’ K d ’ K ; K d onvex losed, lZ2 



2

:z ò K ; z ò K : 1

1

2

2

2

2

2

; ò R

d

%

(1.139a)

℄

K2d # K2 ;

d ¡0

(1.139b)

C

1

C

1d (z1 ; -) Ù 1 (z1 ; -) & 2d (-; z2 ) Ù 2 (-; z2 );

(1.139 )

C

1 % 2 Ù 1 % 2 ; d

#K ;

d ℄ K1 d ¡0

;K

1;

onvex ompa t

(1.139d)

:d ò R :z ò K : LevK1d inf 1d K1d z2 %  d (-; z ) ’ K % ;

d

2

(

2

;

)

1

2

1;

;

(1.139e)

76

Ë

1 Ba kground Generalities

; ò R% ;K

2;

:d ò R% :z ò K d : LevK2d inf 2d z1 K2d %  d (z ; -) ’ K

onvex ompa t

1

;

Then, for all

1

(

;

1

2

)

2;

:

(1.139f)

d ¡ 0, NashK d ,K d (1d ; 2d ) #Ö  and 1

2

Limsup Nash ( d ;  d ) ’ Nash ( ;  ) : 1

K 1d , K 2d

d Ù0

2

1

K1 ,K2

(1.140)

2

Proof. First, let us note that, by Theorem 1.67, the approximate problems always ad-

K1d  K1 and K2d  K2 are onvex ompa t. Hav-

mit Nash equilibria; note that both

;

;

z1d ; z2d ) ò NashK1d ,K2d (1d ; 2d ), by the uniform oer ivity of approximate problems (1.139e) and (1.139f) one an lo alize all onsiderations on a ompa t set K 1 , K 2 and suppose that (possibly after taking a ner net) there is ( z 1 ; z 2 ) ò K 1 , K 2 su h that z1d Ù z1 and z2d Ù z2 for d Ù 0. Our aim is to show that (z1 ; z2 ) òNash K1 ,K2 (1 ; 2 ). d d d d d d d d d d We know that  1 ( z 1 ; z 2 ) ¢  1 ( z 1 ; z 2 ) and  2 ( z 1 ; z 2 ) ¢  2 ( z 1 ; z 2 ) for any d d ( z 1 ; z 2 ) ò K , K . In parti ular, 1 2 ing (

;





;





:( z ; z ) ò K d , K d :  d (z d ; z d ) %  d (z d ; z d ) ¢  d ( z ; z d ) %  d (z d ; z ) : 1

2





1

2

1

1

2

2

1

2

1

1

2



2

(1.141)

2

1



limdÙ  d ( z ; z d ) #  ( z ; z ) and limdÙ  d (z d ; z ) #  (z ; z ). Mored d d d d d over, by (1.139d) also lim d Ù  ( z ; z ) %  ( z ; z ) #  ( z ; z ) %  ( z ; z ). This By (1.139 ),

0

1

1

1

2



0

1

1

2



1

0

2

2

1

2

2

1

1

2

2



1

2

1

2

1

2 

2

allows us to pass to the limit in (1.141), whi h gives

:( z ; z ) ò K d , K d :  (z ; z ) %  (z ; z ) ¢  ( z ; z ) %  (z ; z ) : 1

2





1

1

2

1

2

2

1

2

1

1

2



2

1

2 

(1.142)

Eventually, by (1.133a) with (1.139a) and (1.139b) one an see that (1.142) holds even for any (

z 1 ; z 2 ) ò K1 ,K2 . In parti ular, taking z 1 :# z1 shows that 2 (z1 ; z2 ) ¢ 2 (z1 ; z 2 ) z 2 ò K2 . Analogously, z 2 :# z2 shows that z1 minimizes 1 (-; z2 ) over K1 . 



for any









In view of the (quite restri tive) onditions (1.133b) and (1.139d), it is worth

1 % 2 is onstant  without any loss of 1 % 2 # 0. This means that the players have entirely

onsidering a spe ial lass of games where generality, we an suppose

antagonisti goals in the sense that the prot of one player is just the loss of the other one and vi e versa. In su h situation we speak about a zero-sum game. Putting

 :# 1 # "2 , from (1.131)

one an easily see that the point (

z1 ; z2 ) ò K1 ,K2

is a

Nash equilibrium if and only if

min ( z ; z

z 1 ò K 1

1 

2)

# (z ; z 1

Su h point is also alled a saddle point of

2)

# max (z ; z ) : 1

z2 ò K 2

, and 

2 

(1.143)

is addressed as a payo. Let us

denote the set of all saddle points by

Saddle  :# Nash (; ") # K1 ,K2

K1 ,K2

†( z 1 ; z 2 ) ò K 1 , K 2 ;

1:143) holds‡ :

(

(1.144)

Ë 77

1.5 Basi s from optimization theory

The fa t that (

z1 ; z2 ) ò K1 ,K2 is a saddle point of  is equivalent156 to the fa t that

inf sup ( z ; z 1

z 1 ò K 1 z 2 ò K 2 and



# sup inf ( z ; z

2)

1

z2 ò K 2 z1 ò K 1





(1.145)

2) 

z1 ò K1 and z2 ò K2 are so- alled onservative strategies in the sense that

sup (z ; z ) # inf sup ( z ; z 1

z 2 ò K 2

2

1

z 1 ò K 1 z 2 ò K 2





2) 

inf ( z ; z ) # sup inf ( z ; z ) :

and

1

z 1 ò K 1

2



1

z 2 ò K 2 z1 ò K 1

2





The meaning of a onservative strategy is that a player tries to rea h the highest own prot on the assumption that the only goal of the opponent is to make him or her as highest harm as possible.157 As a plain onsequen e of Theorem 1.67, we an laim that provided



is separately ontinuous and

(-; z2 )

 has a saddle point

is onvex and inf- ompa t while

(z1 ; -) is on ave and sup- ompa t and uniformly oer ive, and K1

and

K2

are on-

vex. Nevertheless, spe ial hara ter of the zero-sum problem makes possible to modify the oer ivity assumptions:

Theorem 1.70 (Saddle point  von Neumann [781℄, generalized).158 Let 

is separately ontinuous ;

(1.146a)

K1 and K2 are onvex;

:z ò K ; z ò K : ;z ò K : ò R : ;z ò K : ò R : Then

1

1

2

2

1

1

2

(1.146b)

(-; z2 ) is onvex; (z1 ; -) is on ave,

2

(1.146 )

LevK1 (-; z ) is ompa t; LevK2 ("(z ; -)) is ompa t :

(1.146d)

2

;

(1.146e)

1

;

SaddleK1 ,K2  #Ö .

Sket h of the proof. We use Theorem 1.67 for

K in # {z i ò K i ;



z i  ¢ n}.

Thus, for a

n, a saddle point of  or, in other words, a Nash equilibrium (u1n ; u2n ) ò Nash K n , K n (  ; "  ) does exist. For z i # z i , we then have 1 2 su iently large

for

z2

"  

# z

for

2

z1

inf (-; z ) ¢ (u n ; z ) ¢ (u n ; u n ) ¢ (z ; u n ) ¢ sup (z ; -)   2

1

2

1

2

1

2

# z

1

1

% :

(1.147)

156

See, e.g., Aubin [35, Proposition 8.1℄ or Aubin and Ekeland [36, Se t. 6.2, Proposition 1℄.

157

If no onvex/ on ave stru ture of the game an be guaranteed (as typi al, e.g. in games with fully

nonlinear systems or pursuer/evader games), it is often a satisfa tory task to nd a onservative strategy of at least one of the players; f. Friedman [334℄, M Millan and Triggiani [525℄, Nikol'ski  [579℄, Warga [791, Chap. IX℄, et .

158

See [781℄ for a spe ial ase that the set of admissible strategies

a nite game, or Nikaid and Isoda [577℄ for

K1

and

K2

K1 and K2 are mixed strategies for

onvex and ompa t. The presented general

version is due to Aubin and Ekeland [36, Se t. 6.2, Thm. 8℄ where even a lower/upper semi ontinuous payo fun tion

 is admitted.

78

Ë

1 Ba kground Generalities

u1n }nòN and {u2n }nòN must be bounded, hen e they live in some ompa t set K 1 , K 2 for m large enough, and thus (up to possibly a subn n m m n n sequen es) ( u 1 ; u 2 ) Ù ( u 1 ; u 2 ) ò K 1 , K 2 and also  ( u 1 ; u 2 ) onverges to some limit, say L . Making a limit passage in (1.147) gives

This implies that the sequen es {

m

m

(u1 ; z2 ) ¢ lim inf (u1n ; z2 ) ¢ lim (u1n ; u2n ) # L ¢ lim sup (z1 ; u2n ) ¢ (z1 ; u2 ) : n Ù

n Ù

n Ù

(1.148) Putting

z1 # u1 and z2 # u2 , we get (u1 ; u2 ) # L. Then (1.148) yields (u1 ; u2 ) that a  on K1 ,K2 .

saddle point of

1 (-; z2 ) and 2 (z1 ; -) possess Gâteaux derivatives, denoted respe tively by and › z 2  2 , from (1.131) we an easily establish the rst-order ne essary onditions for the Nash equilibrium point ( z 1 ; z 2 ), namely If

›z

1 1

›z

1

1 (z1 ; z2 ) ò "N K1 (z1 )

and

›z2

2 (z1 ; z2 ) ò "N K2 (z2 ) :

Conversely, (1.133 ), (1.146b) and (1.149) imply (

(1.149)

z1 ; z2 ) ò NashK1 ,K2 (1 ; 2 );

f. Re-

mark 1.58. Let us now investigate a game-theoreti al problem involving a state equation like in Se tion 1.2d, i.e.

(P )

where

J1 (z1 ; z2 ; y) ; . Nash equilibrium  6 6 J2 (z1 ; z2 ; y) ; 6

(z1 ; z2 ; y) # 0 ; z1 ò K1 ; z2 ò K2 ;

> subje t to 6 6 6 F

J l : Z1 , Z2 , Y Ù

R, l # 1; 2, and  : Z , Z , Y Ù X with Y and X Bana h 1

2

spa es. Like in Se tion 1.1.2d, we suppose that the state equation always a unique solution

y # (z1 ; z2 )

(z1 ; z2 ; y) # 0 has

whi h denes the ontrol-to-state mapping

 : K1 ,K2 Ù Y . Then we dene naturally the set of equilibrium points of (P ) by

Nash(P ) :# Nash( ;  K1 ,K2

1

2)

for

 l (z1 ; z2 ) :# J l (z1 ; z2 ; (z1 ; z2 )); l # 1; 2:

Theorem 1.67 and Proposition 1.69 an be applied straightforwardly to (P ); note that

l  whi h is bi-ane159.

the assumption (1.133 ) about the onvex stru ture of the omposed ost fun tions basi ally for es us to onsider only the ontrol-to-state mapping

It is noteworthy to spe ify the optimality onditions (1.149) for this problem involving the state equation:

Proposition 1.71 (Optimality onditions for (P )). Let J l (z ; z ; -) : Y Ù R, l # 1; 2, and (z ; z ; -) : Y Ù X be Fré het dierentiable, J (-; z ; y) : Z Ù R, J (z ; -; y) : Z Ù 1

1

2

1

2

2

1

2

1

2

This means both  (- ; z 2 ) and  ( z 1 ; -) are ane. In fa t, the uniform onvexity of J 1 (- ; z 2 ; y ) and J2 (z1 ; -; y) may sometimes guarantee (1.133 ) even if (-; u2 ) and (u1 ; -) are slightly non-ane, f.

159

[627, 679℄.

Ë 79

1.5 Basi s from optimization theory

R, 

(- ; z 2 ; y ) : Z 1 Ù X , and  ( z 1 ; - ; y ) : Z 2 Ù X be Gâteaux equi-dierentiable around y ò Y , let the ontrol-to-state mapping  : K1 ,K2 Ù Y as well as all the mappings [› z 1 J 1 ( z 1 ; z 2 ; -)℄( z 1 ) : Y Ù R, [› z 2 J 2 ( z 1 ; z 2 ; -)℄( z 2 ) : Y Ù R, [› z 1  ( z 1 ; z 2 ; -)℄( z 1 ) : Y Ù X , and [›z2 (z1 ; z2 ; -)℄( z 2 ) : Y Ù X be ontinuous, let ›y (z1 ; z2 ; y) ò L(Y; X ) 







have a bounded inverse, and (1.146b) be valid. Then: (i)

If ( z 1 ;

z2 ) òNash(P ) and y # (z1 ; z2 ), then [› z l

for

1 ; 2 ò X *

*

(z1 ; z2 ; y)℄  l " ›z l J l (z1 ; z2 ; y) ò N K l (z l ) ;

*

*

*

l # 1; 2;

(1.150)

satisfying the adjoint equation [› y

(z1 ; z2 ; y)℄  l # *

*

›y

J l (z1 ; z2 ; y); l # 1; 2:

(1.151)

if, for some ( z 1 ; z 2 ) ò K1 ,K2 , the omposed ost fun tions J1 (-; z2 ; (-; z2 )) : K1 Ù R and J2 (z1 ; -; (z1 ; -)) : K2 Ù R are onvex and (1.150) (1.151) hold for y #  ( z 1 ; z 2 ) and for  1 ;  2 ò X , then ( z 1 ; z 2 ) òNash(P ).

(ii) Conversely,

*

*

*

Sket h of the proof. Sket h of the proof. The point (i) is just (1.149) if one evaluates › z 1  1 ( z 1 ; ›z2

z2 ) and 2 (z1 ; z2 ) by means of Lemma 1.59. The su ien y (i.e. the point (ii)) then follows

by the onvex stru ture of the parti ular minimization problems; f. Remark 1.58. For a spe ial ase

J # J1 # "J2 , (P ) be omes the zero-sum game problem involving

a state equation:

J(z1 ; z2 ; y) ;

Minimax . 6 6

(P )

(z1 ; z2 ; y) # 0 ; z1 ò K1 ; z2 ò K2 ;

subje t to > 6 6 F

0

and it is natural to dene the set of saddle points of (P0 ) by

Saddle(P ) :# Saddle  0

for

K1 ,K2

(z1 ; z2 ) :# J(z1 ; z2 ; (z1 ; z2 )) :

Corollary 1.72 (Optimality onditions for (P )). Let J(z ; z ; -) : Y Ù R be Fré het difz ; y) : Z Ù R and J(z ; -; y) : Z Ù R be Gâteaux equi-dierentiable around y ò Y , let the mappings [› z 1 J ( z ; z ; -)℄( z ) : Y Ù R and [› z 2 J ( z ; z ; -)℄( z ) : Y Ù R be ontinuous, let (1.146b) be valid, and let  and  be as in Proposition 1.71.

ferentiable, J (- ;

1

0

2

1

1

2

2

1

2

1

1



2

2 

Then: (i)

If ( z 1 ;

z2 ) òSaddle(P0 ) and y # (z1 ; z2 ), then J z ; z2 ; y) " [›z1 (z1 ; z2 ; y)℄  ò "N K1 (z1 ) ;

(1.152a)

J z1 ; z2 ; y) " [›z2 (z1 ; z2 ; y)℄  ò N K2 (z2 ) ;

(1.152b)

*

›z1 ( 1 ›z

with

 òX *

*

*

2 (

*

*

satisfying the adjoint equation [› y

(z1 ; z2 ; y)℄  # *

*

J z ; z2 ; y) :

›y ( 1

(1.152 )

80

Ë

1 Ba kground Generalities

z1 ; z2 ) ò K1 ,K2 , J(-; z2 ; (-; z2 )) : K1 Ù R is onvex, J(z1 ; -; (z1 ; -)) : K2 Ù R is on ave, and (1.152) hold for y # (z1 ; z2 ) and for  ò X , then (z1 ; z2 ) òSaddle(P0 ).

(ii) Conversely, if, for some (

*

*

2 Theory of Convex Compa ti ations Ar himedes denes a onvex ar ... When in the seventeenth

entury

Ar himedes'

methods

were

taken up again, onvexity ... played still a role, for instan e in the work of Fermat.

Moritz Werner Fen hel ...though

onvex

sets

belong

to

(19051988)

geometry,

they

be ome one of the basi tools of the analyst...

Gustave Choquet Aleksandrov

began

to

onstru t

(19152006)

the

theory

of

ompa t spa es... This on ept ... still today is used

onstantly in various elds of mathemati s.

Evgeniy Frolovi h Mis henko

(19222010)

In various relaxation s hemes the ommon feature is the onvexity of the used

ompa t envelopes of the original spa es. Thus, to give an abstra t and unied viewpoint to parti ular on rete ases, it is worth developing a general theory of what we will all  onvex ompa ti ations. This is, as it sounds, ompa ti ations whi h are simultaneously onvex subsets of some lo ally onvex spa es. The onvexity is

ertainly a onsiderable restri tion and it should be emphasized that not every topologi al spa e admits nontrivial onvex ompa ti ations but, on the other hand, there are topologi al spa es whi h admits a lot of them. It is then ertainly useful to introdu e a natural ordering of onvex ompa ti ations of a given spa e. This will be done in Se tion 2.1. Furthermore, we will nd useful to have a ertain unied (we will say  anoni al) form of an arbitrary onvex ompa ti ation. Imitating the lassi al onstru tion based on the multipli ative means on some ring of ontinuous bounded fun tions, in Se tion 2.2 we will onstru t our onvex ompa ti ations by using the means ( f. Se tion 1.5) on a suitable (we will say  onvexifying) linear subspa e of the spa e of ontinuous bounded fun tions on a topologi al spa e to be ompa tied. An important result is that there is a one-to-one order-preserving orresponden e between all losed onvexifying subspa es and all onvex ompa ti ations. In parti ular, it identies the topology of the uniform onvergen e as de isive for the reated onvex

ompa ti ation in the sense that, on one hand, making a losure of the onvexifying linear subspa e in this topology does not hange the orresponding onvex ompa ti ation but, on the other hand, any further enlargements reate onvex ompa ti ations whi h are a tually stri tly ner. In many of on rete applied problems the spa es to be ompa tied possess, beside a topologi al stru ture, also a bornology al stru ture. It enables us to speak about

https://doi.org/10.1515/9783110590852-002

Ë

82

2 Theory of Convex Compa ti ations

a oer ivity of these problems, whi h eventually lo alizes investigations onto one suf iently large bounded set. Typi ally this set annot be hosen a priori for a given

lass of problems, whi h for es us to modify in Se tion 2.3 our on ept of onvex om-

- ompa t but, - ompa ti ation. It

pa ti ations in su h a manner that the resulting envelope is onvex in general, not ompa t. As su h, it will be alled a  onvex

may be itself a linear manifold, though typi ally it is rather not. Sometimes onvex

- ompa ti ations an have additional important pa tness, metrizability, or so- alled

spe ial properties, as lo al om-

B - oer ivity.

The anoni al form enables us, in Se tion 2.4, to develop an approximation theory of onvex ompa ti ations, whi h forms an abstra t framework for developing a omputer-implementable numeri al s hemes in on rete ases. Also, the anoni al form enables us to formulate simple riteria for mappings (esp. fun tions) to admit an ane ontinuous extensions onto respe tive onvex

- ompa ti ations and also to

investigate their dierentiablity properties. This will be performed in Se tion 2.5.

Convex ompa ti ations

2.1

Let us begin dire tly with the denition of the notion of a onvex ompa ti ation whi h represents the fundamental on ept used for relaxation theory as presented in this book. Let us onsider a topologi al spa e

U

to be ompa tied,

T

being its

topology.

Denition 2.1 (Convex ompa ti ation).

A triple ( K ;

Z ; i) is alled a onvex ompa t-

i ation of a topologi al spa e ( U; T ) if (a)

Z is a Hausdor lo ally onvex spa e,

(b)

K is a onvex, ompa t subset of Z ,

( )

i : U Ù K is ontinuous, and

(d)

i(U) is dense in K .

If

i

is also inje tive (resp. homeomorphi al embedding), ( K ;

(resp.

Z ; i)

is alled a Hausdor

T - onsistent) onvex ompa ti ation.

K ; i), reated from a onvex ompa ti ation (K ; Z ; i) by Z , is a ompa ti ation of U in a usual sense as introdu ed in Se tion 1.1. Also note that, in general, the embedding i is even not required to be inje tive so that some points of the original spa e U an be glued together in the ompa ti ation K . The set of all onvex ompa ti ations of a given topologi al spa e U an be natLet us note that the pair (

forgetting

urally ordered.

Denition 2.2 (Ordering of onvex ompa ti ations).

and ( K 2 ;

Let

us

onsider

Z2 ; i2 ) two onvex ompa ti ations of U . Then we will say that:

(

K1 ; Z1 ; i1 )

2.1 Convex ompa ti ations

(i)

Ë 83

K1 ; Z1 ; i1 ) is a ner onvex ompa ti ation of U than (K2 ; Z2 ; i2 ), and write K1 ; Z1 ; i1 ) ³ (K2 ; Z2 ; i2 ), if there is an ane ontinuous mapping : K1 Ù K2 1 1 1 1 xing U ; the adje tive ane means ( z % z ) # 2 2 2 (z) % 2 ( z ) for any z ; z ò K1 , while xing U  means Ž i1 # i2 . ( K 1 ; Z 1 ; i 1 ) and ( K 2 ; Z 2 ; i 2 ) are equivalent with ea h other, and write ( K 1 ; Z 1 ; i 1 ) Ê ( K 2 ; Z 2 ; i 2 ), if simultaneously ( K 1 ; Z 1 ; i 1 ) ³ (K2 ; Z2 ; i2 ) and (K2 ; Z2 ; i2 ) ³ ( K 1 ; Z 1 ; i 1 ). ( K 1 ; Z 1 ; i 1 ) is stri tly ner than ( K 2 ; Z 2 ; i 2 ), and write ( K 1 ; Z 1 ; i 1 ) ± ( K 2 ; Z 2 ; i 2 ), if ( K 1 ; Z 1 ; i 1 ) ³ ( K 2 ; Z 2 ; i 2 ) and ( K 1 ; Z 1 ; i 1 ) ÊÖ ( K 2 ; Z 2 ; i 2 ). (

(



(ii)

(iii)





K1 ; Z1 ; i1 ) ³ (K2 ; Z2 ; i2 ), we will also say that (K2 ; Z2 ; i2 ) is Z1 ; i1 ), and write (K2 ; Z2 ; i2 ) ² (K1 ; Z1 ; i1 ). Let us emphasize that ( K 1 ; Z 1 ; i 1 ) Ê ( K 2 ; Z 2 ; i 2 ) does not mean that the lo ally onvex spa es Z 1 and Z 2 are isomorphi with ea h other. Also let us note that, if Z 's are forgotten, the orderOf ourse, if (

oarser than ( K 1 ;

ing of onvex ompa ti ations agrees with the usual ordering of ompa ti ations

K1 ; Z1 ; i1 ) ³ (K2 ; Z2 ; i2 ), then the ompa ti ation ( K 1 ; i 1 ) is ner in the usual sense than ( K 2 ; i 2 ). Of ourse, the onverse as introdu ed in Se tion 1.1. For example, if (

impli ation does not hold in general. In parti ular, the ane ontinuous mapping

:K ÙK 1

2

xing

U , used in Def-

inition 2.2, must be always surje tive be ause this holds for usual ompa ti ations, as well.1

(

Let us also agree that we will o

asionally abbreviate e.g. K 1 ³ K 2 instead of K1 ; Z1 ; i1 ) ³ (K2 ; Z2 ; i2 ) when Z1 , Z2 , i1 , and i2 are obvious from a ontext. The set of all onvex ompa ti ations of a given topologi al spa e U has always

the smallest element, i.e. the oarsest onvex ompa ti ation. Indeed, any triple

K # {z} a singleton in a Hausdor lo ally onvex spa e Z and with i z) is the oarsest onvex ompa ti ation of U . Obviously, this

onvex ompa ti ation glues all points of the original spa e U into one point and thus the embedding i : U Ù K # { z } is not inje tive provided U ontains at least two points. Sometimes it an happen that U does not admit any other (up to the equiva-

(

K ; Z ; i)

with

onstant (and equal

len e) onvex ompa ti ation than this  ollapsed oarsest one. For example, this happens when

U

is a dis rete topologi al spa e ontaining only a nite number of

points. Fortunately, in pra ti ally interesting ases the set of all onvex ompa ti ations an be far ri her, f. Theorem 3.41 below. Ex eptionally it an happen that

U possesses also the nest onvex ompa ti aU . However, generally the nest

tion, e.g. in the previous ase of a nite dis rete spa e

onvex ompa ti ation does not exist and, instead of it, we have only guaranteed an

1

z ò K there is a net {u  } ò ’ U su h that i (u  ) Ù z in Z . As K ’ K must have a luster point z ò K . As : K Ù K is ontinuous, is surje tive sin e z ò K was arbitrary.

Indeed, by Denition 2.1d for any

i u  )} ò

is ompa t, the net { 1 (

z

( )

# z and

2

1

2

1

2

1

2

2

1

84

Ë

2 Theory of Convex Compa ti ations

existen e of at least one maximal onvex ompa ti ation.2 This is an essential dieren e between the ordering of the usual ompa ti ations. We already know that, if

U

is a regular topologi al spa e, this ordering always admits a nest ompa ti ation, namely the ƒe h-Stone ompa ti ation

U . The ƒe h-Stone ompa ti ation an be

in simple ases homeomorphi with a maximal onvex ompa ti ation but in general it need not be homeomorphi with any onvex ompa ti ation; e.g. when

U

is

nite dis rete, ontaining at least two points.

Example 2.3 (Maximal onvex ompa ti ations).

Let U be homeomorphi al via i K of a lo ally onvex spa e Z . Then (K ; Z ; i) is obviously a Hausdor onvex ompa ti ation of U . Besides, it is maximal.3 On the other hand, su h K need not be the nest onvex ompa ti ation. This appears in the simple ase 2 indi ated on Figure 2.1 where U ’ R is homeomorphi al both to an ellipse K 1 and to a re tangle K 2 . As we just saw, ea h of them forms a maximal onvex ompa ti ation of U whi h, however, annot be transformed anely one onto ea h other. Therefore,

with a onvex ompa t subset

these maximal onvex ompa ti ations are not equivalent with ea h other and, in parti ular, there does not exist any nest onvex ompa ti ation.

K2

K1

i1

PSfrag repla ements

U

i2 Fig. 2.1:



Two dierent maximal

onvex ompa ti ations.

2.2

Canoni al form of onvex ompa ti ations

The aim of this se tion is to show that every onvex ompa ti ation of alently expressed as a set of means M(F ) on a suitable subspa e of all ontinuous, bounded, real-valued fun tions on

f

norm by C 0 ( U )

2

:# supuòU f(u). Supposing that F

F

U an be equivC0 (U)

of the ring

U endowed with the Chebyshev

ontains onstants, we have de-

See Corollary 2.9 and realize that there always exists at least one onvex ompa ti ation, namely

the oarsest one.

K1 ; Z1 ; i1 ) were a onvex ompa ti ation of U ner than K and were the ane onK1 Ù K xing U , then "1 # i1 Ž i"1 would be ontinuous, whi h shows that K1 is also a oarser onvex ompa ti ation than K . This shows K to be maximal. 3

Indeed, if (

tinuous surje tion

2.2 Canoni al form of onvex ompa ti ations

ned in Se tion 1.2. the set of all means on

F ) :#  ò F * ;  F

M(

# { : F F*

where

# 1 and  ; 1 # 1 Ù R linear;  £ 0 &  ; 1 # 1} ;

denotes the topologi al dual of

f ò F , by

by4

*

(1.21). Moreover, the evaluation mapping and

F

F

e(u); f

U . For this, the following property of F

(2.1)

endowed with the standard dual norm

e :U Ù

M(

F ) has been dened, for u ò U

:# f(u) :

It should be emphasized that, in general, (M(F )

(2.2)

; e) need not be a ompa ti ation of

is de isive, as we an see in Proposition 2.5.

Denition 2.4 (Convexifying subspa es). ing if

Ë 85

’ C (U) is alled onvexify0

F

A subspa e

1 1 :u ; u ò U ; a net {u  } ò ’ U :f ò F : lim f(u  ) # f(u ) % f(u ): 2 2  ò 1

1

2

(2.3)

2

Let us note that any subspa e of a onvexifying subspa e is again onvexifying but not vi e versa. Thus requiring a subspa e

ertain impli it restri tion that

F

F

’ C (U) to be onvexifying represents a 0

must not be too large.

Proposition 2.5 (Properties of M(F )).

be a linear subspa e of

C0 (U) ontaining

onstant fun tions and M(F ) be endowed with the weak* topology of

F * . Then the fol-

Let

F

lowing statements are equivalent with ea h other: (i)

F

(ii)

e(U) is dense in M(F ),

is onvexifying,

(iii) the pair (M(F )

; e) is a ompa ti ation of U ,

(iv) the triple (M(F )

; F ; e) is a onvex ompa ti ation of U . *

e is inje tive if and only if F u1 #Ö u2 ;f ò F : f(u1 ) #Ö f(u2 ).

Besides,

separates points of

U in the sense that :u1 ; u2 ò U

ã(iii)ã(iv) is obvious.

Proof. In view of Proposition 1.16, the equivalen e (ii)

á(ii). Let us put

We will prove (i)

F :#

{nite subsets of

F} , N

(2.4)

£ dened by  # (F ; n ) £  # (F ; n ) i F “ F and n £ n . Clearly,  F is dire ted by £. From (2.3) we an dedu e that for every f ò F 1 1 and n ò N there is  f n ò  su h that  f ( u  ) " 2 f(u ) " 2 f(u ) ¢ n for all  £  f n . As (  ; £) is a dire ted index set, for every  # ( F; n ) ò  F there is   ò  su h that   £  f n for all f ò F . We denote ({ f } ; n ) ò  F by  f n . Putting u  # u   , we get ! ! 1 1 1 :f ò F ; n ò N;  £  f n : !!!!! f( u  ) " f(u ) " f(u ) !!!!! ¢ : 2 2 n and endow it with the ordering 1

1

1

1

2

2

2

1

2

2

1

;

1

2

;

;

;

;

4

For the equality in (2.1), see Proposition 1.16.





1

2

Ë

86

2 Theory of Convex Compa ti ations

u  }òF

for

will play then the role of {  }  ò  .



Now, let us onsider another point we get, for ea h

F

 in Denition 2.4 without any u u3 ò U . Repla ing u1 and u2 with u3 and u  ,

In parti ular, we an take the ommon index set loss of generality  of ourse, {

 ò F , a net {u  } òF



su h that

! ! 1 1 1 :f ò F ; n ò N; £  f n : !!!!! f(u  ) " f(u ) " f( u  ) !!!!! ¢ : 2 2 n 3

;



u

By the diagonalization pro edure, we obtain the net {  }  ò  F su h that

! ! 1 1 1 :f ò F ; n ò N;  £  f n : !!!!f(u  ) " f(u ) " f(u ) " f(u ) !!!! ¢ ! ! 4 4 2 !! !! 1 1 1 !!!! 1 1 !! !! ¢ !!!f(u  ) " f( u  ) " f(u )!! % !!f( u  ) " f(u ) " f(u ! ! 2 2 2 2 2 ! ! ! 1

;

2

3



3

1



!! !! !! !!

2)

¢

3 : 2n

fu

u

In other words, we have got the net {  }  ò  F su h that { (  )}  ò  F onverges to

1 1 1 4 f(u1 ) % 4 f(u2 ) % 2 f(u3 ) for all f ò F . It is now evident that repeating this pro edure k

yields su h a net for every onvex ombination  j #1

"l {m2 ; l ò

N; m #

0; :::; 2l }, and kj#1

a j f(u j ) with k ò N; u j ò U , a j ò L #

a j # 1. M L (F ) the set of all nite means  ò M(F ) in the form  # k  j #1 a j e ( u j ) with a j ò L . Paraphrasing the pre eding on lusion, we an also say that for every  ò M L (F ) there is a net, say { u  }  ò  F , su h that e ( u  ) Ù  weakly*. In other words, e ( U ) is dense in M L (F ). n Sin e L is dense in the interval [0 ; 1℄ and the mapping ( a 1 ; :::; a n ) ÜÙ j #1 a j e ( u j ) n n from {( a 1 ; :::; a n ) ò [0 ; 1℄ ;  j #1 a j # 1} to M(F ) is ontinuous, M L (F ) is dense in the Let us denote by

set of all nite means whi h is dense in M(F ) by Proposition 1.16. It eventually yields that

e(U) is dense in M(F ).

á(i). Sin e M(F ) is 1 1

onvex,  # 2 e(u1 ) % 2 e(u2 ) belongs to M(F ) for ea h u1 ; u2 ò U . As e(U) is supposed to be dense in M(F ), there is a net { u  }  ò  ’ U su h that { e ( u  )}  ò  onverges to  . It 1 1 means pre isely that lim  ò  e ( u  )( f ) # [ e ( u 1 ) % 2 2 e(u2 )℄(f) for all f ò F , whi h is just 1 1 1 1 (2.3) be ause e ( u  )( f ) # f ( u  ) and [ e ( u 1 ) % 2 2 e(u2 )℄(f) # 2 f(u1 ) % 2 f(u2 ). Finally, if u 1 #Ö u 2 and F separates u 1 ; u 2 ò U , then there is f ò F su h that It remains to prove the onverse impli ation, that means (ii)

e(u1 ); f> # f(u1 ) #Ö f(u2 ) # , this means e(u1 ) #Ö e(u2 ), thus e is inje tive. e is inje tive, then u1 #Ö u2 implies e(u1 ) #Ö e(u2 ), whi h means the existen e of f ò F su h that f ( u 1 ) # < e ( u 1 ) ; f > #Ö < e ( u 2 ) ; f > # f ( u 2 ), hen e F separates u1 and u2 whi h were taken arbitrarily. Å


# f (z); note that, be ause i(U) is dense in K , there is pre isely one f ò A (K) su h that f # f Ž i, hen e is well dened. As all  f are ontinuous and ane, so is . Be ause

where 

Let us dene a mapping

for any

uòU

and

i u)); f # f (i(u)) # f(u) # e(u); f

( (

f ò F , we get

Ž i # e. Altogether, we proved that M(F ) is a oarser K . In parti ular, is surje tive. show the inje tivity of . Let us take z 1 ; z 2 ò K ; z 1 #Ö z 2 .

onvex ompa ti ation than Now we are going to

Z is a Hausdor lo ally onvex spa e, there exists a linear ontinuous fun tional f0 ò Z that separates the points z1 and z2 . Putting f # f0 K , we obtain f òA (K) su h that  f (z1 ) #Ö f (z2 ). Then, for f # f Ž i ò F , we obtain < (z1 ); f> # f (z1 ) #Ö f (z2 ) # < (z2 ); f>. In other words, ( z 1 ) #Ö ( z 2 ), thus is inje tive.

Sin e

*

Eventually, realizing that spa e

is a one-to-one ontinuous mapping from a ompa t

K onto a Hausdor spa e M(F ), we an see that also "1 is ontinuous.5

As for the uniqueness of

F

onstru ted above, we have to show that if

F1

and

F2 are two onvexifying, losed linear subspa es of C (U) ontaining onstants su h Ê M(F2 ), then ne essarily F1 # F2 . As there is an ane homeomorphism : M(F1 ) Ù M(F2 ) su h that Ž e1 # e2 , it is easy to see that A (M(F1 )) Ž e1 # A (M(F2 )) Ž e2 be ause obviously A (M(F2 )) Ž # A (M(F1 )), where e l : U ÜÙ Fl* 0

that M(F1 )

l # 1; 2. Simultaneously, we have6 also # A (M(Fl )) Ž e l for l # 1; 2. Altogether, we have obtained

denotes the respe tive evaluation mappings,

Fl

F1

# A (M(F

1 ))

Ž e # A (M(F 1

2 ))

Ž

Ž e # A (M(F

2 ))

1

Že #F : 2

2

’ M(F ) onverging weakly* in F to some  ò M(F ), then the net { is  )} ò ’ K must have a luster point z ò K be ause K is ompa t, but ( z ) #  be ause

ontinuous and the weak* topology on M(F ) is a Hausdor one, hen e z is determined uniquely and " (  )} the whole net {   ò ’ K onverges to it. f. also, e.g., Engelking [284, Theorem 3.1.13℄. 5

Indeed, taking a net {

"1 ( 

  } ò

1

6

See Berglund at al. [108, Corollary 3.7℄ for details.

*

88

Ë

2 Theory of Convex Compa ti ations

The pre eding theorem authorizes us to de lare (M(F )

; F ; e) with F *

Å a onvex-

C0 (U) ontaining onstants as a anoni al form of a onvex ompa ti ation ( K ; Z ; i ) in question whenever (M(F ) ; F ; e ) Ê ( K ; Z ; i ). Let us note that ifying subspa e of

*

this denition does not require

F

to be losed and therefore the anoni al form is not

K ; Z ; i). More pre isely, F C0 (U); f. Theorem 2.8 below. Su h

determined uniquely by a given onvex ompa ti ation ( is determined uniquely only up to the losure in

a denition of the anoni al form ree ts the fa t that in on rete ases we are given by some onvexifying subspa e

F

of

C0 (U) but its losure in C0 (U) usually annot be

ee tively determined. Let us turn our attention to the ordering of onvex ompa ti ations. It is natural to seek still ner and ner onvex ompa ti ations and a natural question in this

ontext is whether there exists maximal onvex ompa ti ations. The following theorem gives an answer in terms of the anoni al form, although, of ourse, by a non onstru tive way via the Kuratowski-Zorn lemma 1.1.

Theorem 2.7 (Maximal onvexifying subspa es). of

C0 (U)

Any maximal onvexifying subspa e

is losed. Moreover, every onvexifying subspa e

some maximal onvexifying subspa e of

C (U).

F

’ C (U) is ontained in 0

0

lim"Ù lim ò f " (u  ) # lim ò f(u  ) proU , that means f " " f C0 U Ù 0, and provided lim  ò  f " ( u  ) does exist for any " ¡ 0. Therefore, if F is onvexifying, in Proof. For any net { u  }  ò  , we have the identity

f

vided a sequen e { " } " ¡0 onverges to

f

0

uniformly on

view of (2.3) we an see that its losure in

(

)

C0 (U) remains also onvexifying. In parti -

ular, no onvexifying subspa e whi h is not losed an be maximal. The rest will be proved by using the Kuratowski-Zorn lemma. Therefore, we are to prove that, for every olle tion {F  }  ò A of onvexifying subspa es of

su h

F2 ’ F1 for any 1 ; 2 ò A, there is a onvexifying subspa e 0 F ’ C (U) su h that F ’ F for ea h  ò A. We want to show that it su es to take F # U òA F . Su h F is a linear spa e. Indeed, for any f 1 ; f 2 ò F there are  1 ;  2 ò A su h that f1 ò F1 and f2 ò F2 . As F1 ’ F2 or F2 ’ F1 , both f1 and f2 are ontained in F2 or in F  1 , respe tively. Hen e any linear ombination of f 1 and f 2 is ontained in F  2 or in F  1 and, in parti ular, also in F . It remains to show that F is onvexifying. Let us take u 1 ; u 2 ò U and the dire ted that

F1

’

C0 (U)

F2

or

# (F; n) ò F there is  # ( ) ò A su h that F ’ F be ause we an always suppose F # {f i }ki#1 with some k ò N and f i ò F i for some  i ò A su h that F  1 “ F  2 “ ::: “ F  k , whi h allows us to put simply ( ) # 1 . As F is onvexifying and F is a universal index set ( f. the proof of Proposition 2.5), there is a net { u  }  ò  F ’ U su h that, for every f ò F and n ò N,  index set (

 # F

as in (2.4). For any

)

(

)

(

(

there is 1

n . As

(

(

)

(

)

 £ (f; n) it holds f(u  ) " 12 f(u1 ) " 12 f(u2 ) ¢ is dire ted, there is  ( F; n ) ò  F su h that  ( F; n ) £  ( f; n ) for any f ò F ;

 # (f; n) ò F

 F

)

)

)

su h that for any

(

)

Ë 89

2.2 Canoni al form of onvex ompa ti ations

1

F is nite. Thus we get the situation that f(u  ) " 2 f(u1 ) " 12 f(u2 ) ¢ 1n for any  £  ( F; n ) and any f ò F . Taking a net { u } ò  F and u # u  F n for # ( F; n ) and realizing that  F ’  F for any ò  F , then, for any f ò F , n ò N, £ ({ f } ; n ), we

re all that

(

(

fu

have got  ( )

;

)

)

" 21 f(u ) " 21 f(u 1

¢

2 )

Å

1

n . It shows that F is onvexifying.

The following assertion together with Theorem 2.6 show that the topology of the

C0 (U) is de isive for M(F ) in the sense 0 that we an always enlarge F up to its losure in the C -topology without any inu-

uniform onvergen e on the subspa es

F

of

en e on M(F ). On the other hand, any further enlargement of F will already inuen e

F ). Also note that, sin e F 's are endowed by the (relativized) C0 -topology, the embedding F1 ’ F2 is of Type (C) a

ording to the lassi ation from (1.43) whi h auses * * the adjoint operator F2 Ù F1 to be surje tive.

M(

Theorem 2.8 (Dependen e of M(F ) on F ).7 Let F ; F 1

spa es of

2

be two onvexifying linear sub-

C0 (U) ontaining onstant fun tions. Then:

’ F implies M(F ) ² M(F ); i.e. the mapping F ÜÙ M(F ) is monotone. (ii) If l C 0 U F # lC 0 U F , then M(F ) Ê M(F ). (iii) If F ’ F but l C 0 U F #Ö l C 0 U F , then M(F ) ° M(F ). F1

(i)

2

(

1

)

1

1

(

2

2

2

)

(

1

1

)

(

)

2

2

1

2

Proof. It is easy to see that the anoni al ane ontinuous surje tion M(F2 ) xing

Ù M(F

1)

U is just Q : F2 Ù F1 , where Q : F1 Ù F2 is the in lusion F1 ’ F2 . Hen e *

*

*

F2 ) is ner than M(F1 ), as laimed at the point (i). 0 To prove (ii), let us take F a onvexifying subspa e of C ( U ) and put F # lC 0 ( U ) F . * Now let Q : F Ù F be the in lusion F ’ F . As above, the adjoint operator Q : * * F Ù F realizes the anoni al ane ontinuous surje tion M(F ) Ù M(F ) xing U . We want to show that the restri tion of Q* on M(F ) is inversely ontinuous, whi h * will show M(F ) Ê M(F ). As F is dense in F , Q : F * Ù F * is inje tive; indeed, for  1 ;  2 ò F su h that  1 #Ö  2 , the restri tions on the dense subspa e F must M(

Q 1 # 1 F #Ö 2 F # Q 2 . As Q *

*

also dier from ea h other, hen e



*

is a one-to-one

ontinuous mapping between two ompa t sets M(F ) and M(F ), the inverse mapping

Ê M(F ). From this ) provided F # F , as laimed

must be also ontinuous.8 Altogether, we have thus proved M(F ) we get immediately M(F1

) Ê M(F ) # M(F ) Ê M(F 1

2

2

1

2

at the point (ii). In the rest of the proof we want to prove that M(F1 )

F2

7

are losed and

F2

The proof of the fa t

of the

C0 (U)- losure

of

“

M(F F

F1

1)

but

F2

° M(F

#Ö

F1 . We want to

° M(F

2 ),

supposing that

prove that there are

F1 ,

1 ; 2 ò

2 ) follows also quite straightforwardly from the uniqueness

for a given onvex ompa ti ation of

U,

f. Theorem 2.6. Nevertheless,

we performed a dire t, hopefully interesting proof whi h is not expli itly supported by Berglund at al. [108, Corollary 3.7℄ used for the uniqueness in Theorem 2.6. In fa t, [108, Corollary 3.7℄ relies on similar arguments as Hahn-Bana h, Riesz and Jordan theorems used in the proof presented here.

8

Let us note that the embedding is of the type (AB) a

ording to the lassi ation on p. 36.

90

Ë

2 Theory of Convex Compa ti ations

M(F2 ) su h that  1 #Ö  2 but <  1 ; f > # <  2 ; f > for any f ò F1 , whi h shows that Q 1 # Q 2 for Q : F1 Ù F2 being the in lusion F1 ’ F2 , hen e M(F1 ) ÊÖ M(F2 ). *

*

As M(F1 )

² M(F

2)

° M(F

has been already proved, this will imply M(F1 )

Ž Let us denote by F

1

the annihilator of

F1Ž

2 ).

*

F1 in F2 , this means9

#   ò F ; :f ò F :

 ; f

*

1

2

# 0: Ž

We show that M(F2 ) is not at in ea h dire tion from F1 in the sense that, for any

ò

F1Ž dierent from zero, there are 1 ; 2 ò M(F2 ) su h that 1 " 2 #  with some #Ö 0. Ž Indeed,  ò F means, in parti ular, that <  ; 1> # 0. Let U be a ompa ti ation of U 1

f ò F admits a ontinuous extension f on U . Then f ÜÙ forms a linear ontinuous fun tional on F # {f ; f ò F } ’ C( U). By the

su iently ne10 su h that every

2

Hahn-Bana h theorem 1.11, this fun tional admits the ontinuous linear extension on

C( U), and then by the Riesz representation theorem 1.32(iii) f d  for any f ò F2 . Then we an there is a measure  òr a( U ) su h that <  ; f > # P U

the whole Bana h spa e





make the Jordan de omposition of

 

%

onto its positive variation



and its negative

* % " " variation  , whi h again belong to r a( U ). Then we dene  ;  ò F2 



for any

f ò F2 .

%; f

#

X

f d  %



and



U

Therefore we an see that



"; f

#

X

by

f d  " 

U

admits a (generally not uniquely deter-

% " " with both %

# £ 0 and " £ 0. We want to prove that " # ¡ 0. Supposing the ontrary, both % and " would be % " identi ally zero sin e the orresponding measures  and  would be non-negative % with <  ; 1> # P d  % # 0 and





zero. Hen e also

 # 0, but we supposed  #Ö 0, a ontradi tion. Therefore we may put 1 #

Obviously,

2 ò M(F2 ).

1 £ 0

For any





and


and

2 #

" : " <  ; 1>

1 ; 1> # 1, hen e 1 ò M(F2 ),

see (2.1), and similarly also

f ò F2 \ F1 there is some  f ò F1Ž su h that #Ö 0 be ause otherwise f

would have to belong to the annihilator of

F1Ž in F2 , whi h is the losure of F1 , hen e

#  f and making the above des ribed de omposition, we obtain the situation <  ; f > #Ö <  ; f > for f ò F \ F , whi h shows that a tually  #Ö  , but, on the other hand, <  ; f > " <  ; f > # <  ; f > # 0 for any f ò F . Å F1 itself. Choosing  1

2

1

2

2

1

1

2

1

F1Ž #Ö {0} be ause otherwise F2 ’ F1 , ontrary to what is supposed. 10 For example, if U is ompletely regular, one an always take for U the ƒe h-Stone ompa ti ation U . If not, one an rene the topology on U be ause, e.g., a dis rete topology is ertainly om9

Let us noti e that

pletely regular.

Ë 91

2.2 Canoni al form of onvex ompa ti ations

Corollary 2.9 (Maximal onvex ompa ti ations).

For every onvex ompa ti ation

K of U , there exists a maximal onvex ompa ti ation K1 of U ner than K . Proof. By Theorem 2.6, we an take a losed onvexifying linear subspa e

F

of

C0 (U)

K Ê M(F ), and by Theorem 2.7 there is a maximal ’ C (U) su h that F ’ F . Let us put K # M(F ). K ² K as a onsequen e of Theorem 2.8. To show that

ontaining onstants su h that

onvexifying subspa e

F1

0

’ F , it holds ( K ; Z ; i ) # (M(F ) ; F ; e F

As

1

1

1

1

*

1

1

1

1

1

1

1)

is maximal, let us take another onvex ompa ti a-

K2 ; Z2 ; i2 ) of U su h that K2 ³ K1 . We want to demonstrate that K2 ² K1 . Let us put F2 # A ( K 2 ) Ž i 2 and F1 # A ( K 1 ) Ž i 1 . Then we have K 2 Ê M(F2 ) and K 1 Ê M(F1 );

tion (





f. the proof of Theorem 2.6. On the other hand, we have also

# A (M(F )) Ž e surje tion :K ÙK

F1

1.

1

2

1

Sin e we supposed

Ži # i

su h that

2

1.

K2 ³ K1 ,

F1

#

F1

be ause11

there is an ane ontinuous

Then obviously

A (K ) Ž ’ A (M(F

2 )).

1

Altogether, we have obtained

However, Hen e

F1

#

F1

was a maximal onvexifying subspa e

F1

# A (K ) Ž i # A (K ) Ž 1

1

1

Ž i ’ A (K ) Ž i #

K2 Ê M(F2 ) # M(F1 ) # K1 has been proved.

2

2

F1

2

F2 :

’ C (U), so that F # 0

2

F1 .

Å

Now we want to relate the standard onstru tion of a ompa ti ation by multipli ative means with the onvex ompa ti ations. Every ompa ti ation, reated from a onvex ompa ti ation by forgetting the lo ally onvex spa e, must be obviously equivalent in the usual sense to some ompa ti ation obtained by the multipli ative means. Considering a onvex ompa ti ation in its anoni al form (M(

F ); F * ; e), as we always an due

to Theorem 2.6, we want now to onstru t the

equivalent standard ompa ti ation expli itly. Besides, we shall see that the homeomorphism that makes them equivalent to ea h other is even ane. Let us denote by

Ring(F ) the smallest losed subring of C (U) ontaining F ’ C (U), i.e. 0

n

0

m

Ring(F ) # lC0 U – H I f ij ; n ; m ò N; f ij ò F — : (

)

(2.6)

i #1 j #1

Theorem 2.10 (Conne tion with multipli ative means). Let F be a linear subspa e of C (U) ontaining onstant fun tions and Ring(F ) the orresponding smallest losed ring ontaining it, and let Q : F Ù Ring(F ) denote the in lusion F ’ Ring(F ). Then: (i) The adjoint operator Q realizes a homeomorphi al embedding of Mmult (Ring(F )) 0

*

into M(F ). (ii)

Q

*

(Mmult (

Ring(F ))) #

F ), i.e. Q*

M(

is a homeomorphism, if and only if

vexifying.

11

See Berglund at al. [108, Corollary 3.7℄ for details.

F

is on-

Ë

92

Q

Proof. Mmult (

2 Theory of Convex Compa ti ations

*

is

Ring(F )))

obviously

ontinuous

and

maps

Ring(F ))

(and

M(

thus

also

Q

into M(F ). Now we will prove the inverse ontinuity of

Ring(F )), this means:

*

if

restri ted on Mmult (

: ò Mmult (Ring(F )) :" ¡ 0 :f òRing(F ) ;Æ ¡ 0 ;k ò N ;{f l }kl# ’ F 1

:  ò Mmult (Ring(F )): max  Q  ; f l " Q  ; f l  ¢ Æ á *

*

¢l¢k





1

!! ! !!  ; f "  ; f !!! 

¢ ":

(2.7)

Ring(F ), there exist m ; n ò N and f ij ò F , i # 1; :::; n, j # 1; :::; m, n m f " ni#1 A m j #1 f ij C 0 U ¢ " /3. Then also <  ; f > " <  ;  i #1 A j #1 f ij > ¢ " /3 and n m <  ; f > " <  ;  i #1 A j #1 f ij > ¢ " /3 be ause  #  # 1. Now we an estimate:

By the denition of su h that 

(

)



!! !!  ; f



!!

"  ; f !!!! ¢ !!!!  ; f " ¼ ; 

!

%

n m n m !! ! !!¼  ; H I f ½ " ¼  ; H I f ½!!! % ; ij ij !! !! ! ! i #1 j #1 i #1 j #1 

1 " % 3

¢

n

n m !! ! H I f ij ½!! !! i #1 j #1

n ! m m !! ! H !!! I  ; f ij " I  ; f ij !!! !! !! i #1 j #1 j #1 

m

¢ L H H !!!!  ; f ij "  ; f ij !!!! % 

i #1 j #1

%

n m !! !!¼  ; H I f ½ ij !! ! i #1 j #1 

!!

"  ; f !!!! 

!

1 " 3

2 "; 3

L is the Lips hitz onstant of the mapping (a1 ; :::; a m ) ÜÙ A m j #1 a j restri ted to the arguments a j with  a j  ¢ max i j f ij C 0 U . Using the fa ts that < Q  ; f ij > # <  ; f ij > and < Q  ; f ij > # <  ; f ij > be ause f ij ò F , we obtain (2.7) when taking Æ # " /(3 nmL ), k # nm, and {f l } # {f ij }. Thus the point (i) has been proved. Now suppose that Q is the homeomorphism of Mmult (Ring(F )) onto M(F ). Let R : U Ù M(Ring(F )) from the evaluation us distinguish the evaluation mapping e R R mapping e : U Ù M(F ). Clearly, e # Q Ž e . Sin e e ( U ) is dense in Mmult (Ring(F )), e(U) is dense in Q (Mmult (Ring(F ))) # M(F ). By Proposition 2.5 (ii) á (i), F must be

where

*

;

(

)

*





*

*

*

then onvexifying. It remains to prove the onverse impli ation, supposing that

F

is onvexifying.

á (ii), we an see that e(U) is dense in M(F ). By the obvious estimate e ( U ) ’ Q (Mmult (Ring (F ))) ’ M(F ) and by the weak* ompa tness of Mmult (Ring(F )) and the weak* ontinuity of Q , we get eventually the surje tivity of Q : Mmult (Ring(F )) Ù M(F ): Å Using Proposition 2.5 (i) *

*

*

Let us remark that the adjoint operator

Q : (Ring(F )) Ù F

ing theorem makes nothing else than the restri tion on fun tionals

Ring(F ) Ù R.

*

*

F

*

from the pre ed-

of the linear ontinuous

Theorem 2.10 showed M(F ) equivalent (as a usual ompa ti ation) with

Ring(F )).

Mmult (

However, it should be pointed out that Mmult (

Ring(F ))

denitely

2.3 Convex



Ë 93

- ompa ti ations

annot serve well as far as onvexity on erns be ause it is extremely bent in

Ring(F ))

(

*

, whi h is made pre ise by the following assertion.

Proposition 2.11.

Let F be a subspa e of C ( U ) ontaining onstant fun tions, 0

the losed subring generated by

Ring (F ) % 12  ò

1 F , and let 1 ; 2 ò Mmult (Ring(F )). Then 2 1

Ring(F )) \ Mmult (Ring(F )) whenever  #Ö 

M(

1

2

2.

1  # 21 1 % 2 2 ò Mmult (Ring(F )). In other # for all f1 ; f2 ò Ring(F ). Taking f # f1 # f2 ò Ring(F ),

Proof. Suppose the ontrary, that means words,


we get

1 1 1 1 2 2 2 2 2 2  ; f %  2 ; f #  1 ; f %  2 ; f #  ; f #  ; f 2 1 2 2 2 2 # Œ 12 1 ; f % 12 2 ; f  # 41 1 ; f 2 % 14 2 ; f 2 % 12 1 ; f 2 ; f :

1 ; f>2 % 2 # 2, whi h means pre isely # <  2 ; f >. As f òRing(F ) is arbitrary, we have obtained  1 #  2 , a ontradi tion. Å

This is true only if
# P g ( x ) v 0 ( u ( x ))(1%  u ( x ) ) d x . In other words,

p m this means < i ( u ) ; h > # P h ( x ; u ( x ))(1%  u ( x ) ) d x for h ò C ( , R R ). We onsider

m weak* topology on r a( , R R ), whi h makes i ontinuous. Thanks to the estimate *

embedding



*







i u  ) r a ,

(

(

R

¢   % u  pLp Rm ;

Rm )

(

;

(3.48)

)

 òr a( ,

iu

the net { (  )}  ò  is bounded, and it must have a luster point 



R

Rm . There)

fore there is a ner net (denoted for simpli ity by the same indi es, f. Example 1.4)

1%u  p } ò , being bounded in L ( ), has a



onverging weakly* to  . Besides, the net {

luster point in

1

r a( ), say , and we may and will assume that our ner net has been

hosen so that (3.46a) holds, as well. Let us dene

T : R Ù r a( ) Ê C( )

*







T v0 ; g 

for any

by

#  ; g ‹ v

0

g ò C( ). In view of (3.48), we have the estimate

!! ! !! Tv 0 ; g !!! 

¢ v

0

#

C 0 (Rm )

!! ; g !! 

‹v

0

!

!!!

!!

!!

# !!!! lim X g(x)v (u  (x))(1% u  (x)p ) dx!!!! 0

!  ò

lim X g(x)1%u  (x)p  dx # v  ò



!

0

C 0 (Rm ) X





g(x)(dx):

(3.49)

T v0 ℄(A) ¢ v0 C0 Rm (A) for any Borel subset A ’ , whi h , and thus by the Radon1 Nikodým theorem it admits a representation by a density Tv 0 ò L ( ;  ), that means 1 [ T v 0 ℄( A ) # P [ Tv 0 ℄( x )  (d x ). The mapping T : R Ù L ( ;  ) is linear, and in fa t it is A

In parti ular, it implies [ means that

Tv0 



(

)



is absolutely ontinuous with respe t to







Ë

148

3 Young Measures and Their Generalizations

T:RÙL (3.49) and of the duality between L ( ;  ) and L ( ;  ). bounded (with the norm being equal 1) as an operator 1



(

; ) be ause of







Obviously, (3.46b) follows easily from the denitions, namely

X g ( x ) v 0 ( u  ( x ))(1%  u  ( x )

p

)

d x # i ( u  ) ; g ‹ v # Tv ; g #

Ù  ; g ‹ v

X



0

R R

0

0

g(x)[Tv0 ℄(x)(dx) :

R

0 m ) is separable, hen e m is metrizable. As Suppose now that R ’ C ( R m is separable and dense in m , we have m separable. As  is regular, R R  deC( ) ‹ C( R m ) is dense in L1 ( ; ; C( R m )).40 Therefore, the fun tional  1 m ned above an be extended ontinuously onto L ( ;  ; C ( )). By Dunford-Pettis R 1 m ))* Ê L  ( ;  ; r a( m )), whi h assigns   an element theorem, L ( ;  ; C ( w R R m )).  ò L  ( ;  ; r a( w* R m and  repla ing the Lebesgue meaIt is easy to verify (3.17) for  with S # R p m  £ 0. Also, as sure. Indeed, as i ( u  ) £ 0 for any u  ò L ( ; ), in the limit we get  p  ; g ‹ 1>. Simultanealways 1 ò R , we have P g ( x )(1%  u  ( x ) ) d x # < i ( u  ) ; g ‹ 1> Ù < 

p  ously, P g ( x )(1%  u  ( x ) ) d x Ù P g ( x )  (d x ), whi h shows that P P



A Rm x (ds)(dx)

R

R



R

R



R





R



R

R



R R

# (A) for any Borel subset A ’

, R Rm Ù .

R

or, in other words,





is the proje tion of  via





S’

Then by Lemma 3.5, whi h holds true not only for able ompa t

S#

R

R

m , we obtain that  ò Y( ;  ;

R



R

Rm but also for the metriz-

m ). Moreover, for any g ò C ( ),

it holds

X X



R

Rm

g(x)v0 (s)  x (ds)(dx) #



()

; g ‹ v0

#  ; g ‹ v

0

#

X



g(x)[Tv0 ℄(x)(dx) ;

Lw ( ; ; r a( R Rm )) Ù L1 ( ; ; C( also Lemma 3.4. As (3.50) holds for any g , (3.47) has been proved.

where

denotes the isomorphism







Let us agree to denote the set of all Radon measures on

,

R

R

(3.50)

Rm

*

)) , f.

Å

Rm onstru ted in

the previous proof by

p m  òr a( , R R m ); DMR ( ; R ) # Ž ;{u  } ò bounded in L p ( ; Rm ) :

If

R

40

’C

0

Rm

(



w*-

) is separable, we will also work with the set

See Warga [791, Thm. I.5.25℄.

lim i(u  ) #  DZ:  ò

(3.51)

3.2 Various generalizations

DMR ( ; Rm ) # Ž( ;  ) òr a% ( ) , Y( ; ; p







;{u  } ò bounded in L

p

(

Rm ;

; Rm : )

R

(3.46a)(3.47) holdDZ

)

Let us agree to address the elements of both

Ë 149

p m DMR ( ; R )

and

:

(3.52)

DMR ( ; Rm ) p

as



DiPerna-Majda measures. It should be emphasized that  is not a Young measure onstru ted in Theorem 3.19. A tually, the relation between the DiPerna-Majda and the Young measures is stated in the following assertion. Let us note that the ontributions of a given DiPernaMajda measure supported at innity (this means on the remainder

Rm Rm ) are

R p "1 used in (3.54) vanishes at innity. forgotten be ause the weight (1% s  )

Theorem 3.26 (DiPerna and Majda).41 Let {u k }kòN L p ( ; Rm ) and R a separable omplete subring of C

0

be

a

(

)

bounded

Rm , p ò

sequen e

1; %).

[

\

in

Then there

is a subsequen e, denoted again by { u k } k òN , onverging to a DiPerna-Majda measure

(

 ;  ) ò DMR ( ; Rm ) in the sense p



lim X g(x)v(u k (x)) dx k Ù

for any

g ò C( )

and any

#

X X



R

Rm

v(s) # v0 (s)(1%sp )

g(x)v0 (s)  x (ds)(dx) ;

with

v0 ò R.

sequen e onverges in the sense of (3.34) to a Young measure

x (d s )

# d  (x)

(3.53)

Simultaneously, this sub-

ò Yp ( ; Rm ) given by

 x (d s ) 1%sp

(3.54)

 is absolutely ontinuous (with respe t to the Lebesgue measure on ) with d  being its density.42

provided

Proof. Obviously, (3.53) is a mere ombination of (3.46b) and (3.47) together with the fa t that the separability of

R

R

R

implies metrizability of the weak* topology of

C( ,

m )* restri ted on bounded subsets, whi h eventually allows us to work in terms

of subsequen es. Let us show that

from (3.54) is a tually a parametrized probability measure (i.e

a Young measure). Let us put

X g(x) dx

#

v0 (s) # (1%sp )"1 into (3.53). Realizing that v # 1, we get

X g ( x ) v ( u k ( x )) d x

Ù

X g ( x ) ¤X

 x (d s ) R

Rm

1%sp

¥ d  (x) dx ;

whi h implies

41

We refer to DiPerna and Majda [266, Corollary 4.3℄. The assertion here is, however, slightly gener-

42

In fa t, formula (3.54) holds even if

absolutely ontinuous part of

R

L ( ; m ) has been admitted for (3.54) in [266℄.  is not absolutely ontinuous  then d  is the density of the

alized be ause only sequen es bounded in

 in its Lebesgue de omposition.

150

Ë

3 Young Measures and Their Generalizations

 x (d s )

”X R

for a.a.

1%sp

Rm

#1

• d  (x)

x ò . Realizing that s ÜÙ (1%sp )"1 vanishes on the remainder

obtain

R

Rm Rm , we \

 x (d s )  x (d s ) # d  (x)X # 1; m 1%sp 1%sp RR x is a probability measure for a.a. x ò be ause the fa t that

x (d s )

X

Rm

whi h shows that

# d  (x)X

Rm

x is

non-negative is lear from its denition (3.54).

u

Simultaneously, we know from Theorem 3.19 that a (sub)sequen e { k } determines a Young measure, let us denote it by

1

oin ides with

. It remains to show that

g ò C( ) and v ò C p (Rm ), we know from (3.34) and (3.53) that P g(x)v(u k (x)) dx 1 

onverges to P P m g ( x ) v ( s ) x (d s ) d x and to P P Rm g(x)v0 (s) x (ds)(dx), respe R

1

. For



R

tively. Therefore, we have

X X

Rm

x (d s ) d x

g(x)v(s)

#

1

#X



#X



X R

X

Rm

Rm

g(x)

g(x)

X X



R

v(s)

1%sp

v(s)

1%sp

Rm

g(x)v0 (s)  x (ds)(dx)

 x (d s ) d  ( x ) d x

 x (d s ) d  ( x ) d x

#X



X

Rm

Of ourse, we used the denition (3.54) and the fa t that remainder

R

g(x)v(s)

x (d s ) d x :

v(s)/(1% sp ) vanishes on the

Rm Rm . Sin e g and v have been taken arbitrarily, the identity \

1

#

is proved.

u

For illustration, let us give examples of some sequen es { k }

Å

’ L p ( ; Rm ) and the

orresponding DiPerna-Majda measures (  ;  ) generated by them. As a rst, trivial p m ) and onsider   # i ( u ) òr a( , R m ). example, let us have a fun tion u ò L ( ; Then the orresponding DiPerna-Majda measure (  ;  ) takes the form

R

R



(dx) # 1%u(x)p  dx ;



x

# Æu x ; (

(3.55)

)

and we an see that (ex eptionally!)  oin ides with the orresponding Young measure when one repla es

 by the Lebesgue measure.

A bit less trivial example is the situation outlined on Figure 3.3, namely for two

u1 ; u2 ò L p ( ; Rm ) let us onsider an os illating sequen e { u k } ’ L p ( ; Rm ) m  # ( i ( u 1 ) % i ( u 2 ))/2 ò r a( , whi h onverges weakly* to  R R ), and the orresponding Young measure is x # ( Æ u 1 x % Æ u 2 x )/2. In view of (3.46a) and (3.54), we fun tions





( )

(

)

an easily establish the on rete form of the orresponding DiPerna-Majda measure (

 ;  ), namely

1 p p (2 %  u ( x ) %  u ( x ) ) d x ; 2 1%u (x)p 1%u (x)p # Æ u1 x % Æ : p p 2 % u (x) % u (x) 2 % u (x)p % u (x)p u2 x

(dx) # x

1

2

1

2

(

1

2

)

(

1

2

)

/ 7

(3.56) ? 7 G

3.2 Various generalizations

All the rest examples will use

R€ %

Ë 151

# (0; 1), m # 1, and R from (3.45) so that

R Ê 1

R

; "}. They deal with the situations when some non-vanishing part of the u p is a tually arried to innity. The right-hand part of these gures is to  ) ’ , R R1 # [0 ; 1℄ , (R € {% ; "}). Let us note that in ea h ase indi ate supp(  the orresponding Young measure ò Y((0; 1); R) is homogeneous, namely x # Æ0 for a.a. x ò (0 ; 1). p The example shown in Figure 3.7 presents a sequen e { u k } k òN su h that  u k  onp

entrates, onverging just to a Dira distribution (multiplied by the fa tor a b ) at a point x 0 ò . The resulting DiPerna-Majda measure is given by {



energy  k 



 # 1 % a p bÆ x0 ; where

x

#



Æ0 Æ%

if if



 is determined obviously from (3.46a) while

x #Ö x 0 ; x # x0 ;

(3.57)

 is determined from (3.53) with

help of the identity

lim

1

k Ù

X 0

g(x)v(u k (x)) dx # lim

x 0 " b/(2 k ) ”X g(x)v0 (0) dx

k Ù 0 x 0 % b/(2 k ) %X g(x)v0 (ak1/p )(1 x 0 " b/(2 k )

repla ements 1

% a p k) dx % 1

# X g(x)v (0) dx % a p bg(x )v (%) # X X 0

0

0

0

S

1

0

%

uk

1

g(x)v0 (0) dx• X x 0 % b/(2 k ) g(x)v0 (s)  x (ds)(dx):

R€{%;"}

%

ak1/p

k

Ù

b/k

0 Fig. 3.7:



0

x0

x0

The DiPerna-Majda measure reated by a on entrating sequen e.

The next example shows a homogeneous DiPerna-Majda measure (i.e. both

 do not depend on

 and

x) reated by an os illating sequen e s hemati ally outlined on  ;  ) is the following:

Figure 3.8. The resulting DiPerna-Majda measure (

(dx) # 1 % a p b "1  dx

and

x

#

ap b Æ0 % Æ p

%a b

% ap b %



;

(3.58)

whi h an be determined respe tively from (3.46a) and (3.53) similarly as in the pre eding example.

Ë

152

PSfrag repla ements

S

1

3 Young Measures and Their Generalizations

% u k

%

ak1/p

k

Ù

b/k2

0

c k

Fig. 3.8:



0

2c 3 c k k

The DiPerna-Majda measure reated by an os illating sequen e.

The last example shows a DiPerna-Majda measure reated by an os illating sequen e whose os illations on entrate near a point DiPerna-Majda measure (

 ;  ) is the following:

 # 1 % 3a p bÆ x0

and

x

#



Æ0 2 Æ% 3

x0 ò ; f. Figure 3.9. The resulting

if

% Æ" 1



3

if



x #Ö x 0 ; x # x0 :

(3.59)

PSfrag repla ements

S

1

%

uk

%

ak1/p

uk

k

0

x0

Ù



0

x0



b/k

"ak

1/

p

" Fig. 3.9:

" The DiPerna-Majda measure reated by an os illating/ on entrating sequen e.

Let us noti e that the DiPerna-Majda measures ( possess the omponent

 ;  ) from (3.55), (3.56), and (3.58)

 absolutely ontinuous, while those from (3.57) and (3.59) do  abso-

not. Obviously, (3.58) demonstrates that even DiPerna-Majda measures with

lutely ontinuous an still arry some part of energy to innity. Therefore, a stronger kind of regularity is still worth to be onsidered: let us all a DiPerna-Majda measure (

; )

to be

onverging to (

;

p-non on entrating

weakly ompa t in

L1 ( ).

u

if there is a bounded net {  }  ò 

 ) in the sense of (3.53) su h that the set

{

u

p  ;

’ L p ( ; Rm )

 ò } is relatively p-non on-

Let us only remark that the property of being

entrating is quite natural be ause solutions of oer ive relaxed optimization prob-

Ë 153

3.2 Various generalizations

p-non on entrating,

lems are typi ally

as we will see later in Propositions 4.46(iv),

4.76(iv), 4.116(iv), 5.21 and 7.15(i), and Remark 6.23.

Lemma 3.27 (Non on entrating DiPerna-Majda measures). Let the omplete R ’ C (Rm ) be separable and ( ;  ) be a DiPerna-Majda measure. Then:

subring

0

(i)

(

 ;  ) is p-non on entrating if and only if its energy is not supported at innity

(i.e. on the remainder

R

Rm Rm ) in the sense that \

X X



R

Rm \Rm

 x (d s )  (d x )

# 0:

(3.60)

 ;  ) is p-non on entrating, then  is absolutely ontinuous and any bounded sep m quen e { u k } k òN ’ L ( ; R ) onverging to (  ;  ) does not on entrate energy (i.e. p 1 the set { u k  ; k ò N} is relatively weakly ompa t in L ( )) and the orresponding

(ii) If (

Young measure onstru ted in Theorem 3.26 is fully ee tive in the sense that (3.34)

v ò C p (Rm ).

holds even for any

’ L p ( ; Rm ) be a bounded sequen e onverging to ( ;  ) in the

Proof. Let { u k } k òN sense of (3.53).

u

First, let us suppose that { k 

p;

N

kò

} is relatively weakly ompa t in

L1 ( ).

1%u k k ò N}, and by Theorem 1.28(ii) it is also uniformly :" ¡ 0 ;r " ò R% : p  ;

Therefore, so is also set {

integrable, whi h means:

sup X k òN

For

{

x ò ;  u k ( x )£ r " }

r ¡ 1, let us dene v0r # 1 " v r

with

v0 ò R , so that we an estimate r

X X



m m R R \R

 x (d s )  (d x )

¢

1%u k (x)p dx ¢ " :

vr

dened by (3.38). Let us note that always

X X



m RR

v0r " (s)  x (ds)(dx)

r" p X v 0  u k ( x ))(1%  u k ( x )  d x k Ù

# lim

¢ sup X k òN

As

1%u k (x)p dx ¢ " :

{

x ò ;  u k ( x )£ r " }

" ¡ 0 was arbitrary, (3.60) has been proved. Let us prove the onverse impli ation. Supposing the DiPerna-Majda (

es (3.60) and putting

lim X X r Ù%

R

B r # {s ò Rm ; s ¢ r}, we get Rm \ B r

 x (d s )  (d x )

#

X X



R

Rm \Rm

 x (d s )  (d x )

# 0;

m  òr a( , -additivity of the measure  RR ) Theorem 3.25. Let us now take " ¡ 0. Then for r

whi h follows simply by means of the assigned to (

 ;  );

f. the proof of



su iently large we have got

X X



R

Rm

 ;  ) satis-

v0r (s)  x (ds)(dx) ¢

X X



R

Rm \ B r

 x (d s )  (d x )

¢

"

2

:

Ë

154

3 Young Measures and Their Generalizations

Moreover, there is some

!! !!X X !!!

m RR

k r ò N su h that, for every k £ k r , !! r p X v 0 ( u k ( x ))(1%  u k ( x ) ) d x !!! !!

v0r (s)  x (ds)(dx) "

Altogether we obtained P



¢

"

2

:

v0r (u k (x))(1%u k (x)p ) dx ¢ " for any k £ k r , and therefore

also

1%u k (x)p dx ¢

X

x ò ;  u k ( x )£ r %1} p The nite set {1% u k  ; k {

L1 ( ),

P

{

# 1; :::; k r }

r p X v 0 ( u k ( x ))(1%  u k ( x ) ) d x

is obviously relatively weakly ompa t in

hen e uniformly integrable, whi h means that for some

x ò ;  u k ( x )£ r 0 }

¢ ":

r0

su iently large

1%u k (x)p ) dx ¢ " for any 1 ¢ k ¢ k r . Altogether, we got for any k ò N

(

X {

x ò ;  u k ( x )£max( r 0 ; r %1)}

1%u k (x)p dx ¢ " :

" ¡ 0 was arbitrary, we have proved that the whole sequen e {1%u k p }kòN is uni1 formly integrable, hen e also relatively weakly ompa t in L ( ), so that, by the denition, the DiPerna-Majda measure (  ;  ) is p -non on entrating. This nishes the point

As

(i).

 ;  ) is p-non on entrating, it must full (3.60) and then we saw in the pre eding part of the proof that any { u k } k òN onverging to (  ;  ) p in the sense of (3.53) has the property that { u k  ; k ò N} relatively weakly ompa t in L1 ( ). In parti ular, also {1%u k p ; k ò N} enjoys this property. In view of (3.46a), we p 1

an see that the limit of {1% u k  } k òN is in L ( ), hen e  is absolutely ontinuous. Fi1 nally, let us observe that { v ( u k ); k ò N} is relatively weakly ompa t in L ( ) provided p m v ò C (R ). Then (3.34) follows by Lemma 3.20. Å Let us go on to the point (ii). Sin e (

If the ring

DMR ( ; Rm ) p



R

is separable, even a omplete hara terization of elements of

an be established. We present it here rather for an illustration how

parti ular generalizations of lassi al Young measures an be, in fa t, fairly ompli ated.

Proposition 3.28 (Chara terization of DiPerna-Majda measures).43 Let the ring R be m ) ò r a( ) , L w ( ;  ; r a( R R )). Then the following two state-

separable and (  ;







*



ments are equivalent with ea h other: (i)

The pair (  ;  ) is a DiPerna-Majda measure, i.e. (  ;  )

(ii)

 dened by   (d x ) # „P m ,  and  R

p m ò DMR ( ; R ).

 x (d s )…  (d x ) satisfy

 òr a% ( ) ;

(3.61a)



43

Let us note that the absolute ontinuity of

on the remainder

R

R

m

\

R

m

provided



 , laimed in 

on entrates at the point

the oarsest (i.e. Alexandro 's one-point) ompa ti ation of DiPerna and Majda [266, Formula (4.18)℄.

(3.61 ), for es

Rm

x

x

to be fully supported

in question. For the ase of

, this observation has been made by

3.2 Various generalizations

R

 òr a% ( ) 

is absolutely ontinuous and





: where

d 

Rm

 ò Y( ;  ;

a.a.

xò :

:

and

)

d  (x) #

a.a.

xò :

Rm

Rm

 x (d s )

¥

1%sp

¡ 0;

(3.61b) (3.61 )

"1

 x (d s )

¤X

X

Ë 155

X

Rm

 x (d s ) ;

(3.61d)

 with respe t to the Lebesgue measure. is the density of 

á(ii). The rst

Proof. (Partly by Kruºík [457, 458℄.) Let us start with the impli ation (i)

part of (3.61b) with (3.61a) has been already shown in Theorem 3.26. To show that

P m  x (d s ) R

¡ 0 for a.a. x ò , let us realize that the Lebesgue measure is absolutely 

ontinuous with respe t to  , having the density d ò L ( ;  ) given by44 1



d  (x) # P dx Ai

# PA i PRm

x (d s )  (d x )

:

(3.62)

A  # {x ò ; PRm

 x (d s )

# 0: This proved (3.61b).

ò Yp ( ;

Let us take a Young measure

1%sp

Rm

This density inevitably vanishes on the set

p "1  (1% s  )

 x (d s )

X

Rm

# 0} so that A   #

) generated by a sequen e whi h attains

 ;  ) in question and dene the absolutely Ž 1 measure  by means of the density d  Ž ò L ( ) given by

the DiPerna-Majda measure (

d Ž (x) # 1 % X

Rm

Using (3.34) and (3.53) with passage for

rÙ

s

 

p

ontinuous

x (d s ) :

v(s) # (1%sp )v r (s) with v r from (3.38) and making a limit

, we obtain by the Lebesgue dominated- onvergen e theorem 1.30

the identity

Ž (A) # A % X

X

Rm

A

for any Borel subset

s

 

x (d s )d x

p

#X

A

X

Rm

 x (d s )  (d x )

#  (A)

 . In parti ular,   A ’ , whi h shows that Ž # 

is absolutely

ontinuous, as laimed in (3.61 ). Denoting the Lebesgue measure by Nikodým derivative45

d  # 

By the very denition of  , we have

P m R

p "1  (1% s  )

0 ,

we an use the formula for the Radon-

d d d # : d d d 0

(3.63)

0

d /d # PRm  x (ds). By (3.62), we have d /d # 0

x (d s ). Plugging it into (3.63) just gives (3.61d).

à(ii). Let us put

Let us go on to the onverse impli ation, i.e. (i)

x (d s )

# d  (x) ”X

44

This formula follows from (3.53) with

45

See Halmos [374℄ for details.

Rm

v # 1.

 x (d s )•

"1 [   m ℄(ds) x R

1%sp

:

(3.64)

Ë

156

3 Young Measures and Their Generalizations

"1

# d  (x) „PRm  x (ds)… PRm (1%sp )"  x (ds) # 1 as a dire t onsequen e of (3.61d). Also, x ÜÙ x is weakly measurable46 so that ò Y( ; Rm ). Moreover, PRm sp x (ds) # PRm (1%sp ) x (ds) " 1 # " d  (x) „PRm  x (ds)… PRm x (ds) # d  (x) " 1, whi h belongs to L ( ) as a fun tion p m of x . Therefore even ò Y ( ; R ). x (d s )

x is positive and PRm

Obviously,

1

*

1

Besides,

 ;  ) if tested by fun tions p. Indeed, for g ò C( ) and v ò C p (Rm ) one has

gives the same result as (

stri tly less that

X X

Rm

1

with a growth



x (d s ) d x

g(x)v(s)

#X

g(x)

X

Rm

#X

X

Rm

v(s)

g(x)

d  (x) P m R

v(s)  x (d s ) d x  x (d s ) 1% s  p

 x (d s )  (d x )

1%sp

#X



X R

Rm

g(x)v 0 (s)  x (ds)(dx)

v 0 ò C( R Rm ) being a ontinuous extension of v0 ò C0 (Rm ) dened by v0 (s) # v(s)/(1%sp ); note that v 0 vanishes on the remainder R Rm \ Rm . Let us now take the sequen e { u k } k òN onstru ted in Proposition 3.22, i.e. it generp 1 and simultaneously the set { u k  ; k ò N} is relatively weakly ompa t in L ( ); ates

with

f. Remark 3.23. Our aim is now to modify this sequen e suitably to attain the original

 ;  ). both

DiPerna-Majda measure ( As

R

is separable,

therefore, for every

Pl #

{



j1 l V

j J(l) l } j #1 of

j2 l # i diam( )   l

m

and

R

Rm Rm \

are metrizable ompa t sets and

N, there exist nite partitions Pl # il Ii#l of

Rm su h that il1 V il2 # ; 1 ¢ i   i ¢ I l

l ò

R

R



{

\



1

}

( )



1

and

( ) and

2

j l are measurable with

; 1 ¢ j1   j2 ¢ J(l) 1/l and diam( lj )   1/l for all i and j. Besides, we may suppose that, for any l ò N, the partitions P l %1 and P l %1 are respe tively renements of the partitions P l i and P l and that int( ) #Ö  for all i . We shall denote by v 0 the ontinuous extension l m m of v 0 ò R on R R , i.e. v 0 ò C ( R R ). We an dene i and moreover all and l







a lij # X

il

Let us hoose

X

,

46 47

R

Rm

1 ¢ i ¢ I(l); 1 ¢ j ¢ J(l):

x ij ò int( il ), 1 ¢ i ¢ I(l), 1 ¢ j ¢ J(l), x ij1 #Ö x ij2 , j1 #Ö j2 , s j ò

 òr a( ,

l a measure 

X  x (d s )  (d x ) ; j l



R

Rm

I(l) J(l)

X

v0 (s)  (ds)g(x) (dx) % H H v 0 (s j )g(x ij )a lij

Rm

i #1 j #1

In view of (3.64), this follows from the measurability of

R

C( ); v0 ò R } is dense in C( ,



l j and dene

) by the following formula47

g(x)v 0 (s)  l (dxds) # X

We used also the fa ts that

(3.65)

m

is a Borel subset in

m RR

).

d 

R

R

and the weak

m

*

(3.66)

measurability of

and that the linear hull {

g

.

‹ v ; g ò 0

3.2 Various generalizations

for any

,(

l g ò C( ) and any v0 ò R . In other words, 

aggregates the part supported on



R

R

m

\

Rm

Ë 157

) so that it has the form

I(l) J(l)

l   l # 

,Rm % H H a ij Æ i #1 j #1

(

x ij ; s j )

;

 òDMR ( ; Rm ) orresponds to ( ;  ) ò DMR ( ; Rm ) in question. m is dense in m Now let us take l ò N xed. As R R R , for any j there is a sek k m m su h that lim quen e { s } k òN ’ R k Ù s j # s j in R R , whi h means pre isely j p

where 

p







limkÙ v (s kj ) # v (s j ) for any v ò R . Inevitably, limkÙ s kj  # %. We an k dene neighbourhoods N of points x ij for k ò N, 1 ¢ i ¢ I ( l ) and 1 ¢ j ¢ J ( l ) by ij k l N ij # x ò ; x " x ij    (a ij /s kj p B(1)) n  where B(1) is the Lebesgue measure of the k k n unit ball in R . Let us note that, sin e  s  Ù % for k Ù , N are pairwise disjoint j ij k i and N ’ , 1 ¢ i ¢ I ( l ) and 1 ¢ j ¢ J ( l ) whenever k is large enough. ij l

that



0

0

0



1/



u

Let us now modify the sequen e { k } k òN by putting

u lk (x) # ® 

I(l)

J(l)

x ò \ U i#1 U j#1 N ijk ; k if x ò N : ij

u k (x) s kj

if

This pro edure is illustrated on Figure 3.10 whi h ounts

m # n # 1,

R

R Ê R€ & {

},

% ; "}

and the limit energy supported on both omponents of the remainder {  spatially homogeneous with the densities equal to

repla ements

a p b:

u lk

uk



ak

S

1/

p

b/k

1

0

"ak Fig. 3.10:

A modi ation of a sequen e {



0

1/

1/l

p

u k }kòN sending a pres ribed energy to the reminder &.

Now we are to prove that the modied sequen e {

 l in the u lk }kòN ’ L p ( ; Rm ) attains  

sense that

lim X v( u lk (x))g(x) dx k Ù



for any

#X

,

R

Rm

g(x)v 0 (s)  l (dxds)

g ò C( ) and v(s) # v0 (s)(1%sp ) with v0 ò R. We have

158

Ë

3 Young Measures and Their Generalizations

lim X v( u lk (x))g(x) dx k Ù



# lim

k Ù

”X Il Jl

\Ui#1 Uj#1 N ijk ( )

( )

g(x)v0 (u k (x))(1%u k (x)p ) dx I(l) J(l)

% H H X g(x)v (s kj )(1%s kj p ) dx• 0

i #1 j #1 k N ij

#X

X

I(l) J(l)

v0 (s)  x (ds)g(x) (dx) % H H v 0 (s j )g(x ij )a lij

Rm

i #1 j #1

N ijk  # a lij /s kj p , whi h is implied by the fa t that the volume of the ball of the n n radius r in the spa e R is given by the formula B (1) r and therefore be ause 

lim N ik j (1%s kj p ) # a lij ;

k Ù

;

and be ause48

lim X k Ù N ijk

g(x)v0 (u k (x))(1%u k (x)p ) dx # 0:

(3.67)

 Ê ( ;  ) for l Ù

l Now we want to show that  approa hes  for any

 g ò C( ) and any v0 ò R , we have

lim X l Ù ,

m RR

Indeed, denoting by and

g(x)v 0 (s)  l (dxds) # X



X R

Rm

v 0 (s)  x (ds)g(x) (dx):

M v 0 ; M g : R% Ù R% respe tively the moduli of ontinuity49 of v 0

g and using (3.61a) and (3.65)(3.66), this onvergen e follows from:

!! !!X !! ! ,

m RR

!!

g(x)v 0 (s)  l (dxds) " X

X

!! I ( l ) J ( l ) !! H H v ( s j ) g ( x ) a l 0 ij ij !! ! i #1 j #1

"X



# ¢

48

 in the sense that,



m RR



I(l) J(l)

H H X X  v 0 ( s j ) g ( x ij ) j i l i #1 j #1 l

To show (3.67), one is to realize that {

X

v 0 (s)  x (ds)g(x) (dx)!!!! ! !!

m m R R \R

v 0 (s)  x (ds)g(x) (dx)!!!! !

" v (s)g(x)  x (ds) (dx) 0

u k p ; k ò

N

} is relatively weakly ompa t in

L1 ( ), hen e also

equi- ontinuous due to the Dunford-Pettis theorem 1.28(iii). Moreover, it is only a simple observation

# limkÙ a lij /s kj p # 0: Let us take " ¡ 0. Due to the equi- ontinuity we an nd k ò N su h that, for any k £ k , we have PN k (1%u k (x)p ) dx   " g "C0 v "C0 Rm and then we have

that

limkÙ





N ijk 



0

the following estimate 

0

P k N ij

1

ij

g(x)v0 (u k (x))(1%u k

(

p ( x ) ) d x 

¢ g C0 v (

)

)

0

0

1

(

)

%u k (x)p ) dx  

C0 (Rm ) PN k (1 ij

" ; whi h proves (3.67). This means that  v 0 ( s 1 ) " v 0 ( s 2 ) ¢ M v (  ( s 1 ; s 2 )) and  g ( x 1 ) " g ( x 2 ) ¢ M g ( x 1 " x 2 ), where 0 (-; -) denotes some metri indu ing the (metrizable) ompa t topology of R m and, of ourse, lim"Ù0% M v 0 (") # lim"Ù0% M g (") # 0 be ause v 0 and g are uniformly ontinuous.

49

R

3.2 Various generalizations

¢

I(l) J(l)

H H X X  M v 0 Œ j i l i #1 j #1 l

¢ M v 0 Œ

1 l

 g

C ( )

1 l

% MgŒ

 g C 0 ( i ) l

% Mg Œ

1 l

1 l

Ë 159

 v 0 0 j ‘  x (d s )  (d x ) C ( l)

 v 0 C 0 (Rm ) ‘  ( )

Ù 0

l Ù ):

(for

Now we are in the situation that

lim lim X v( u lk (x))g(x) dx l Ù k Ù



#X



X R

Rm

v 0 (s)  x (ds)g(x)(dx)

g ò C( ) and v(s) # v0 (s)(1%sp ) with v0 ò R . By a suitable diagonalization l p m pro edure, one an hoose the net { u }  ò  ’ L ( ; R ) su h that k for any





l

lim X v( u k (x))g(x) dx # X  ò







As the whole net {

l

u k  } ò 

% P P {

l



R

v 0 (s)  x (ds)g(x)(dx) :

Rm

L p ( ; Rm )

and both

C( )

and

R

are sep-

N dire ted by the standard ordering. The men-

 #

u k } ò

m m  x (d s )  (d x ) R R \R p m ). in L ( ;

u k }kòN

R

is boundedin

arable, we an even suppose tioned boundedness of {

X

follows from the estimate

¢ C p % P P R Rm

p

u lk L p 

m  x (d s )  (d x ) \R

(

;Rm )

¢ u k pLp Rm (

;

)

  %, where C bounds Å

Remark 3.29 (Embedding of Yp ( ; Rm ) into DiPerna-Majda measures). Having ò Yp ( ; Rm ), we an dene an absolutely ontinuous  ò r a( ) by the density d  ò L ( ) given by p (1% s  ) x (d s ) d  (x) # 1 % X sp x (ds) and  x (ds) # (3.68)

1

d  (x)

Rm

Rm while  vanishes on R Rm Rm . It is easy to see that  ;  satises (3.61), so p m thanks to Proposition 3.28. Obviously, (3.60) is also satised that  ;  ò DM R ; R p m  ò DM so that  ;  is p -non on entrating by Lemma 3.27. The measure  R ; R m  is then given on , R by the formula

orresponding to  ; on

\

(

)

(

(

(

)

)



)

(

(

)



)



 (d x d s ) # (1% s  p ) 

,(

x (d s ) d x ;

Rm Rm . Besides,  Ê

(3.69)

 ;  ) give the same result as when tested by fun tions with the growth stri tly less than p in the sense m that, for any v ò C p (R ) and g ò C ( ), it holds

while it vanishes on the remainder

R



\

)

(



X X

Rm

v(s)

x (d s ) g ( x ) d x

# #

X

,

X X



R

Rm

m RR

g(x)v(s)  (d x d s )  1%sp v(s) 1%sp

 x (d s ) g ( x )  (d x ) :

In our last theorem we will show that every DiPerna-Majda measure (

(3.70)

 ;  ) whi h does

Ž Ž not satisfy (3.60) an be modied to a DiPerna-Majda measure (  ;  ) satisfying (3.60)

Ë

160

3 Young Measures and Their Generalizations

Ž

 ;  ) and (Ž ;  ) give the same results when tested by fun tions with the growth less than p in the sense that and, simultaneously, both (

X X



for any sure

v(s)

p m R R 1% s 

 x (d s ) g ( x )  (d x )

#X



v(s)

X

p m R R 1% s 



Ž

x (d s ) g ( x ) 

Ž (dx):

(3.71)

v ò C p (Rm ) and g ò C( ). It is natural to address su h DiPerna-Majda meaa p -non on entrating modi ation of (  ;  ). The following assertion es

Ž Ž ( ;  )

tablishes an expli it formula for it.

Proposition 3.30 (Non on entrating modi ation  Kruºík [457℄). Let the ring R be p m separable and let (  ;  ) ò DMR ( ; R ) be given. Furthermore, let us dene an absoŽ lutely ontinuous  òr a( ) by means of the density d  Ž ò L ( ) given by

1



d Ž (x) # ” X

Rm

and 

Ž

ò Lw ( ; ; r a(  *

R



Rm

))

 x (d s )

1%sp

Ž

x (d s )

#



Rm

)

Rm

 x (d s )

 x Rm
(d s ) X

Rm

Ž ;  Ž ) ò DM p ( ; R

X

(3.72a)

by



Then ( 

"1

•

is a

 x (d s )

:

(3.72b)

p-non on entrating modi ation of ( ;  ).

Proof. Let { u k } k òN be a generating sequen e of (  ;  ) bounded in

L p ( ; Rm ).

On

R ò ò Yp ( ; Rm ) denes a p-non on entrating DiPerna-Majda measure, p Ž Ž m Ž Ž let us denote it by (  ;  ) ò DM ( ; R ), whi h satises (3.70) with (  ;  ) in pla e of R Yp ( ;

the other hand, this sequen e generates

m ) satisfying (3.70). Due to Re-

mark 3.29 this



 ;  ). Altogether we an see that (3.71) is fullled. The formula (3.72a) is just (3.61d) for  #  Ž whi h was veried in the proof of Proposition 3.28. Then (3.72b) an be obtained  Ž "1 p by putting x from (3.64) into the formula  x # d Ž ( x )(1% s  ) x (d s ); f. (3.68). 

(

As the set

p m DMR ( ; R ) is onvex and lo ally ompa t ( f. Example 3.70 below),

its extreme points and rays are of importan e (see Klee's theorem 1.15). We an see that the rays are intimately onne ted with on entration ee ts. Let us still introdu e the shorthand notation

rem( ,

R

Rm # )

% ( ,

 òr a †



R

Rm ; supp  ’ , )

(

)



(

R

Rm Rm \

)‡

:

Ë 161

3.2 Various generalizations

p m Proposition 3.31 (Geometri al properties of DMR ( ; R ); mostly by Kruºík [458℄). p m  òDM ( ; R ) is an extreme point if and only if   # i ( u ) for some u ò L p ( ; R m ). (i)  R p m In parti ular, extreme DiPerna-Majda measures in DMR ( ; R ) must be p -non on





entrating.

p p m  % t  ; t ¡ 0} with some   òDM ( ; R m ) ( ; R ) has the form {  DMR R m  òrem( , R R ). In parti ular, there is no ray in the set of p -non on entraand  p p m m ting DiPerna-Majda measures from DM ( ; R ). Also, any ray in DM ( ; R ) is R R

(ii) Any ray in



0



0











p-non on entrating

omposed from DiPerna-Majda measures that have the same modi ation.

p m ( ; R ). DMR p  # i ( u ) for some  % t  ; t ¡ 0} in DM ( ; R m ) is extreme if and only if  A ray {  R p m m m u ò L ( ; R ) and  # Æ x s for some x ò and s ò R R \ R .

(iii) There is no straight line in (iv)

0





0





(

; )



Proof. (Kruºík [458℄.) First, let us realize that an extreme point an be found only

 ò DMR ( ; Rm ) not p-non #

on entrating we an write  %  with #   ,Rm and  ò rem( , R Rm )  lies on the ray {  Ž % t  ; t ¡ 0} and therefore it nonvanishing. It follows that su h  among

p-non on entrating

p

measures. Namely, for any 

 Ž



 Ž









annot be an extreme point.

 is p-non on entrating but not an extreme point in DMR ( ; Rm ), then  1,  2 ò not an extreme point in the p -non on entrating measures, i.e. there exist  p m  1 #Ö   2 su h that   # ( 1 %   2 )/2. The ( ; R ), both p -non on entrating,  DMR p



If  is





onverse impli ation is trivial.

Û  : Yp ( ; Rm ) Û { ò p-non on entrating} ( f. (3.69)), the extreme points in Yp ( ; Rm )

Sin e there is a one-to-one ane mapping

p DMR ( ;

R

m );  

p-non on entrating DiPerna-Majda

are thus mapped uniquely onto extreme points in measures, hen e onto extreme points in in

p m ( ; R ). However, the extreme points DMR

Yp ( ; Rm ) has been already des ribed by Proposition 3.24(i), whi h proves (i). p m By Proposition 3.24(ii), Y ( ; R ) does not ontain a ray and therefore also p -non-

on entrating DiPerna-Majda measures annot ontain a ray.

R

p  2 %(1" t )  1  ;  2 òDMR ( ; m ) and suppose that  (t) # t  p m i #  Ž %    and belongs to DM ( ; ) for any t ¡ 0. Let us make the de omposition  i i R  Ž ( t ) are p -non on entrating and  ;    ( t ) ò rem( ,  Ž (t) %    ( t ) where  Ž ;   (t) #  i i m ), i # 1 ; 2. It implies that   Ž (t) # t   Ž2 % (1" t )   Ž1 and    (t) # t   2 % (1" t )   1 for any R To prove (ii), let us take  1



R



R

t ¡ 0. We saw that there in no ray in p-non on entrating DiPerna-Majda measures, so  Ž1 #   Ž2 . Thus we obtain that   (t) #   Ž1 % t   2 % (1" t )  1 #   1 % t( 2 "   1 ). Putting that   # 2 "   1 and  0 #   1 , we get the desired result.  p 0 % t  ; t ò R} be a line in DM ( ; R m ). To prove (iii), let us suppose that L # {  R p 0 % t  ; t ¡ 0} and {  0 % t  ; t   0} would be rays in DM ( ; R m ), whi h Then both {  R m  and "   belong to rem( , implies by the point (ii) that both  R R ). This is possible  vanishes, so that L is a singleton. only if  































Ë

162

3 Young Measures and Their Generalizations



Let us go on to the point (iv). The end point  0 of the ray in question belongs

p m DMR ( ; R ) and therefore50 the extreme ray must rise from an extreme point of p m DMR ( ; R ). By the point (i),  is of the form i(u) for some u ò L p ( ; Rm ). p  % t  ; t ¡ 0} in DM ( ; R m ) is extreme if and only Due to the denition, a ray {  R p  ;  òDM ( ; R m ): if the following impli ation holds for  R to



0





0



1

2



;t ò R% ;r ò (0; 1) su h that  % t  # r  % (1"r ) âá :r ò (0; 1) ;t ò R% su h that  % t  # r  % (1"r) : 

0

0

0

0

0

1

0

2



0

1

2

m  Ž1 %   1 and  2 #   Ž2 %   2 with  1 ;   2 òrem( ,  # R R ). Using this Ž Ž 1 #  2 #   0 (note that representation, we obtain from the above impli ation used for  m) 0 #   Ž0 ) that it holds for  1 ;   2 òrem( ,  R R

We an write  1

















;t ò R% ;r ò (0; 1) su h that t  # r  % (1"r ) âá :r ò (0; 1) ;t ò R% su h that t  # r  % (1"r) : 

0

0



0

0



0

1



2





1

2

t  ; t ¡ 0} is an extreme ray in rem( ,

 The last impli ation says pre isely that { 





R

Rm , )

Å

whi h is possible if and only if  is the Dira measure.51 

Remark 3.32 (More detailed representations).

In parti ular ases, a more detailed

representation of DiPerna-Majda measures an be established. In ase

R

p # 1 u

and

from (3.45), Alibert and Bou hitté [15, Thm. 2.3℄ proved that any sequen e { k } k òN

L1 ( ; Rm ) ontains a subsequen e (denoted, for simpli ity, by the same 1 m m "1 ) , r a% ( ), one has indi es) su h that, for some ( ; ;  ) ò Y ( ; R ) , Y( ;  ; S bounded in









h Ž u k Ù h DZ % (h2 DZ 

) -



weakly* in 

r a( )

h ò Car1 ( ; Rm ) su h that h(x ; s)/(1%s) admits a ontinuous extension on

, R Rm and h2 : , S m"1 Ù R is dened by h2 (x ; s) :# limtÙ s"1 h(x ; ts); note that for h # g ‹ v we get just h 2 # g ‹ v 2 with v 2 from (3.45). One an verify 1 m that, for a DiPerna-Majda measure (  ;  ) ò DMR ( ; R ), the Alibert-Bou hitté rep"1 from (3.54), resentation ( ; ;  ) is given by x ( s ) :# a ( x )  x ( s ) where s ÜÙ s m "1 Ù m m denotes the natural homeomorphism S R R \ R , and  :# a , where  a(x) :# P Rm Rm x (ds); note that is dened  -a.e. By Lemma 3.27(i), ( ;  ) is pfor any

























R

\

non on entrating if and only if Another representation of

 # 0.52  ò DM1R ( ; Rm ) 



 ) with ; ;  ò Y ( ; R  ; S m "1 ) being the Fonse a measure on

Y( ;  1

(



50

with

R

 È  ;  ) ò r a% ( ) , to  , is suggested by

from (3.45), namely

m ) given again by (3.54) and with

orresponding

(



At this point the reader is referred to Köthe [436, Se t. 25℄.

51

Cf. Köthe [436, Se t. 25℄.

52

This was also proved dire tly by Alibert and Bou hitté [15, Thm. 2.6℄. On the other hand, no attain-

ability of su h pairs (

 ;  ) has been studied in [15℄.

3.2 Various generalizations

Ë 163

the formula (3.112) bellow. Su h representation has been proposed by Fonse a, Müller and Pedregal [324℄.

p ò [1; %), a similar de omposition, namely ( 1 ; 2 ; ) ò , Y( ; ; S m"1 ) , r a% ( ) with 1x ò r a% (Rm ), has been proposed

For a general

Lw ( ; r a(Rm )) 

*



by DiPerna and Majda [266, Thm. 1℄ who showed the onvergen e

h Ž u k Ù (1%0 )h1 DZ

1

% (h DZ

2

2



) -

weakly* in

r a( )

h ò Carp ( ; Rm ) in the form h(x ; s) # h1 (x ; s)(1%sp ) % h2 (x ; s/s)sp h1 (x ; -) ò C0 (Rm ), where 0 ò L1 ( ) denotes the absolutely ontinuous part of .

for any

with

Remark 3.33 (Testing dis ontinuously a

ording A. Kaªamajska [411, 412℄). A generam Ù R only pie ewise ontinuous with a-priori lization for fun tions h with h ( x ; -) : R xed hypersurfa es of possible dis ontinuities has been proposed in [411, 412℄, assuming some nite partition of taken a separable ring

Rm on some open subdomains ( alled bri ks) and then

R of fun tions whose restri tions on these bri ks is ontinuous.

Thus extended DiPerna-Majda measures are supported on ompa ti ations of ea h bri ks separately. For further investigation of the set of all DiPerna-Majda measures we refer to Examples 3.47 and 3.70 below.

3.2.d

Fonse a's extension of L1 -spa es

It is worth ompleting the previous generalizations by a similar onstru tion by Fonse a [316℄ who developed an extension of

L1 ( ; Rm ) whi h an handle positively ho-

mogeneous integrands. As su h integrands form a separable (in a natural topology) linear subspa e, we an work in terms of sequen es.

Theorem 3.34 (I. Fonse a [316℄, here modied). Let {u k }kòN be a bounded sequen e in L ( ; Rm ). Then there is a subsequen e, denoted again by {u k }kòN for simpli ity, a mea% m " ) (where S m " denotes again the sure  òr a ( ), and a Young measure  ò Y( ;  ; S 1

1

unit sphere in

R

1

m ) su h that

lim X h(x ; u k (x)) dx k Ù

#X

X

S m"1

h(x ; s)  x (ds)(dx) ;

(3.73)

h ò C0 ( , Rm ) su h that h(x ; s) # 0 for x ò :# bd( ) and h(x ; as) # ah(x ; s) m % for any ( x ; s ; a ) ò , R , R . for any



Sket h of the proof. Let us dene the embedding

i ( u ) ; h 0

#

X h 0 œx ;



i : L1 ( ; Rm ) Ù r a( , S m"1 ) by

u(x)   u ( x )d x u(x)

(3.74)

164

Ë

3 Young Measures and Their Generalizations

h0 ò C0 ( , S m"1 ), i.e. the set of ontinuous fun tions vanishing on , i(u) is a positive Radon measure with the variation equal to P i ( u )( d x d s ) # u L1 ;Rm . Having a bounded sequen e {u k }kòN in L1 ( ; Rm ),

, S m"1 m "1 ) is bounded as well, so that we an sele t a weakly* its image via i in r a( , S for any

S m"1 . Note that

(

)

onvergent subsequen e:

i(u k ) Ù 

weakly* in

r a( , S m" ) : 1

(3.75)

Let us denote53

F( ; Rm ) # †  òr a( , S m" ); 1

;{u k }kòN ’ L ( ; Rm ) : (3:75) holds ‡ : 1

Let us dene the mapping

T : C(S m"1 ) Ù r a( ) Ê C0 ( )

*



T v0 ; g 

#  ; g ‹ v

0

(3.76)

by

(3.77)

v0 ò C(S m"1 ) and g ò C0 ( ). Likewise (3.49), we an estimate here < T v0 ; g> ¢ v 0 C S m"1 P  g ( x )  (d x ), where  # w*- lim k Ù  u k  in r a( ); note that this limit

m "1 ). does exist thanks to (3.75) tested by the fun tions of the form g ‹ 1 ò C 0 ( , S Analogously as in the proof of Theorem 3.25, we an dene Tv 0 ò L ( ;  ) and estabm "1 ) su h that lish a Young measure  ò Y( ;  ; S for any (





)



X X

S m"1

 ; g ‹ v 0 # X g ( x )[ Tv 0 ℄( x )  (d x ); g(x)v0 (s)  x (ds)(dx) # 

(3.78)

 vanishes on a set of positive Lebesgue C0 ( ) ‹ C(S m"1 ) m "1 ), (3.78) also implies P P  ; h 0 > for any h 0 ò in C 0 ( , S h (x ; s)  x (ds)(dx) # 0 for s # 0 ; F

f. also (3.50) and realize that it holds also if

measure, whi h an a tually happen here. By the density argument of

we an write, when using (3.74), (3.75), and the identity

lim X h(x ; u k (x)) dx k Ù

# lim

X h 0 Œx ; k Ù

#  ; h As su h

h

0



h ,S m"1 # h0 ,

u k (x)  u k ( x )d x # lim i ( u k ) ; h 0 u k (x) kÙ

#X

h0 (x ; s)  x (ds)(dx) X

S m"1



#X

h(x ; s)  x (ds)(dx): X

S m"1

an range all ontinuous positively homogeneous integrands, (3.73) has

Å

been proved.

F( ; Rm )

 ;  ) ò r a( ) , Lw ( ; ; S m"1 ) 1 m generated in the sense of (3.73) by some sequen e in L ( ; R ); the elements of both F( ; Rm ) and F( ; Rm ) will be addressed as Fonse a measures. Let us denote by

53

Note that any sequen e {

u k }kòN



the set of all pairs (

satisfying (3.75) must be bounded in

*

L1 ( ;

Rm

).

3.2 Various generalizations

Remark 3.35 (Chara terization of Fonse a measures).

Ë 165

We have the simple omplete

hara terization:54

F( ; Rm ) #

(  ;  );

 òr a% ( );

Also note that Theorem 3.34 determines  only

 ò Y( ;  ; S m "1 ) :

-a.e.

(3.79)

so that it is arbitrary on



\

supp() whi h may be nonempty; this is a dieren e from the DiPerna-Majda measures where always supp(  ) # .

Remark 3.36 (Non on entrating Fonse a measures).

Likewise we did for DiPerna-

Majda measures, we an dene also here the notion of

 ;  ), whi h will

1-non on entrating Fonse a

u  ;  ) in the sense (3.73), su h that the set {u k ; k ò N} is relatively weakly ompa t in L1 ( ). Here one an show that ( ;  ) ò F( ; Rm ) 1-non on entrating means pre isely  absolutely ontinuous with respe t to the Lebesgue measure. However, ontrary to

measure (

indi ate the existen e of a sequen e { k } k òN , generating

(

DiPerna-Majda measures, even sequen es whose energy is not relatively weakly om-

1-non on entrating Fonse a measure. Indeed, the sequen e from Figure 3.8 (with p # 1) generates the homogeneous Fonse a measure " dx and  # Æ whi h an equally be generated by a on(  ;  ) given by  (d x ) # ab x " " stant sequen e u k # ab or also by an sequen e os illating between 0 and 2 ab pa t in

L1 ( )

an onverge to a

1

1

1

1

with the ratio 1:1, et . Thus we an see that the Fonse a measures an re ord mu h lesser information than the DiPerna-Majda measures, indeed.

Remark 3.37 (Properties of F( ; Rm )). In view of Remark 3.35, we an observe that m % m " ). In fa t, the triple F( ; R m ) ; r a( , S m " ) ; i  with simply F( ; R ) # r a ( , S i from (3.74) forms a onvex - ompa ti ation of L ( ); of ourse, r a( , S m" ) is m

onsidered in the weak* topology. Obviously, F( ; R ) is also lo ally (sequentially) 1

1

1

1

ompa t, and thus it must ontain a ray. It is evident that every ray has just the form

0 ;   òF( ; R m ),   #Ö 0. Also, F( ; R m ) annot ontain any  % t  ; t ¡ 0} with some 

{0

line. For further investigations of the set of all Fonse a's measures we refer to Examples 3.49 and 3.72.

54

The in lusion 

’ just follows from Fonse a's theorem 3.34. The onverse in lusion in (3.79) must

be proved by a dire t onstru tion: taking a partition a pie e-wise homogenization of a given (

k

 ;  ), we

Pk

of

as in the proof of Theorem 3.3 and making  k ò r a% ( )

get some pie e-wise homogeneous

ò Y( ; S m" ) not uniquely dened, however, be ause we must rst use a suitable extension Ê ( ;  ) to make possible testing by only pie e-wise ontinuous fun tions. Taking the Young k ò Y ( ; Rm ) dened by k ( s ) #  k ( x )"m  k (  k ( x )" s ) for s from the sphere of the radius measure x x k  (x) and vanishing elsewhere (if  k (x) # 0, then kx # Æ ), we an onstru t the sequen e onverging 

and of

1



1



to

k

(and also to (

k ; 

k

0

) if tested by positively homogeneous integrands) as in the Steps 2b- of the

proof of Theorem 3.6. Then passing

k Ù % and making a suitable diagonalization pro edure, we get  ;  ).

the sought sequen e onverging to (

166

Ë

3.3

A lass of onvex ompa ti ations of balls in

3 Young Measures and Their Generalizations

L p -spa es

In Se tions 3.1 and 3.2 we ould see in fa t several on rete onvex ompa ti ations of (bounded subsets in) Lebesgue spa es. The reader might anti ipate that all of them (and also many others) an be overed by a unied way by a suitable general setting. The aim of this se tion is just to onstru t a su iently large lass of onvex ompa ti ations that will in lude the previous ones.

3.3.a

Generalized Young fun tionals YHp % ( ; S) ;

U , the topologi al spa e % ò R% in L p ( ; S) with p ò [1; %℄, i.e.

Let us rst onsider radius

U # B% #

† uòL

p

(

to be ompa tied, as the ball of the

; S);



u L p

(

¢ %‡ ;

;S)

(3.80)

where S will again denote a separable Bana h spa e, though mostly we will use merely

S # Rm

in appli ations.

Let us denote by

Carp ( ; S) the linear spa e of all Carathéodory fun tions55 h :

, S Ù R with at most p-growth, i.e. 

p

h(x ; s) ¢

®

a h (x) % b h s S p a h (x) % b h ( s S ) 

with some

a h ò L1 ( ), b h ò

R% and b h ò C R% (



for p ò [1; %) ; for p # % ;

(3.81)

) nonde reasing. We will onsider

Car ( ; S) as a lo ally onvex spa e endowed by the seminorm  - % dened by p



!!

!!

h% # sup !!!!X h(x ; u(x)) dx!!!! : uòB % !

Whenever we will want to emphasize that

(3.82)

!

Car p ( ; S) is endowed with this topology, we

p will write Car% ( ; S) to distinguish it from a ner lo ally onvex topology introdu ed p on

Car ( ; S) later in Se tion 3.4.

Furthermore, we dene the mapping

[

% h℄(u) #

% : Carp ( ; S) Ù C0 (B % ) by

X h ( x ; u ( x )) d x

for h òCarp ( ; S); u ò B % :

(3.83)

Nh : L p ( ; S) Ù L1 ( ) is bounded and ontinuous (see Theorem 1.24), we an see that % h # [ % h ℄(-) is bounded and ontinuous on p p 0 the ball B % in L ( ; S), hen e % a tually maps Car ( ; S) into C ( B % ). Obviously,

As the Nemytski mapping

55

Re all that the adje tive Carathéodory means that

and

h(x ; -) : S Ù

R

are ontinuous for a.a.

x ò .

h(-; s) :

Ù

R

are measurable for all

sòS

3.3 A lass of onvex ompa ti ations of balls in

Lp

-spa es

Ë 167

p

h% # % h C0 B % and the onvergen e h  Ù h in Car% ( ; S) means pre isely p % h  Ù % h in C0 (B % ). Let us note that % is not inje tive, hen e Car% ( ; S) is



(

)

not a Hausdor spa e, f. also (3.7) above. Furthermore, let us dene the embedding

i : L p ( ; S) Ù Carp ( ; S)

*

by

i(u); h For a linear subspa e

#

X h ( x ; u ( x )) d x :

(3.84)

H ’ Carp ( ; S) we dene a linear subspa e FH % ’ C0 (B % ) by ;

FH ; %

# % (H) %  onstants on B % 

(3.85)

e H : B % Ù FH % as the restri tion e H (u) # e(u)FH % of the evaluation mapping e : B % Ù C0 (B % ) . Moreover, the embedding i H : L p ( ; S) Ù H is dened again by the formula (3.84) for h ò H , i.e. i H ( u ) # i ( u ) H is a restri tion of i on H . The weak* p

losure of i H ( B % ) in H will be denoted by Y H % ( ; S), i.e. *

and

;

;

*

*

*

;

p

YH

;

# †  ò H ; ;{u  } ò ’ B % :  #

% ( ; S)

*

Convention 3.38 (Generalized Young fun tionals).

w*-lim

 ò

i H (u  ) ‡ :

The elements of

p

YH

;

(3.86)

% ( ; S) will be

alled generalized Young fun tionals. Let us note that for a spe ial hoi e

H # L1 ( ; C(S)), p # % and S ’ S # Rm

the

%, the generalized Young fun tionals oin ides with Young fun tion-

ball of the radius

als as stated by Convention (3.1), whi h justies the adje tive generalized. Later we will make the meaning of generalized Young fun tionals even a bit wider; f. Convention 3.65 below.

Theorem 3.39.

Let

H be a linear subspa e of Carp ( ; S), p ò [1; %), and B % and FH

;

%

given respe tively by (3.80) and (3.85). Then: (i)

The linear subspa e

FH ; %

of

C0 (B % )

is onvexifying ( f. Denition 2.4) and thus

FH ; % ); FH ; % ; e H ) is a onvex ompa ti ation of B % . *

(M(

(ii) If

H

is endowed with a topology whi h makes it a lo ally onvex spa e and whi h

is ner than that indu ed from

ompa ti ation of mapping

% .

Carp% ( ; S), then (YHp % ( ; S); H ; i H ) forms a onvex *

;

B % whi h is equivalent with (M(FH

% ) ; F H ; % ; e H ) via the adjoint *

;

*

(iii) If one of the following onditions are satised: 1.

S # Rm , p ò (1; %) and H “ C( )‹(Rm )* ontains also the integrand h( x ; s ) # sp ,

or 2.

; M dense in L p ( ; S) :u ò M : h u ò H , where h u (x ; s) # s " u(x) Sp , then the

onvex

ompa ti ation

M(

FH ; % )

is

norm- onsistent;

i.e.

iH

is

a

(strong,weak*)-homeomorphi al embedding. Proof. To show that {

u  } ò

su h that

FH ; %

u1 ; u2 ò B % a net p # %, we an use dire tly

is onvexifying, we must onstru t for any

lim ò e H (u  ) # 12 e H (u ) % 12 e H (u 1

2 ).

If

168

Ë

3 Young Measures and Their Generalizations

the onstru tion from the proof of Theorem 3.3 ( f. also Figure 3.3) adopted for the ase of a separable Bana h spa e S instead of

L ( ; S) from L ( ; S)  B % . 

density of

Rm , while if p ò 1; %

), we an employ56 the

[

p in L ( ; S) in order to approximate

u1 and u2 by some fun tions



Sin e, by the very denition (3.85),

FH ; % always ontains onstants, M(FH ; % ) is a

B % ; f. Proposition 2.5. The point (i) has thus been proved. Let us onsider % restri ted as H Ù F H % . In view of the onsidered topology on H , % is ontinuous and linear. Then a tually % : FH % Ù H . Moreover, % xes B % in the sense that % Ž e H # i H be ause the identity

onvex ompa ti ation of

;

*

*

*

*

;

*

[ %

*

Ž e H ℄(u); h # e H (u); % h # [ % h℄(u) # i H (u); h

h ò H and u ò B % . In parti ular it shows L p -norm topology on B % to the weak* topology on H

holds for every

that

*

the

iH

(3.87)

is ontinuous from

e H : B % Ù FH % *

be ause

;

is ontinuous (thanks to the ontinuity of the respe tive Nemytski mappings) and

% , being linear, is also ontinuous in the weak* topologies. Furthermore, we want to prove that the adjoint operator % : F H Ù H is inje tive and has a weakly* ontinuous inverse if restri ted on M # {  ò F H % ; <  ; 1> # 1}. Indeed, for any  1 ;  2 ò F su h that %  1 # %  2 and <  1 ; 1> # 1 # <  2 ; 1> we H % be ause

*

*

*

*

*

;

*

*

*

;

have the identity

 1 ; % h

%

# %  ; h %  ; 1 # %  ; h %  ; 1 #  ; % h %

*

*

1

1

2

2

2

ò R. Due to the denition (3.85) of FH % , it just means that 1 # 2 . Let us now suppose that we have a net {  } ò in M and  ò M su h that { %   } onverges to %  weakly* in H . This implies hòH

valid for any

*

and

;

*

  ; % h

*

%

# %   ; h %   ; 1 Ù %  ; h %  ; 1 #  ; % h %

*

*

h ò H and ò R, whi h means pre isely that   Ù  weakly* in FH . As M(F H % ) ’ M by the very denition of M(F H % ) and % Ž e H # i H by (3.87), we an onp p

lude that M(F H % ) Ê Y H % ( ; S), the ane homeomorphism M(F H % ) Û YH % ( ; S) being just % . *

for any ;

;

;

*

;

;

;

*

Let us go on to the point (iii). In the rst possibility, the weak* onvergen e

i H (u  ) means that onverges for any g ò C( ; Rm ) whi h is dense in p L ( ; Rm ) Ê L p ( ; Rm ) and also that , being for h(x ; s) # sp equal to p p m u  p L ;Rm , onverges. Sin e 1   p   %, the spa e L ( ; R ) is uniformly onvex, and therefore { u  }  ò  must onverge also in the strong topology thanks to the of





*

(

)

Fan-Gli ksberg theorem.

Ù i H (u) means, in par# Ù # u " u pLp S for any u ò M . i

u

For the se ond possibility, the weak* onvergen e H (  )

u "

ti ular, that 

56

p u L p ( ;S) 

For details we refer to Warga [791, Thm. I.5.18℄.



(

;

)



Lp

3.3 A lass of onvex ompa ti ations of balls in

Ë 169

-spa es

u " u L p S ¢ u  " u L p S % u " u L p S ¢ 2 u " u L p S % " for  ò  large enough (depending on u and "). As we may take u p arbitrarily lose to u and " Ù 0, we get eventually u  Ù u in L ( ; S). Å

Moreover, one an ertainly estimate  ;

(



(

;

)

;

(



)

)

(



;

)





H , we have a large freedom in the hoi e of the topolH . Let us noti e that, by Theorem 3.39, taking a ner p topology on a given H an enlarge only H but not Y H % ( ; S) ’ H . Quite typi ally, parti ular H will be endowed by some norm - H stronger than the seminorm from (3.82), % that means  h  % ¢ C h H for all h ò H and some C ò R xed. Examples of su h norms We have seen that, for a given

ogy of the linear topologi al spa e

*

*

;

will be given later, f. (3.98) or (3.108). The following assertion points out the topology

Carp% ( ; S) as the a tually limit topology at least if one onsiders the " subspa es H ontaining % ({ onstants on B % }) # { g ‹ 1 ò L ( ; C (S))}.57 indu ed from

1

Theorem 3.40.

1

0

H be a linear subspa e of Carp ( ; S) endowed with a lo ally onvex p topology ner than that indu ed from Car% ( ; S), p ò [1 ; %), and F H % be given by p (3.85). Moreover, let H be the losure of H in Car% ( ; S). Then: p p (i) Y H % ( ; S) Ê YH % ( ; S). Let

;

;

(ii) If

;

H 

Carp ( ; S) endowed with a topology ner than H “ H “ {g ‹ 1 ò L ( ; C (S))} but H #Ö H , then

is another linear subspa e of

p that indu ed from Car% ( ; S) and p p Yp ( ; S) ± YH ; % ( ; S). H;%

Proof. Let us onsider the in lusion

1

0





H ’ H

Q : H Ù H . This H and H . Therefore, the

as a linear operator

operator is ontinuous thanks to the hosen topologies on

H into H . Let us show that Q Ž i H # i H . Indeed, we have the obvious identity

adjoint operator

Q

*

maps

*

*

*

Q

*

i H (u); h # i H (u); Qh # % h
(u) #

h ò H and u ò B % . p p to prove that Y H % ( ; S) Ê YH % ( ; S).

i H (u); h

valid for every We want

p YH ; % ( ; S)

;

;

By Theorem 3.39(ii) we have

Ê M(FH % ). Thanks to the proper topology of Carp% ( ; S), FH % is just the losure of F H % in C ( B % ), and therefore by Theorem 2.8(ii) we obtain M(F H % ) Ê M(F H % ). p Using again Theorem 3.39(ii) we get eventually M(F H % ) Ê Y ( ; S). This proved the H % ;

;

0

;

;

;

;

;

point (i).

“ FH % p #Ö FH % . The losedness of H in Car% ( ; S) means pre isely the losedness

Let us go on to (ii). By the assumptions and denition (3.85), we have F p H;% but

Fp H;%

;

;

S#

Rm

, one an show as before that % h is onstant u ò B % ; :a.a. x ò : u(x) S ¢ r} if and only if [ % h℄(x ; -) is onstant on the ball B r ’ S for a.a. x ò . Then passing r Ù , we get that % h is onstant on B % if and only if [ % h℄(x ; -) is onstant on S. If S is not lo ally ompa t, the maximum and the minimum of h ( x ; -) need not be attained but we

an work with " -a

ura y, as well. If p # , it su es to take r # % .

57

This an be shown similarly as (3.7): In ase

on {

170 of

Ë

3 Young Measures and Their Generalizations

FH ; % in C0 (B % ). Then M(Fp H;%) ±

get

p ( ; S) H;%

FH ; % ) by Theorem 2.8(iii). By Theorem 3.39, we

M(

± YHp % ( ; S) Ê YHp % ( ; S).

Yp

Å

;

;

The following assertion shows that the lass of onvex ompa ti ations built up by the above pro edure is fairly ri h although, as we will see later in Theorem 3.42, it still does not ontain all onvex ompa ti ations of

Theorem 3.41.

B% . N

p ò [1; %). Then there exist at least 22 dierent onvex ompa tip  ations of B % in the form M(F H % ) with H being a linear subspa e of Car ( ; S). Let

;

S is home N, the ardinality of S is surely at least58 the ardinality of N, whi h N 0 2 is known59 as 2 . Then we an take s ; s 0 ò S \ S and put R ( s ) # { v 0 ò C (S); ; v 0 ò 0 C( S) : v0 # v 0 S & v 0 (s) # v 0 (s0 )}; then R (s) is a omplete subring of C (S). In other words, the ompa ti ation R s S is reated from S if the points s and s 0 are glued N 2 dierent manners, whi h gives to ea h other. We an hoose s ò S \ S at least by 2 N 2 at least 2 dierent ompa ti ations R s S of S. Proof. As S ertainly ontains a dis rete ountable subset whose losure in

omorphi with

( )

( )

Then we put

H(s) # C( ) ‹ Ôp (R (s)) ;

(3.88)



where Ô

p

:C

0

Rm Ù C p Rm

(

)

(

) is dened by

[Ô

p

v℄(s) # v(s)(1% sp ) :

(3.89)

s1 ; s2 ò S, s1 #Ö s2 implies M(FH s1 % ) ÊÖ M(FH s2 % ).  the set N , {nite subsets of C0 (S)} dire ted by the ordering ¢ , ’. It is lear that, for any  # ( k ; { v 1 ; ::: v L }) ò  , the set N  ( s j ) # { s ò S; max1¢l¢L v l (s) " v l (s j ) ¢ 1/k} forms a neighbourhood of s j in S, j # 0; 1, with v l standing for the ontinuous extension of v l . Therefore, for any  ò , we an nd s j  ò N  (s j ) and then, obviously, the net {s j  } ò onverges to s j in S. We an even suppose s j  ò S be ause S is dense in S. As s j ò S \ S, the set B % # { s ò S;  s  £ % } is also a neighbourhood of s j so that lim  ò   s j   # %. The image of s 1 via the anoni al surje tion S Ù R s 1 S is glued with the image of s 0 while R s 2 S glues only s 2 with s 0 so that the image of s 1 via the anoni al surje tion S Ù R s 2 S remains separated from the image of s 0 provided s 1 #Ö s 2 , as supposed. This means that lim  ò  [ v ( s 1  ) " v ( s 0  )℄ # 0 for every v ò R( s 1 ) while lim ò [v(s1  ) " v(s0  )℄ #Ö 0 for some v ò R(s2 ).

We want to show that, for any

(

);

(

);

Let us take as the index set

;

;

;

;

(

(

(

)

;

;

58

)

)

;

;

More pre isely, it is even equal to the ardinality of



N

if

S is separable, as supposed; f. Engelking

[284, Thm. 3.5.3℄.

59

See Bourbaki [144, Chap.IX, Exer ise 1.12℄ or Engelking [284, Corollary 3.6.12℄ for details.

Lp

3.3 A lass of onvex ompa ti ations of balls in

Choose some x 0 ò and, for any  ò  , a neighbourhood N j  N j    # % p /(1%s j  p ) and, denoting j  # N j   , let us put ;



;

;

;

u j  (x) #

sj 



;

0

of

x0

su h that

;

if

;

Ë 171

-spa es

if

x ò j  ; x ò \ j  ; ;

;

j # 0; 1. Note that u j  L p ;S # %s j  /(1%s j  p )1 p   % so that ea h u j  belongs to B % . As lim  ò   j   # 0, we may also suppose lim ò  diam( j  ) # 0. p n Let us take some h # g ‹ Ô v with g ò C ( ) and v ò R( s 1 ). As ’ R is ompa t, g % % is uniformly ontinuous on so that there is the modulus of ontinuity M g : R Ù R of g , i.e. lim " Ù0 M g ( " ) # 0 and  g ( x 1 ) " g ( x 2 ) ¢ M g ( x 1 " x 2 ) for any x 1 ; x 2 ò . Then for

;

(

)

;

/

;

;

;

;







!! ! !! e H ( s ) ( u 1 ;  ) " e H ( s ) ( u 0 ;  ) ; % h !!! 1 1 !! !! # !!!!X g(x)1%s1;  p v(s1;  ) dx " X g(x)1%s0;  p v(s0;  ) dx!!!! ! 1  !

0  !! !! ¢ !!!!X g(x0 )1%s1;  p v(s1;  ) dx " X g(x0 )1%s0;  p v(s0;  ) dx!!!! ! 1  !

0  ;

;

;

;

%

H X g(x) j #0 ; 1 j 

" g(x

1% s j ;  

0 )

p !! )! v ( s j ;  ) d x !

;

¢ % p !!!!g(x

!  v ( s 1 ;  )" v ( s 0 ;  )!!!

0)

onverges to zero if



ranges

%

H % j #0 ; 1

p

M g (diam( j  ))v(s j  )

;

the same luster points in M(F H ( s 1 ); % ). On the other hand, taking

h#1

e H(s

2)

(

v ò R(s2 )

su h that

‹ Ôp v, we obtain obviously

u1  ) " e H ;

(

;

. For a general h ò H(s1 ) we an obtain the same result e u1  )} ò and {e H s1 (u0  )} ò must have

analogously. This shows that the nets { H ( s 1 ) (

putting

;

s2) (u0;  ); % h

#

(

lim ò [v(s

1;

)

)

;

" v(s

0;

 )℄

#Ö 0 and

p X 1% s 1 ;   v ( s 1 ;  ) d x

1  ;

"X

p 1% s 0 ;   v ( s 0 ;  ) d x

0 

# % p v(s

1;

)

" v(s

0;

 )

;

;

whi h does not approa h zero. In other words, the luster points of the nets {

eH

(

s 2 ) ( u 1 ;  )}  ò  and { e H ( s 2 ) ( u 0 ;  )}  ò  an be separated in M(F H ( s 2 ); % ). This shows that

denitely M(F H ( s 1 ); % )

ÊÖ M(FH s2 (

);

Å

% ).

Let us omplete the properties of the ordering of the lass of onvex ompa ti ations of

B%

presented here. Let us emphasize that two onvex ompa ti ations

need not possess a supremum in the lass of all onvex ompa ti ations of a given

B%

( f. Example 2.3), so that the situation stated in the following theorem is rather

ex eptional.

Theorem 3.42 (Properties of the ordering). M(F H ; % );

The lass of onvex ompa ti ations

H a linear subspa e of Carp ( ; S)

(3.90)

172

Ë

3 Young Measures and Their Generalizations

of the ball

B%

²

ordered by the relation   is a latti e and, for two subspa es

Carp ( ; S), it holds

sup M(FH1 % ); M(FH2 % ) # ;

;

inf M(FH1 % ); M(FH2 % ) # ;

;

H1 ; H2 ’

FH1 %H2 ; % ) ;

M(

FH 1 H 2 ; % ) ;

M(

p where H j denotes the losure of H j in Car % ( ; S). Moreover, the lass (3.90) possesses the nest element, namely M(FCarp ( ;S); % ), whi h is, however, not any maximal onvex

ompa ti ation of

B % in general.

H1 ; H2 linear subspa es of Carp ( ; S), it is obvious that both H 1  H 2 p and H 1 % H 2 are also linear subspa es of Car ( ; S). Both subspa es generate via 0 (3.85) some onvexifying subspa es of C ( B % ); as for H 1 % H 2 , it is essential that p % (Car ( ; S)) itself is a onvexifying subspa e of C0 (B % ). 1 0 Let us put H 0 # { g ‹ 1 ò L ( ; C (S))}.60 We will show that M(F H % ) ² M(F H % ) implies H ’ H % H 0 . First, thanks to Theorem 2.8 we have M(F H % ) Ê M(F H % H % ) be0 0

ause F H % H % is the losure in C ( B % ) of F H % . Therefore, M(F H % H % ) ² M(F H % H % ). 0 0 0 Proof. For



;



;

;

;

As both

FH %H0 ; %

and

FH %H0 ; % 

;

;

are losed, we an dedu e that

;

FH %H0 ; %



;

’ FH %H0 % ; f. 

;

H % H0 # %"1 (FH %H0 % ) and H % H0 # %"1 (FH %H0 we an also dedu e that H % H 0 ’ H % H 0 , and therefore H ’ H % H 0 , as well. 

the proof of Corollary 2.9. As



;



;

% ),



Let us now prove that M(F H 1 % H 2 ; % ) is the supremum (i.e. least upper bound) of M(

FH1 ; % )

³ M(FH j ; % ) for j # 1; 2 FH1 %H2 ; % “ FH j ; % due to the denition (3.85). that M(F H ; % ) ³ M(F H j ; % ) for j # 1 ; 2 implies M(F H ; % ) ³

and M(F H 2 ; % ). First, it is lear that M(F H 1 % H 2 ; % )

thanks to Theorem 2.8 be ause obviously Se ondly, we have to show

H j ’ H % H . As H % H is a linear spa e, we have got also H % H ’ H % H . This implies M(FH1 %H2 % ) ² M(F H % H % ) Ê M(F H % ) due to Theorem 2.85 be ause F H % H % is the losure in C ( B % ) 0 0 p of F H % thanks to the appropriate topology on Car% ( ; S). M(

FH1 %H2 ; % ).

Indeed, we have already demonstrated that 1

2

0

;

0

;

;

0

0

;

;

Let us now prove that M(F H

1 H 2 ; %

) is the inmum (i.e. greatest lower bound) of

) ² M(F H j ; % ) for j # 1 ; 2 1 H 2 ; % FH 1 H 2 ; % ’ FH j ; % and be ause M(FH j ; % ) Ê 0 M(F H j ; % ) sin e F H ; % is the losure in C ( B % ) of F H j ; % . Se ondly, we have to show that j M(F H ; % ) ² M(F H j ; % ) for j # 1 ; 2 implies M(F H ; % ) ² M(F H H ; % ). Indeed, we have 1 2

FH1 ; % )

M(

and M(F H 2 ; % ). First, it is lear that M(F H

thanks to Theorem 2.8 be ause obviously

H ’ H j % H0 for j # 1; 2, and therefore also H ’ H 1  H 2 % H0 . This implies M(F H % ) ² M(F H H % H % ) Ê M(F H H % ) due to the denition of H 0 and 1 2 0 1 2 already demonstrated that ;

;

;

(3.85).

p ò [1; %) and # 1 , 2 for j some n j ¡ 0, j # 1; 2; n # n1 % n2 . It is known that L p ( ; S) p p is isometri ally isomorphi with L ( 1 ; L ( 2 ; S)) via the mapping T dened by p [ Tu ℄( x 1 ) # u ( x 1 ; -). Let us put S # L ( 2 ; S), whi h is again a separable Bana h Let us now onsider a spe ial ase

domains in

Rn j

with



60

Note that

H0

# %"

1

({ onstants}) and thus it is losed in

Carp% ( ; S) be ause % is ontinuous.

Lp

3.3 A lass of onvex ompa ti ations of balls in

Carp ( ; S) and apply our theory dire tly.

spa e, so that we an speak about the spa e

1



% (Car ( 1 ; S)) is a onvexifying subspa e of C0 (B % ), : Car ( 1 ; S) Ù C (B % ) is dened by [ % h ℄(u) # P h (x1 ; Tu(x1 )) dx1

This implies in parti ular that

p

%

where

Ë 173

-spa es



p





0









1

h ò Car ( 1 ; S) and u ò B % . We want to show that the onvex ompa ti % (Carp ( 1 ; S))) of B % is stri tly ner than the onvex ompa ti ation p p M( % (Car ( ; S))) # M(FCarp ;S % ). Indeed, having h ò Car ( ; S) we an always p nd h ò Car ( 1 ; S) su h that % h # % h , namely h ( x 1 ; s ) # P h(x1 ; x2 ; s (x2 )) dx2

2 p for s ranging L ( 2 ; S). Indeed, by the Fubini theorem, for any u ò B % we have for

p



ation M(







(

);

















€ % h  (u) 



# X h (x ; Tu(x 1



1

1 ))

dx # X 1

”X h ( x 1 ; x 2 ; u ( x 1 ; x 2 )) d x 2 • d x 1

2

1

# X h(x ; u(x)) dx # % h
(u) :

Let us also note that a tually

L p (

2

h ò Carp ( 1 ; S) 



be ause the ontinuity of

h (x1 ; -) : 

R follows by the standard properties of the Nemytski mappings (see

; S) Ù

Theorem 1.24) and be ause of the estimate



h (x1 ; s ) ¢ 



¢

X h(x1 ; x2 ;

2

s (x2 ))dx2

X a h (x1 ; x2 )

2



% b h s (x 

a h ò L1 ( 1 ) is dened by a h (x1 ) #

where





p S

dx # a h (x ) % b h s Sp ;

2 )

2

1

a h (x1 ; x2 ) dx2

P

2



and

b h # b h

with

ah

and

b h the oe ients from (3.81). This shows that % (Car ( 1 ; S)) “ % (Car ( ; S)), from % (Carp ( 1 ; S))) ³ M( % (Carp ( ; S))) follows by Theorem 2.8. 1 2 Let us take two sequen es { u } k òN and { u } k òN in B % . The former one has the k k 1 properties that u ( x 1 ; -) is onstant on 2 for a.a. x 1 ò 1 and takes only the values k s ; "s ò S with some s #Ö 0 su iently small, and the whole sequen e {u1k }kòN on2 2 1 p verges weakly in L ( ; S) to 0. The latter sequen e { u } k òN is dened by u ( x ) # u ( x ) k k k 2 1 if x $ ( x 1 ; x 2 ) ò 1 , 2 and u ( x ) # " u ( x ) if x $ ( x 1 ; x 2 ) ò 1 , 2 , where 2 and 2 k k are some disjoint measurable parts of 2 of a positive measure su h that 2 € 2 # 2 . Let us note that both sequen es are bounded even in L ( ; S), so that we an interp



whi h M(

p





















pret them by means of suitable Young measures; f. Example 3.44 or 3.45 below. Then, roughly speaking, both sequen es onverges to the homogeneous Young measure

# 21 Æ s % 12 Æ"s in the representation of M(FCarp S % ). On p the other hand, on the representation of M( % (Car ( ; S))), the rst one onverges to the homogeneous Young measure ò Lw ( ; rba(S)) given by x1 # 12 Æs1 % 21 Æ"s1 with s ( x ) # s , while the se ond one onverges to ò Lw ( ; rba(S)) given by 1 2 1 2 x 1 # 2 Æs % 2 Æ "s with s ( x ) # s if x ò and s ( x ) # " s if x ò . It is lear that p #Ö be ause they an be separated by any h ò Car ( ; S)) with the properties h (x ; s ) # h (x ; " s ) #Ö h (x ; s ) # h (x ; " s ). This shows that M( % (Carp ( ; S))) ± M( % (Carp ( ; S))) so that M( % (Carp ( ; S))) is not a maximal onvex ompa ti ation of B % . Å ò Lw ( ; rba(S)) given by 

*

x

(

1



1



1 

2

2 

1



2



2

2

2



2

1



1







2

2

1

1





1

2





1

 *

2

1

2

1





2



2





);

1

1

*

;



1



Ë

174 3.3.b

3 Young Measures and Their Generalizations

The omposition h DZ 

u ò B % and a Carathéodory integrand h, we spoke about a omposition h Ž u h Ž u℄(x) # h(x ; u(x)). We saw in (3.3) that there is a natural generalization if, instead of u , we onsider a Young measure # { x }xò . Then the omposition h DZ results to a fun tion h DZ : x ÜÙ PS h(x ; s) x (ds); note that if x # Æ u x , then P h ( x ; s ) x (d s ) # h ( x ; u ( x )) # [ h Ž u ℄( x ) so that the omposition h DZ a tually extends S the omposition h Ž u . If the Young measure is onsidered61 as a Young fun tional 1  ò L ( ; C(S)) , the reader an easily verify that this fun tion, denoted by h DZ , an be alternatively dened by the identity < h DZ  ; g > # <  ; g - h > for any g ò L ( ), where g - h abbreviates (g ‹ 1) - h, i.e. For a given

dened by [

(

)

*



[

g - h℄(x ; s) # g(x)h(x ; s) :

To make possible a study of lo al properties of generalized Young fun tionals, we

an perform this onstru tion even in general situations, the result being a ertain

, however. For a linear subspa e C( ) ’ G ’ L p subspa e H ’ Car ( ; S) is G -invariant if

measure on





(

), we will say that the

G-H # H;

:g ò G :h ò H : g- h ò H . For h ò H G  omposition h DZ  ò G by whi h means

(3.91) and

òH

*

, let us then dene the

*

h Let us note that a tually

h

G

DZ  ; g #  ; g- h :

(3.92)

G

h DZ i H (u) # Nh (u) # h Ž u be ause, for any g ò G,

G

DZ i H ( u ) ; g # i H ( u ) ; g - h #

The dual to the intermediate spa e

G

X g ( x ) h ( x ; u ( x )) d x

# h Ž u ; g :

orresponds to a ertain spa e of measures,

though we do not want to spe ify su h measures in details; f. also Example 1.33. Of ( ), these measures are just r a( ) vba( ), respe tively; f. Theorem 1.32. Yet, these limit ases may bear sometimes

ourse, in the limit ases or

G # C( )

or

G # L





disadvantages as the former one does not allow multipli ation by dis ontinuous fun tions while the latter one reates a nonmetrizable weak* topology on bounded sets in

G

*

. For these reasons, a usage of an intermediate spa e

C ( ) #Ö G #Ö L





(

) may ap-

pear a tually advantageous espe ially for development of a numeri al-approximation theory; f. As

G # lG0 from (3.164) below.

G

G will be mostly lear from a ontext or the result g DZ h will not depend on G,

DZ

G

DZ

we will write simply   instead of  . Let us note that to ensure a tually

61

Cf. (3.4) together with the Convention 3.1.

h DZ òG

*

we

3.3 A lass of onvex ompa ti ations of balls in

must require, in addition to (3.91), the ontinuity of the mapping

Lp

-spa es

Ë 175

g ÜÙ g - h : G Ù H .

However, to guarantee an a tually good sense of this  omposition, we will have to impose even a bit stronger assumption:

:g ò G; h ò H :



g - h H ¢ C g L



(

) h H

:

(3.93)

Proposition 3.43 (Properties of the omposition  DZ ). spa e of

L



(

Let G “ C ( ) be a linear sub ) and H be a G-invariant normed spa e with the norm - H ner than  - %

and satisfying (3.93). Then: The bilinear mapping ( h ;

(i)

,

) ÜÙ h DZ  : H , H Ù G *

*

*

to the weak* topology

G . p p If  ò YH % ( ; S) is attainable by a net { i H ( u  )}  ò  su h that the set { u  ;  ò  } is S 1 1 relatively weakly ompa t in L ( ), then h DZ  ò L ( ) for any h ò H . p If  ò YH % ( ; S) is arbitrary but h satises additionally the growth ondition on

(ii)

is jointly ontinuous from the

H,H

(strong weak*)-topology on the bounded subsets of *

;

(iii)

;

;a h ò L q ( ) ;b h ò R% :



p/ q

h(x ; s) ¢ a h (x) % b h s S

(3.94)

1   q ¢ %, then h DZ  ò L q ( ). q If, for 1   q   , h ò H satises g - h ò H for any g ò L ( ) and g - h H ¢ C h g L q q with some C h ò R, then h DZ  ò L ( ) even for any  ò H .

for some (iv)





(

)

*

h ò H and  ò H , we want to show that: :" ¡ 0 :g ò G :R ò R% ;Æ ¡ 0 ;h0 ò H :h1 ò H :1 ò H , 1 H ¢ R *

Proof. For any

*

*

max   "  ; h 1

This is a tually true for

0

 ; h 1

" h H  ¢ Æ

âá



h1 DZ 1 " h DZ  ; g  ¢ " :

h0 # g - h and 0   Æ ¢ 2"1 " min(1; (CR g L



(

"1 ). Indeed,

) )

we an estimate 

h1 DZ 1 " h DZ  ; g  ¢

#





h 1 DZ  1 " h DZ  1 ; g  %

1 ; g - (h1 " h)  %



h DZ  1 " h DZ  ; g 

1 "  ; g - h  ¢ C 1 H g L h1 " h H " " %  1 "  ; h0  ¢ ÆCR g L % Æ ¢ % ¢ " : 

*



(

2

)



(

)

2

The point (i) has been thus proved.

u

p { u  ; S

B%

lim ò i H (u  ) # 

H and  ò } is relatively weakly ompa t in L ( ). As the mapping  Ù Ü h DZ  was shown to be ontinuous, we an see that lim ò  h DZ i H ( u  ) # h DZ  . Simultaneously, we p p p have the estimate  h DZ i H ( u  ) #  h Ž u   ¢ a h % b h u  with a h and b h from (3.81), S p 1 whi h shows that the set { h DZ i H ( u  ) ;  ò  } is relatively weakly ompa t in L ( ), p as well. Therefore the limit of the net { h DZ i H ( u  ) }  ò  , whi h is just h DZ  , must live in L1 ( ), proving thus (ii). Taking a net { u  }  ò  in B % su h that lim ò  i H ( u  ) #  weakly* in H , the assumpConsider a net {  }  ò  in

su h that

weakly* in

*

1

*

tion (3.94) allows us to estimate

h DZ i H (u  ) L q

"

(

)

"

# hŽu  L q ¢ a h L q % b h """"" u Sp q """""L q (

)

(

)

/

# a h L q % b h u pL pq S : /

(

)

(

)

(

;

)

Ë

176

3 Young Measures and Their Generalizations

h DZ i H (u  )} ò is bounded in L q ( ), hen e it must have a weak q q (or weak* if q # %) luster point in L ( ). As L ( ) is naturally embedded into G q ( f. Example 1.34) and this net onverges in G to h DZ  , this luster point in L ( ) must q

oin ide with h DZ  , whi h shows that h DZ  ò L ( ). Thus (iii) was shown. This shows that the net {

*

*

The point (iv) will be shown if one realizes that, thanks to the estimate

h DZ  ; g

#  ; g - h ¢  H g - h H ¢ C h  H g L q *

g ÜÙ is q itself must belong to L ( ). the mapping

Let us remark that bounded in

H

*

p

YH

;

*

a ontinuous linear fun tional on

% ( ; S) is always a bounded subset of

H



(

)

;

L q ( ) 

so that

Å

i

*

hDZ

be ause H (

B % ) is

thanks to the estimate



i H (u) H ¢ sup *



h H ¢1

!! ! !![ % h ℄( u )!!!

¢ C sup !!!![ % h℄(u)!!!! ¢ C 

(3.95)

h  % ¢1

C is here the onstant from the assumed estimate h% ¢ C h H . Therefore, the h ; ) ÜÙ h DZ  stated in Proposition 3.43(i) is parti ularly relevant for p p  ranging YH % ( ; S). Also let us remark that, supposing  ò YH % ( ; S), h DZ  has a good p sense not only for h ò H but even for h belonging to the losure H of H in Car% ( ; S). p p This is lear by Theorem 3.40, whi h says in parti ular that Y H % ( ; S) Ê YH % ( ; S).

where

joint ontinuity of (

;

;

p

;

;

 ÜÙ h DZ  : YH % ( ; S) Ù G is, in fa t, a onp 1 tinuous ane extension of the Nemytski mapping N h : L ( ; S) Ù L ( ) generated by h ; f. Example 3.96. Let us also remark that the mapping

*

;

We an also generalize the notion homogeneous introdu ed so far only for las-

p

 ò YH

% ( ; S) hoh DZ  is onstant in for every h ò H su h that h(-; s) is onstant for ea h p s ò S, or equivalently for every h ò H in the form h # 1 ‹ v. Analogously,  ò YH % ( ; S) will be alled pie e-wise homogeneous on a given partition of if h DZ  is pie e-wise

onstant on this partition whenever h ò H is su h that h (- ; s ) is pie e-wise onstant for any s ò S. si al Young measures. We will all a generalized Young fun tional

;

mogeneous if

;

3.3.

Some on rete examples

It is time to present some examples viewed from the perspe tive of the theory of generalized Young fun tionals.

Example 3.44 (Classi al Young measures). Let us take p # , S # Rm , and H # Car ( ; Rm ). Let us note that, in fa t, we an equally work with Carp ( ; Rm ) for any p £ 0 be ause Carp ( ; Rm ) ,S % # Car ( ; Rm ) ,S % for S % # {s ò Rm ; s ¢ %} the ball 



3.3 A lass of onvex ompa ti ations of balls in

in

Rm of the radius %. Hen e, we will better take H # Car

0

h

norm Car0 ( ;Rm )

-spa es

Ë 177

; Rm ) endowed with the

# P supsòRm h(x ; s)dx. This strong topology is a tually ner than Car ( ; Rm ) by the seminorm  - % be ause of the estimate 0

the topology indu ed from



(

Lp

h% # % h C0

(

B% )

!! !! !! ! !!X h ( x ; u ( x )) d x !!! !  u ( x )¢ % !

# sup

¢ X sup h(x ; s)dx # h L1 S % ¢ h Car0 Rm : (

 s ¢ %

;

)

(

;

(3.96)

)

B% # U S # S % . The set Y H % ( ; Rm ) then ontains just the Young fun tionals m (i.e. Y H % ( ; R ) Ê Y ( ; S % ), whi h justies the notion of generalized Young fun tion0 0 m m als for a general H dierent from Car ( ; R ). Moreover, H # Car ( ; R ) is L ( )Then it is obvious that we have re overed the situation from Se tion 3.1 with 

given by (3.1) for

;

 ;



invariant and (3.93) is satised due to the obvious estimate



g - h H #

X sup  g ( x )

 s ¢ %

¢ g L



(

-

h(x ; s)dx #

) sup  h (- ; s ) L 1 ( )  s ¢ %

X  g ( x ) sup  h ( x ; s )d x

 s ¢ %

# g L



(

) h H

:

Example 3.45 (Fattorini's generalization). Let us take p # %, S a separable Bana h H # L ( ; C (S)) endowed with the norm h L1 C0 S # P supsòS h(x ; s)dx. This strong topology is again ner than the topology indu ed from Car ( ; S) be ause spa e,

1

0

(

;

(

))

0

the estimate (3.96) applies here as well. Then it is obvious that we have re overed the

B % # U given by (3.1) for S the ball in S of the radius

)-invariant, satisfying (3.93). Besides, we ould also take H # Car0 ( ; S) whi h is possibly larger than L1 ( ; C0 (S)). It would reate also a onvex ompa ti ation of B % # U given by (3.1) for S the ball in S. The question whether it is equivalent with the onvex ompa ti ation 1 0

reated by H # L ( ; C (S)) is open, however.

situation from Se tion 3.2.a with

%. Again, su h H is L



(

Example 3.46 (S honbek's generalization: the L p -Young measures). p ò [1; %), S # Rm , and H # C( ) ‹ C p (Rm )

with

Let

us

take

(3.97)

C p (Rm ) dened by (3.33). We an endow H by the norm

h H #

sup

m ( x ; s )ò ,R



h(x ; s) : 1%sp

This strong topology is ner than the topology indu ed from the estimate

(3.98)

Carp% ( ; Rm ) be ause of

Ë

178

!! !! !! h !!%

3 Young Measures and Their Generalizations

!! ! !!X h ( x ; u ( x )) d x !!! !! !! ! u L p ;R m ¢ % !

# % h C0 B % # (

)

sup (

¢

)

sup

X (1% u ( x )

u L p ;R m ¢ % (

)



)

¢ (  % % p )



sup

m ( x ; s )ò ,R

h(x ; u(x)) dx 1%u(x)p

h(x ; s) # 1%sp

(

 % % p ) h H :

(3.99)

H is C( )-invariant, satisfying also (3.93). We want to show that the orrespondp m

onvex ompa ti ation Y H % ( ; R ) is equivalent with a ertain sets of Young

Su h ing

p



;

measures, namely

Y% ( ; Rm ) # p

–

ò Y( ; Rm );

X X s

Rm

p

x (d s ) d x

¢ %— :

(3.100)

Y% ( ; Rm ) ontains just those ò Y( ; Rm ) su h that # w*- lim ò Æ(u  ) for some net {u  } ò su h that u  L p ( ;Rm ) ¢ %, where  m the embedding Æ : B % Ù L w* ( ; r a(R )) is dened again by (3.15). Let us also rem m * mind that r a(R ) Ê C 0 (R ) by the Riesz theorem 1.32(iv) and, by the Dunford m 1 m * Pettis theorem, we an see that L w* ( ; r a(R )) Ê L ( ; C 0 (R )) . Of ourse, the p m  m natural topology on Y % ( ; R ) is just the weak* topology of L w* ( ; r a(R )). Let us p 1 m m abbreviate H 0 # L ( ; C 0 (R )). By the very denitions Y % ( ; R ) # w*- l Æ ( B % ) and p p p Y H0 ; % ( ; Rm ) # w*- l i H0 (B % ), we an see that Y% ( ; Rm ) Ê Y H0 ; % ( ; Rm ) via the mapm * ping : L w* ( ; r a(R )) Ù H dened by ( f. also (3.14)) In view of the proof of Proposition 3.22,



(

)

; h #

p

X X

Rm

h(x ; s)

x (d s ) d x :

R

R

p m ) Ê Y p ( ; m ) provided we ( ; ;% H; % m m) show the losures of H 0 and H ) to be equal to ea h other. As C 0 ( m ) is dense62 in L 1 ( ; C ( m )) in the standard norm of is separable, C ( ) ‹ C 0 ( 0 1 m L ( ; C0 ( )), hen e in the seminorm  - % , as well. Now we have C( ) ‹ C0 ( m ) ’ H , so that we are to show that C( ) ‹ C ( m ) is dense in H in the topology of Furthermore, by Theorem 3.40 we an get

R



R

p in Car% ( ;

Y H0

R

R

R





0

R

R

Carp% ( ; Rm ). In other words, it su es to show that every h # g ‹ v with g ò C( ) m m and v ò C p (R ) an be approximated by some h " ò C ( ) ‹ C (R ) in the seminorm  -  % . Let us take h " ( x ; s ) # g ( x ) v ( s ) v " ( s ) with the ut-o fun tion v r ( s ) dened by m % % (3.38). Sin e v ò C p (R ), there is a : R Ù R ontinuous su h that  v ( s ) ¢ a ( s ) and limtÙ% a(t)t"p # 0. We an additionally require limtÙ% a(t) # %. Still there is a % %

ontinuous fun tion b : R Ù R su h that lim t Ù% b(t)/t # % and b(a(t)) ¢ Ct p

0



1/







62

We an onsider

Rm

C0 (

) as a losed subspa e of

(Alexandro ) ompa ti ation of Thm. I.5.25℄.

Rm

C(

Rm

) where



Rm

denotes here the one-point

, whi h is a metrizable ompa t, and then apply Warga [791,

3.3 A lass of onvex ompa ti ations of balls in

Lp

-spa es

Ë 179

C ò R% . Therefore, by the Dunford-Pettis and the de la Vallée-Poussin theorems 1.28(ii+iv), the set { a ( u ); u ò B % } is uniformly integrable in the sense for some

lim

sup

k Ù% u L p ;Rm (

)

X

¢%

{

x ò ; a ( u ( x ))£ k }

a(u(x)) dx # 0 :

Then we an estimate 

h " h " % #

!! !!X g ( x ) v ( u ( x ))(1 !! u L p ;R m ¢ % !

sup (

¢ g C (

¢ g C (

¢ g C (

whi h tends to zero with

"v

1/

)

)

)

)

sup



u L p ;Rm



u L p ;Rm



u L p ;Rm

(

X

)

¢%

)

¢%

)

¢%

sup (

sup (

X {

X {

!! ! " ( u ( x )) d x !!! !

a(u(x))(1 " v1

x ò ;  u ( x )£1/ " }

/

" ( u ( x )) d x

a(u(x)) dx

x ò ; a ( u ( x ))£ a (1/ " )}

a(u(x)) dx ;

" Ù 0 be ause a(1/") approa hes %.

Y% ( ; Rm )

Altogether we have thus showed that the set of Young measures

p equivalent with Y H ; % ( ;

H

Moreover,

p Y H ; % ( ;

R

R

m ) for

H from (3.97).

from (3.97) an be enlarged so that

m ) with

H # L



(

Y% ( ; Rm ) p

p

is equivalent also to

) ‹ C p (Rm ) ;

(3.101)

whi h an be proved by the same arguments using also the obvious in lusion

C0 (R

m)

’L



(

) ‹ C0 (R

m)

’ L ( ; C 1

R

0(

is

m )).

C( ) ‹

Moreover, this onvex ompa ti ation is not norm- onsistent.63

Example 3.47 (The generalization by DiPerna and Majda).

Rm , R a omplete subring of C Rm , and 0

(

Let us take

p ò [1; %), S #

)

H # C ( ) ‹ Ô p (R ) ;

(3.102)



p from (3.89). Again we an endow

H by the norm h H dened by (3.98), whi h p m indu es a ner topology than the topology indu ed from Car ( ; R ) be ause of (3.99). p m Then the respe tive onvex ompa ti ation Y H % ( ; R ) is equivalent with a ertain p m subset of DM ( ; R ) from (3.51), namely the set R

with Ô

;

p m  DMR % ( ; R ) # † òr a( ,

;

R

Rm ; )

;{u  } ò ; u  L p Rm ¢ % : (

63

)

w*-

lim i(u  ) #  ‡  ò

(3.103)

u k }kòN from Figure 3.7 with ab p # %. Then ¡ 0 so that u k does not onverge to 0 in the L p -strong topology but Æ(u k ) Ù Æ(0)

To see this, it su es to take a sequen e {

u k Lp ( ;Rm ) # % p weakly* in Y % ( ;

;

Rm

).

1/

Ë

180

3 Young Measures and Their Generalizations

i # (JR )"1 Ž i p *

where

was dened in the proof of Theorem 3.25. Of ourse, the ane

Rm

p m DMR % ( ; R ) is just the mapping Ô p Ž JR : r a( , R Rm ) Ù H where Ôp : H Ù C( ) ‹ R is the isometri al isomorphism h ÜÙ h/(1%sp ). The veri ation that it xes B % , whi h means here Ôp Ž JR Ž i # i H , is

homeomorphism between *

p

YH

;

% ( ;

*

) and

;

*





*

*

an easy exer ise. This onvex ompa ti ation is also norm- onsistent.64 Let us emphasize that the onvex ompa ti ation generated by (3.102) is stri tly ner than that one generated by (3.97). To see it, we an take the net indi ated on

i 0 i 0 p for example, the integrand h ( x ; s ) #  s  . Let us note that H from (3.102) is C ( )-invariant, satisfying also (3.93). Yet, it is not G-invariant for any G “ C( ), G #Ö C( ). This is a ertain disadvantage of the DiPerna-

Figure 3.7, whi h annot be distinguished from H ( ) in the latter onvex ompa ti-

ation while it an be separated from H ( ) in the former onvex ompa ti ation by,







Majda measures, whi h auses di ulties espe ially within an approximation theory ( f. Se tion 3.5) and whi h may sometimes make this onvex ompa ti ation insu iently oarse.

Example 3.48 (A renement of DiPerna-Majda measures).

To put o the disadvan-

tage mentioned in the last example, we are tempted to enlarge a bit the spa e (3.102). Having some ring

G su h that C( ) ’ G ’ L





(

H from

), we an put

H # G ‹ Ô p (R ) :

(3.104)

G - G # G, we have ensured that H is G-invariant. In view of Proposition 3.77(ii), H from (3.104) is the smallest G -invariant linear spa e ontaining H from (3.102). Again we an endow H from (3.104) by the norm h H dened by (3.98), whi h indu es a p m ner topology than the relativized topology from Car% ( ; R ) and guarantees (3.93). In parti ular, we an take G # L ( ). Often it is advantageous to have G separable, so that it has a good sense to onsider G smaller than L ( ), f. (3.161) below. p m The orresponding onvex ompa ti ation Y H % ( ; R ) is stri tly ner than the

As





;

onvex ompa ti ation obtained by the hoi e (3.102). To avoid te hni alities, let us show it only for a spe ial ase ality, we an suppose that

# (0; 1)

Without loss of gener-

ontains a fun tion g dis ontinuous at some x ò , limxÿx0 g (x) and limx÷x0 g (x) do exist. Let us now take

G

and for simpli ity the limits

m # 1.

and 0

0

0

0

¡ 1, the norm- onsisten y follows dire tly from Theorem 3.39(iii). For p # 1 we an rst B % ’ L ( ; Rm ) by the Nemytski mapping N : u ÜÙ uu"" %" (L -strong,L %" strong)-homeomorphi ally onto the ball B 1 1%" ’ L %" ( ; Rm ) and then to make the onvex om% % " m m q   pa ti ation DM r %1 1%" ( ; R ) of this transformed ball, where R # { v ò C (R ); ; v ò R : v ( s ) # R v(ss"" %" )} is a omplete subring of C (Rm ). If " ¡ 0, this latter onvex ompa ti ation is L %" %" m p p # C ( )‹ norm onsistent by Theorem 3.39(iii) and therefore (DM p ŽN) with H r %1 1%" ( ; R ) ; n H ; i H R % " m q Ô (R ) forms an L -norm onsistent onvex ompa ti ation of the ball B % ’ L ( ; R ) whi h m is equivalent with the original onvex ompa ti ation DM R % ( ; R ) via the adjoint mapping to p. N:HÙH 64

If

p

1

transform the ball

/(1

/(

)

1

1

0

;

/(1

1

)

1

/(

)

0

)

1

1

*



;

1

/(

)

1

1

1



;

3.3 A lass of onvex ompa ti ations of balls in

two sequen es {

u k (x) # (k%) 2

1/

p

u1k }kòN

(

Lp

u2k }kòN in B % dened by u1k (x) # (k%)1 p  x0 x0%1 k and , where  M denotes the hara teristi fun tion of M ’ . Ap/

and {

x 0 "1/ k ; x 0 )

Ë 181

-spa es

(

;

/

)

plying the hoi e (3.102), these sequen es annot be separated; more pre isely, they have the ommon limit, whi h is the DiPerna-Majda measure from Figure 3.7 (with

a # %1

/

b # 1). On the other h # g0 ‹ sp one has

p and

Indeed, for

hand, they an be separated by

x 0 %1/ k X k%g0 (x) dx k Ù x 0

lim i H (u k ); h # lim 1

k Ù while

x0 k%g0 (x) dx X k Ù x 0 "1/ k

lim i H (u k ); h # lim 2

k Ù

H

from (3.104).

# % lim g (x) xÿx0

0

# % lim g (x) : x÷x0

0

These limits has been supposed to be dierent from ea h other. Contrary to Example 3.47, we will not interpret su h stri tly ner onvex ompa ti ation in terms of DiPerna-Majda-like measures, though it might be possible.

Example 3.49 (Fonse a's extension of L ( ; Rm )). 1

For

p # 1 one an take

H # Žh : , Rm Ù R; ;h0 ò C0 ( , S m"1 ) : h(x ; s) # h0 x ; Let us note that

s  s DZ : s

(3.105)

 

H ’ Car1 ( ; Rm ) is C( )-invariant. A natural norm on H is now



h H #

(

max

x ; s )ò , S m"1



h(x ; s)

(3.106)

whi h generates a ner topology than the relativized topology from

Car% ( ; Rm ) be1

h ¢ % h H . Equipped with this norm, H is isometri ally isoC0 ( ; Sm"1 ) and thus it is separable. 1 m m m   Then Y H % ( ; R ) is equivalent with F % ( ; R ) := {  òF( ; R ); P , S m"1  (d x d s ) ¢ m %} with F( ; R ) being the set of Fonse a's measures dened by (3.76); the ane homem "1 ) is just the adjoint operator to h ÜÙ h : C ( ; S m "1 ) Ù omorphism H Ù r a( ‹ S 0 0 H with h dened by h(x ; s) # h0 (x ; ss"1 )s. 1 m This onvex ompa ti ation is not L -norm onsistent.65 Also, F % ( ; R ) is stri tly oarser onvex ompa ti ation of B % than the DiPerna-Majda measures 1 m DMR % ( ; R ) provided R is greater than the ring from (3.45).66

ause of the estimate   % morphi with ;

*

;

65

To see it, the reader an onsult Remark 3.36 where various sequen es, distant from ea h other in

R

L1 -norm, onverge to the same limit when embedded into F( ; m ). m"1 ) is dense in C ( , S m"1 ), the spa e H from (3.105) ontains densely H # 66 As C 0 ( ) ‹ C ( S 0 0 m C0 ( ) ‹ {v ò C( ); ;v0 ò C(S m"1 ) : v(s) # v0 (s/s)s} whi h is obviously ontains in C( ) ‹ m ) Ê Y 1 ( ; m ) Ê Y 1 m) ² 1 Ô (R ) with R from (3.45). Using Theorem 3.40(i), we get F% ( ; H; % H0 ; % ( ;

R

Y1



C( )‹Ô1 (R); %

(

;

Rm

R

)

Ê DMR % ( ; Rm ). In view of Remark 3.36, even 1

;

R

R F% ; Rm ° DMR % ; Rm (

)

1

;

(



).

182

Ë

3 Young Measures and Their Generalizations

Example 3.50 (Coarser onvex ompa ti ations I).

Let us take

p ò [1; %℄, S #

Rm ,

and

H # L p ( ) ‹ (Rm ) ; 

Rm

where (

*

)

*

(3.107)

denotes naturally the spa e of linear fun tions

Rm Ù R; note that p



is

p, f. (1.35). Su h H is obviously L ( )-invariant. Let us p m note that a tually H ’ Car ( ; R ). Indeed, as every h ò H takes the form h ( x ; s ) # m g ‹ id # l#1 g l (x)s l with s # (s1 ; :::; s m ) ò Rm and id : Rm Ù Rm denoting the 

the onjugate exponent to

identity on

Rm , we an always estimate by the Hölder inequality: 

h% #

m !! ! !!X H g ( x ) u ( x ) d x !!! l l !! !! ! u L p ;R m ¢ % !

l #1

sup (

¢

)

sup



u L p ;Rm (

)

¢%



u L p

(

;Rm ) g L p ( ;Rm )

It is then natural to dene the norm on





# % g L p



(

;Rm )

:

H as

h H # g ‹ id H # g L p



(

(3.108)

;Rm )

h # g ‹ id ò H . It makes (3.93) satised. Then Y H % ( ; Rm ) is equivalent for 1   p ¢ % with a ball of the radius % ò R% in L p ( ; Rm ) endowed with the weak* m topology67 while for p # 1 it is equivalent with the ball of the radius % in L ( ; R ) Ê m vba( ; R ) endowed by the weak* topology, i.e. the ball in the bi-dual spa e of L1 ( ; Rm ). The ane homeomorphism is via the mapping H Ù €L p ( ; Rm ) adp m joint to the operator g ÜÙ g ‹ id : L ( ; R ) Ù H . As the resulting onvex ompa ti ation is equivalent with B % endowed with the weak* topology provided p ¡ 1, we obtained obviously a minimal Hausdor onvex

ompa ti ation of B % . On the other hand, for p # 1 it is not true. To see this, we an adopt another hoi e of H , for example p

for

;



*

*

*





H # C( ) ‹ (Rm ) : *

(3.109)



H , we an take again the norm (3.108). For 1   p ¢ %, this hoi e H from (3.109) is dense 1 m in the norm (3.108). However, Y ( ; R ) with H from (3.109) is H%

As for the norm on

does not hange M(F H ; % ) in omparison with (3.107) be ause in

H

from (3.107)

;

% in C( ; Rm ) Ê r a( ; Rm ), whi h is a stri tly

oarser Hausdor onvex ompa ti ation of B % . To see this, we an paraphrase the

onstru tion from Example 3.48: supposing # (0 ; 1) and m # 1 to avoid te hni ali1 2 1 ties, we take two sequen es { u } k òN and { u } k òN in B % dened by u ( x ) # k% x 0 x 0 %1 k k k k 2 and u ( x ) # k% x 0 "1 k x 0 . Applying the hoi e (3.109), these sequen es annot be k equivalent with the ball of the radius

*





(

(

/

;

;

/

)

)

separated; more pre isely, they have the ommon limit, whi h is the Dira measure

67

For

1   p   % it is merely the weak topology be ause of the reexivity of L p ( ; Rm ).

3.3 A lass of onvex ompa ti ations of balls in

Lp

-spa es

Ë 183

Æ x0 ò r a( ). On the other hand, they an be separated by H from (3.107): indeed, limkÙ # 0 while limkÙ # % ¡ 0 provided h #  0 x0 ‹ id.





(

;

)

It demonstrated that the hoi e (3.107) still does not provide a minimal Hausdor onvex ompa ti ation if

p # 1.

It is noteworthy that there exists still stri tly oarser onvex ompa ti ations of the ball in

L1 ( ; Rm ) than the ball in r a( ; Rm ). One of them an be reated by

H # C0 ( ) ‹ (Rm )

*

with

C0 ( )

(3.110)

denoting the spa e of ontinuous fun tions vanishing on the boundary

bd( ). This gives Y H % ( ; Rm ) anely homeomorphi with the ball of the radius % in C ( ; Rm ) Ê r a( ; Rm ). 1

;

0

*

Example 3.51 (Coarser onvex ompa ti ations II). pa ti ations an be obtained by mapping

B%

Another lass of onvex om-

homeomorphi ally onto a ball (possi-

bly of a dierent radius) in another Lebesgue spa e and then ompa tify this latter ball. E.g., if we map

uu""/(1%")

B % ’ L1 ( ; Rm )

onto

B %1

1%")

/(

’ L

%" ( ;

1

Rm

) via the mapping

uÙ Ü and then ompa tify this latter ball by means of H from (3.107) with p # 1 % ", we obtain the same ee t as if we ompa tify B % by means of H # L1%1

/

"

(

; Rm ) ‹ {v}

(3.111)

v ò C1 1%" (Rm ; Rm ) dened by v(s) # ss"" 1%" . It is left as an exer ise to show 1 1 m m that H ’ Car ( ; R ) and, for " ¡ 0, su h hoi e of H yield Y H % ( ; R ) equiva1 1% " 1% " m lent to the ball of the radius % in L ( ; R ) endowed with the weak topol1% " m ogy; the ane homeomorphism H Ù L ( ; R ) is the adjoint operator to g ÜÙ h g "" 1%" . This onvex ompa ti ation is stri tly oarser than with h g ( x ; s ) # g ( x ) - s  s  1 m Y ( ; R ). with

/(

)

/(

)

;

/(

)

*

/(

Remark 3.52.

Let us note that

)

H from (3.107) is L



(

)-invariant while H from (3.109) p ¡ 1. Similarly it

is not, though they generate equivalent onvex ompa ti ations if

H from (3.101) and (3.97), respe tively. It shows that sometimes the fa t H is not G-invariant for a given C( ) ’ G ’ L ( ) may be only arti ial and an be removed by a suitable enlargement of H , whi h is possible up to the losure of H p in Car% ( ; S) without hanging the reated onvex ompa ti ation; f. Theorem 3.40. On the other hand, sometimes the la k of G -invariantness for G greater than C ( ) may be essential, e.g. for the ases (3.109) with p # 1 or (3.102). holds also for



that





Y% ( ; Rm ) is not a norm- onsistent ompa ti ap m  m tion of the ball of L ( ; R ). Nevertheless, Y % ( ; R ) is T - onsistent if T is the relap m tivized strong topology of L ( ; R ) with p ò [1 ; %). In parti ular, having a bounded

Remark 3.53.

We mentioned that

p

184

Ë

3 Young Measures and Their Generalizations

’L

u ÙÆu

; Rm



; Rm ) and also Ù 0 for any p   %.

sequen e { k } k òN ( ) onverging weakly* to some PSfrag repla ements ( k) ( ) weakly*, we an laim68 that k L p ( ;Rm )

Æu

PSfrag repla ements

Summary 3.54.

For

1 p 

u "u

uòL



(

 the relations among the above examples an be dis-

played by the diagram (3.107) (3.109)

DiPerna,Majda (refined) (3.104)

DiPerna,Majda

Young

(3.102)

Lebesgue (3.97)

(3.99)

Ê (3.109)

p # 1 this diagram is enhan ed by another row:

(3.45)

while for

DiPerna,Majda (refined) (3.104)

DiPerna,Majda

Young

(3.102)

Lebesgue (3.97)

G # L ( ) and if R ontains (3.45) R ontains (3.45) finitely additive Radon measures Fonseca measures (3.107) on

(3.109)

L1%"

(3.111)

if

if



Radon measures on (3.110)

(3.105)

where ea h arrow goes from a ner onvex ompa ti ation to a oarser one. Moreover, ea h terminal onvex ompa ti ation is stri tly oarser than the initial one ex ept the

ase (3.104)

Ù(3.102) if G # C( ). Besides, no other arrow an be added; it means the

onvex ompa ti ations, whi h are not onne ted by a hain of arrows, are a tually

p # 1 and R given by (3.45), the relation between the 1 DiPerna-Majda, Fonse a, and L -Young measures is pretty exa t in the sense that69 not omparable. Moreover, for

DMR % ( ; Rm ) # sup „ F% ( ; Rm ); Y% ( ; Rm ) … 1

1



(3.112)



;

with F(

; Rm )

being dened by (3.76) but with



repla ed by

. In

fa t, we proved

only some of these relations, the rest being left as an exer ise.

Remark 3.55 (Convex ompa ti ations of L -balls are universal). Let us note that "p p S ' : h ÜÙ h , dened70 by h (x ; s) # h(x ; s s S ), is an isometri al isomorphism p Car% ( ; S) Ù Car% p ( ; S); indeed, one has the identity 1

(1



)/



1



h% #

!! ! !!X h ( x ; y ( x )) d x !!! !! !! ! y L p ;S ¢ % !

sup (

)

where the initial seminorm on erns

Car ( ; S). 1

68

#

!! ! !!X h ( x ; y ( x )) d x !!! ! !! ! p ! y L 1 ;S ¢ % !

sup (



# h% p ;

)

Carp ( ; S)

Then, having some linear subspa e

while the terminal one on erns

H ’ Carp ( ; S), H # S ' (H) ’ 

This is, in fa t, a well known result; see, e.g., Da orogna [241, Corollary 6.2℄ or Málek et al. [512,

Thm. 2.91℄.

69

p

The formula (3.112) follows from Theorems 3.40(i) and 3.42 if one realizes that, for p # 1, C ( ) ‹ R ) is dense in H % C ( ) ‹ C p ( m ) with H as in (3.105) but with C ( , S m"1 ) in pla e of C 0 ( , S m"1 ). (1" p )/ p Cf. also (3.185) with S1 # S2 # S and ' ( x ; s ) # s s . S

Ô (

70



R





3.3 A lass of onvex ompa ti ations of balls in

Lp

-spa es

Ë 185

Car ( ; S) determines a onvex ompa ti ation Y pH % p ( ; S) of a ball B % p ’ L ( ; S). The adjoint mapping ( S '  H ) : H Ù H is a homeomorphism and maps the onvex p p ( ; S); this means

ompa ti ation Y H % ( ; S) of the ball B % ’ L ( ; S) onto Y p H %p 1

1

1

;

*

*

*



1

;

;

p „ Y H ; % ( ; S) ;

H ; iH … Ê

H ; ip H Ž N' 

ΠY p p ( ; S) ; H;% 1

*

*



H # S ' (H) and N' : L p ( ; S) Ù L1 ( ; S) is the Nemytski mapping generated 1" p p . by ' dened by ' ( x ; s ) :# s s S

where



(

)/

Thus there is a one-to-one order-preserving orresponden e between onvex om-

p

B % ’ L p ( ; S) of the form Y H % ( ; S) with H ò Carp ( ; S) and 1

onvex ompa ti ations of a respe tive ball, namely B % p ’ L ( ; S), of the form 1 1 Y H % p ( ; S) with H ò Car ( ; S). Thus, the rst diagram in Summary 3.54 an be embedded into the se ond one provided " # p " 1, its image being denoted by gray pa ti ations of a ball

;

;

boxes.

Remark 3.56 (Convex ompa ti ations of Orli z spa es). Let us onsider an in reas% % ing onvex ontinuous fun tion M : R Ù R su h that M (0) # 0, lim a Ù% M(a) # %, and, for some k ; a ¡ 0 and every a £ a , M(2a) ¢ kM(a). The subset 

0

0

L M ( ) # of

L1 ( )

– uòL

1

(

);

X M ( u ( x )) d x

  % —

is alled an Orli z spa e.71 If equipped with the so- alled Luxemburg norm

M(u(x)/r) dx ¢ 1}, it be omes a separable Bana h spa e. The u ò L M ( ); P M(u(x)) dx ¢ 1}. In parti ular, for p M M(a) # a we have obviously L ( ) # L p ( ). The purpose of this generalization is to handle nonlinearities with non-polynomial growth, as M ( a ) # (1 % a )log (1 % a ) " a or M(a) # a p (1%log(a)). In parallel with the theory for L p -spa es, we an dene here



u L M

(

)

unit ball

# inf {r ¡ 0 : B1

P

is then just equal72 to {

the relevant spa e of integrands

CarM ( ; Rm ) # † h : , Rm Ù R Carathéodory; ; a h ò L ( ); b h ò R% : !!!!h(x ; s)!!!! ¢ a h (x) % b h M s ‡ 1

!

!

h # sup u LM ¢% !!!!P h(x ; u(x)) dx!!!! ; note that h Ž u is a 1 1 tually integrable be ause  h Ž u  ¢ a h % b h M ( u ) and both a h ò L ( ) and M ( u ) ò L ( ) M M provided u ò L ( ). For a subspa e H ’ Car ( ; R), we an dene a onvex ompa tiM M  ation „ Y H 1 ( ; R) ; H ; i H …, where the embedding73 i H : L ( ) Ù H is dened equipped with the seminorm   %





(

)

*

*

;

71

Su h spa es were introdu ed in thirties by Orli z [583℄. More details an be found, e.g., in mono-

graphs by Appell and Zabrejko [23℄, Krasnoselski  and Ruti ki  [441℄, and Kufner, John and Fu£ík [467℄.

72

See Krasnoselski  and Ruti ki  [441, Thm. 9.5℄.

73

By the ontinuity of the Nemytski  mapping

Thm. 17.6℄, the embedding

iH

Nh : L M ( )

is (strong,weak*)- ontinuous.

Ù L ( ) for any h ò Car M ( ; R), f. [441, 1

Ë

186

3 Young Measures and Their Generalizations

i u ; h> # P h(x ; u(x)) dx and Y HM 1( ; R) is the weak* losure of the unit M ball B 1 in L ( ) embedded via i H . Likewise in Remark 3.55, we an dene the isometriM 1 "1 (s)/s).

al isomorphism Car ( ; R) Ù Car ( ; R) by h ÜÙ h with h ( x ; s ) # h ( x ; sM M Then S is a homeomorphism between the onvex ompa ti ation Y H 1 ( ; R) of the 1 M unit ball B 1 in L ( ; R) and the onvex ompa ti ation Y p ( ; R) of the ball again by < H ( )

;





*

;

BM 1 (

H ; M (1)

)

in

L1 ( ). In parti ular, it shows that Y HM 1 ( ; R) is onvex and ompa t in H

*

;

.

Example 3.57 (Extension of a norm). Assuming 1 ‹  - p ò H ’ Carp ( ; Rm ), we an p m extend the norm on L ( ; R ) ontinuously by

p y <  ; 1‹

 :#

p

-  >

R

m );



 ¢ %}

(3.113)

(-) #

This ts with the abstra t Example 2.29 with

p Y H ( ;

:

p are onvex and equal to Y H ; % ( ;

R

p

(-) . In parti ular, {

 ò

m ). This is in parti ular the

ase of (3.28) and (3.30).

3.3.d

Coarse polynomial ompa ti ation by algebrai moments

H # G ‹ V with a niteV omposed from polynomials of the order ¢ 2k. The generalized p m Young fun tionals  ò Y ( ; R ) then orresponds to their algebrai moments, i.e. H Sophisti ated onstru tions exist for the spe ial ase of dimensional spa e

m #  DZ (1 ‹ s 1 s 2 - - - s mm ) 1

(3.114)

2

 # (1 ; 2 ; - - - ;  m ) is the multi-index of non-negative integers 1 % 2 % - - - %  m ¢ 2k. Namely, for any h ò H , it holds that

where

#

 ; h

H  ; g    ¢2 k

‹ s 1 s 2 - - - s mm # 1

2



H X g  (x) ¢2k

 :#

su h that  

m  (x) dx :

(3.115)

 

This potentially gives a han e to work e iently with su h oarse onvex

- ompa ti-

 ations. Yet, to this goal, one needs an e ient hara terisations of these moments.

m # 1. m £ 1, denoting m # (m  )  ¢2k , we dene the so- alled Henkel

This is not trivial and satisfa tory haraterisation exists only for In the general ase matrix

Hk m (

 

) as

Hk m :# „m1%1 (

)



;--- ;

 m % m … 

0 0

¢ 1 % - - - %  m ¢ k : ¢ 1 % - - - %  m ¢ k 

(3.116)



It is used parti ularly e iently in the one-dimensional situations where

m% ℄k  #

plies to [



;



1

Hk m (

) sim-

:

Lemma 3.58 (Polynomial moments of probablity measures).74 It holds

l –m # ŒX s i d (s) R

74

i #0 ; 1 ; : : : ; 2 k

;

òr a% (R)— # m ò R k ; 2

1

Hk m £ 0 : (

)

For this lassi al result see e.g. the monograph J.A. Shoat and J.D. Tamarkin [727℄.

(3.117)

3.3 A lass of onvex ompa ti ations of balls in

Lp

-spa es

Ë 187

H£0

The ordering of matri es in (3.117) is the so- alled Löwner ordering, i.e. {

}

is the losed one of all positive semidenite matri es. Disregarding the restri tion on balls in this subse tion, Lemma 3.58 gives:

Proposition 3.59 (Polynomial onvex - ompa ti ation of L p ( )). Let V ’ C k (R) i be the linear hull of { s ; i # 0 ; 1 ; :::; 2 k } and H # C ( ) ‹ V . Then, for p ¡ 2 k , the p p

onvex  - ompa ti ation Y H ( ; R) of L ( ) is equivalent to 2





m # (mi )i#

0;1; : : : ;2

k;

mi ò L p i ( ); Hk (m(x)) £ 0 : x ò  : /

(3.118)

a.a.

To relax oer ive optimization problems, the hara terisation (3.118) is needed for

p # 2k, in whi h ase it holds for p-non on entrating 's. Also the following onsequen e is useful:

Corollary 3.60. Let '(t) # #k  t  be a one dimensional,

k ¡ 0. Then, any solution  of the semi-denite program: 2

0

oer ive polynomial, i.e.

2

Minimize

2k H   #0

-m#

m

Hk m ³ 0 with m # 1 and m # a ;

subje t to

(

)

0

is omposed of the algebrai moments of a measure

(3.119)

1

solving the following abstra t

optimization problem dened in measures: Minimize X

R

Conversely, if

'(s)

ds)

(

subje t to X

R

ds) # a ;

s

(

òr a% (R) :

(3.120)

1

solves (3.120), then its algebrai moments solve (3.119).

Remark 3.61 (Ve torial problems).

The ve torial situation

m ¡ 1

is unfortunately

mu h more ompli ated and an be handled only approximately, using the asymptoti s for ountable number of higher-order momenta. To this goal, as devised for global optimization of polynomials with moments [380, 476, 477, 480, 594℄, one is to use also the so- alled lo alizing matrix

Lk m # „ m1%1 (

2

)



;--- ;

 m % m " 

m1 %1% 

2 ;--- ;

Lk m (

) dened, for some

 m % m " - - -

" m 1 %  1





;--- ;

 ¡ 0, as

 m %  m %2 … 

0 0

¢ 1 % - - - %  m ¢ k"1 : ¢ 1 % - - - %  m ¢ k"1 



u ò L ( ; Rm ); u(x) ¢ % :a.a. x ò } H # C( ) ‹ V with V # span{s11 s22 - - - s mm ;  # (1 ; 2 ; - - -  m ) ¢ 

Then, the onvex ompa ti ation of the ball {

Y H ( ; R 2k ; s ¢ %} an be approximated for û Ù  by the onvex sets in

m ) with





M # † (m )  ¢2k ; ;(m )2k%1¢  ¢2 : û

 

:a.a. x ò : m

 

(0 ; 0 ; : : : ; 0)(

û

m ò L ( ; Rm )

x) # 1;



H m x £ 0; û(

( ))

with

onsidered for û simple

M

û

for

 # 0; :::; 2û;

and

L m x £0 û(

m # (m )  ¢  

2û

( ))

‡

(3.121)

£ k. In view of Proposition 3.59, the situation for m # 1 is parti ularly M # M k for any û £ k. The usage

is independent of û and, in parti ular,

û

Ë

188

3 Young Measures and Their Generalizations

of su h result is for optimal- ontrol problems with a-priori bounded admissible ontrols or for oer ive variational problems after an approximation by a dis retisation of



whi h, for a xed dis retisation, an be expe ted to have solutions in su iently

big

L



-balls. Then û is to be taken su iently big and represents another approxima-

tion parameter, realizing an outer approximation of the semi-dis retised problem. Su h approximated semi-dis retised problems lead to a semi-denite mathemati al

programming (SDP) for whi h e ient numeri al methods and software pa kages exist, f. [305, 380, 381, 429, 799℄.

3.3.e

Compatible systems of Young fun tionals on B(I; L p )

The above onstru tions are appli able rather to stati problems or evolution problems whi h are in some sense quasistati with spe ial properties, f. Remark 8.7. For general evolution problems, an interesting and elegant onstru tion takes into a

ount ertain nonlo al intera tions like we already presented on an abstra t level in Se t. 2.6, although it should openly be said that its appli ability and interpretation is rather doubtful be ause of too big generality. Nevertheless, it develops a su iently wide lass of onvex ompa ti ations of the spa e of bounded mappings

I Ù L p ( ; Rm ) with I a ompa t interval of R, denoted by B(I; L p ( ; Rm )). Having in

mind some uniform a-priori estimates usually available, it su es to ompa tify only the ball

B  :#  u òB(I; L p ( ; Rm )); : t ò I : u(t) L p

(

;Rm )

¢ :

(3.122)

Coarse ompa ti ations handle spatial os illations/ on entrations on parti ular time levels separately but ner ompa ti ations an handle possible orrelations of su h os illations/ on entrations at various time instan es. Always, a nite (although not a-priori given) number of those time instan es su es to be in orrelation. To this goal, we systemati ally exploit the theory of inverse systems of onvex ompa ti a-

π # (t1 ; t2 ; :::; t# π ) a nite partition of I ’ R t1   t2   :::   t# π where #(π) denotes the number of elements of π and where t i ò I for all i # 1; :::; #(π). Let us denote by F(I) the olle tion of all su h partitions ordered by in lusion. It makes F( I ) dire ted. Let us further take, for any π ò F( I ), some p # π ,m ). Let us further dene normed linear subspa e H π ’ Car ( ; R

tions from Se tion 2.6. Let us denote by with

(

(

)

)

(

eπ : B(I; L p ( ; Rm )) Ù Hπ *

eπ (u); h

)

by

:# X h(x ; u(t ; x); u(t ; x); :::; u(t# π ; x)) dx :

1

2

We onsider simply the Cartesian produ t AπòF( I )

Hπ *

Hπ by (eπ )πòF I *

( )

Hπ . Then we embed B  *

and dene

YH;  (I; L p ( ; Rm )) :# l eπ (B  )πòF(I) ; p

(3.123)

equipped with the Tikhonov

produ t topology here ounting the weak* topologies of ea h into AπòF( I )

)

(

(3.124)

3.3 A lass of onvex ompa ti ations of balls in

Lp

-spa es

Ë 189

where the losure refers to the Tikhonov produ t topology ounting the weak* topolo-

Hπ and where H abbreviates the olle tion (Hπ )πòF(I) . Also, p p m )) # the losure of e ( B ), whi h is a ompa t we an onsider Y (I; L ( ; π  Hπ ;  p * #(π),m ) dened by (3.86). Dening the subset of H π . It is exa tly the set Y ( ; Hπ ;  p m )) Ù L p ( ; #(π), m ) Ê L p ( ; m )#(π) by j ( u ) # mapping j π : B( I ; L ( ; π *

gies on ea h parti ular

R



R R

R

R

u(t1 ; -); :::; u(t# π ; -)), we an see that eπ # i Hπ Ž jπ with i Hπ dened in (3.84) with p Hπ in pla e of H . As the triple (Y Hπ  ( ; R# π ,m ); i Hπ ; Hπ ) forms a onvex ompa tm p p p # π ,m ); max i ation of the set { u ò L ( ; R i #1 # π P j#1 u ij (x) dx ¢ % } just as (

(

)

)

(

*

;

)

(

p p explained in Se tion 3.3, ( Y H ;  ( I ; L ( ; π

Rm

;:::;

)

(

; eπ ; Hπ ) makes a onvex ompa ti ation of B  . If H π are ri h enough (as, e.g., in Theorem 3.39(iii)) ea h i H π is inje tive but e π is not be ause j π is not inje tive (ex ept a trivial ase that I itself is nite). Let p p m us note that, in spite of it, ( e π )πòF I is inje tive. Also note that Y H  ( I ; L ( ; R )) ’ p p AπòF I Y H  ( ; R# π , m ) and, as ea h Y H  ( ; R# π , m ) is ompa t, by Tikhonov's π π p # π ,m ) and thus also Yp (I; L p ( ; Rm )) itself is omTheorem 1.5, AπòF I Y Hπ  ( ; R H p p m pa t, too. Hen e, (Y ( I ; L ( ; R )) ; i ) forms a ompa ti ation of B  . This ompa tH 

*

))

( )

(

( )

;

(

)

;

)

;

(

( )

)

;

;

;

i ation is not metrizable ex ept trivial ases.

π1 Let us now assume existen e of the olle tions ( π2 )π1 ; π2 òF( I ) of linear operators

P

satisfying (2.44) with

Hπ in pla e of Fπ .

1 :# [Pπ π2 ℄ : H π1 Ù H π2

π1 π2

*

: Y Hp π

π1 π2

is surje tive, and then also

1 ;

surje tion, just showing that

p

Y Hπ

1 ;

(

By (2.44a), the adjoint mapping

(

*

)

(3.125)

p

; R# π1 ,m ) Ù Y Hπ (

p

; R# π1 ,m ); Hπ1 ; eπ1  ³ Y Hπ (

*

*

2 ;

(

)

2 ;

(

; R# π2 ,m ) (

)

; R# π2 ,m ); Hπ2 ; eπ2 ; (

)

*

is a

(3.126)

R

p

#(π1 ),m ) of B is ner than the onvex om( ; % 1 ; p π1 # (π2 ), m pa ti ation Y H ). By (2.44a), is

ontinuous, and by (2.44b) it sat( ; π2 π2 ;  π2 π1 π1 π ises π3 Ž π2 # π3 , and eventually (2.44 ) ensures π # identity. The olle tion p p m )) then satises the property that (  π )πòF( I ) of Y (I; L ( ; H;  i.e. the onvex ompa ti ation

R

Y Hπ

R

π2 #

π1 π2  π1 whenever

f. (2.47). Altogether, the operators

p

S # Œ(Y Hπ

1 ;

(

π1 “ π2 :

π1 π 2 play the role of the bonding mappings and thus

; R# π1 ,m ); Hπ1 ; eπ1 ); (

)

*

π1 π2  π1 ; π2 òF( I ); π1 “π2

is an inverse system (in the sense of Se t. 1.1) of onvex ompa ti ations of

YH;  (I; L p ( ; Rm )) is its limit, i.e. p

YH;  (I; L p ( ; Rm )) # lim S p

By Proposition 2.39,

e

ØÚÚ

YH;  (I; L p ( ; Rm )) p

with

S

(3.127)

B  , and

from (3.127).

itself is a onvex ompa ti ation of

B  ; the

embedding is ( π )πòF( I ) and the linear spa e indu ing its onvex stru ture is now

Ë

190

AπòF( I )

3 Young Measures and Their Generalizations

Hπ . By (2.39), also *

YH;  (I; L p ( ; Rm )) Ê sup Y Hπ ;  ( ; R#(π),m ): p

p

πòF( I )

p p H;  ( I ; L ( ;

The threads, i.e. the elements of Y

Rm

)), f. (1.3), are also alled

systems of Young fun tionals. For a spe ial hoi e of the system

ompatible

H as in Example 3.63

p # 1, su h systems have been invented in [248, Se t.7℄ (under the name

below and

 ompatible systems of generalized Young measures) and further used in [249, 250, 306,308℄. A general ansatz based on Example 3.63 below has been s rutinized in some variant also in [466℄.

Example 3.62 (Non- orrelated threads).

A rather standard but oarse onvex om-

pa ti ation is obtained simply by opying the onstru tion from Se tion 3.3 onstantly at ea h time instant, obtaining thus the onvex ompa t subset of the produ t (

H

*

I ) into whi h

B%

is embedded simply by

p

YH

;

% ( ;

Rm

)

I

u ÜÙ (i H (u(t)))tòI . Up to an

equivalen e of onvex ompa ti ations, we an obtain this onvex ompa ti ation in the above framework, too. To this goal, let us put

Ü H#k#π1 h k (x ; s k ); h k ò H DZ : Hπ :# Ž(x ; s1 ; :::; s# π ) Ù (

(

)

(3.128)

)

Carp ( ; R# π ,m ) if H is a subspa e of Carp ( ; Rm ). # π and we an dene π Thus, due to the spe ial hoi e (3.128), here H π Ê ( H ) Ê ( H ) I the linear inje tive mapping : (H ) Ù AπòF I Hπ by  ÜÙ (π )πòF I whi h is also a homeomorphi al embedding. Dening the system H :# ( H π ) πòF I by taking p Hπ from (3.128), we obtain the onvex ompa ti ation YH  (I; L p ( ; Rm )) by (3.124). π1 π1 The bonding mappings are dened as π2 with P π2 : H π2 Ù H π1 given by Hπ

Obviously,

(

is a subspa e of

)

*

*

*

*

(

)

*

( )

( )

( )

;

π Pπ12 h
(x ; s1 ; :::; s# π1 ) :# h(x ; s j1 ; :::; s j# π2 (

)

(

)

(3.129)

)

j : π2 Ù π1 is just the in lusion π2 ’ π1 . The mapping rep p Y H % ( ; Rm )I then realizes the homeomorphism between Y H % ( ; Rm )I and p YH  (I; L p ( ; Rm )), whi h makes these onvex ompa ti ations equivalent to ea h

where here stri ted on

;

;

;

other.

Example 3.63 (Correlated threads based on DiPerna-Majda's measures).

Based

on

the DiPerna-Majda measures, the onstru tion of threads nontrivially orrelated have essentially been invented in some variant in [466℄. For

p DMR #π (

)

(

; R

#(π),m ) with a separable ring R #(π)

’C

0

R

π ò F(I),

one an use

(

#(π),m ) orresponding either

the one-point Alexandro ompa ti ation of the ompa ti ation by a sphere

S# π ,m"1 . The bonding mappings are again determined as in Example 3.62 by means of (3.129). Here, for any π2 ’ π1 ò F( I ), it is important that (

)

:g ò C( ) :v ò R # π2 : (



with Ô

)

p p # π1 1 Pπ π2 ( g ‹ Ô ( v )) ò C ( ) ‹ Ô (R

p from (3.89), whi h indeed holds true for (3.129).

(



)

)

(3.130)

3.4 A lass of onvex



- ompa ti ations of

Remark 3.64 (Threads with a bounded variation). d(s1 ; s3 ) ¢ d(s1 ; s2 ) % d(s2 ; s3 )

:π ò F(I); #(π) £ 2 :

”dπ

Ë 191

-spa es

The onstru tion from Se t. 2.6 an

Rm Ù R% , i.e. the triangle inm is satised for all s ; s ; s ò R . If

be applied here when onsidering the distan e equality

Lp

d :

2

(

)

1

: (x ; s ; :::; s# π ) ÜÙ 1

(

)

2

3

#(π)

H d ( x ; s i "1 ; s i )• ò H π i #2

;

(3.131)

f. (2.49), like in [248, Def.8.1 and 8.6℄ we an dene the dissipation of a thread

I

with respe t to the distan e

d

( )

#(π)

{

;

}

(

Rem.8.3℄, one an write also

dòH

is

 ; dπ > # i#2 ; note that al-

from (3.131). It holds < π

ways

 over

Dissd (; I) :# supπòF I where dπ

by

{

)

{

( )

;

}

t i"1 ; t i } due to (3.131). The Helly sele tion prin iple as in Proposition 2.41

an be applied here for sequen es of

's. The ase p ¡ 1 is however not ompatible

with (2.53). Thus a weakened variant of both (2.53) has naturally to be used, based on

p-non on entrating threads. We all a thread  ò YH  (I; L p ( ; Rm )) pp m non on entrating if there is a net { u  }  ò  ’ B % ’ B( I ; L ( ; R )) attaining the thread p  su h that {u  (t ; -) ;  ò  ; t ò I} is relatively weakly ompa t in L1 ( ). Analogously, p p m a sequen e of threads {  k } k òN ’ Y H  ( I ; L ( ; R )) is alled equi- p -non on entrating p

the notion of

;

;

if there are nets { {

u k } ò k ’ B % ’ B(I; L p ( ; Rm )) attaining the parti ular  k

su h that

u k (t ; -)p ;  ò  k ; t ò I; k ò N} is relatively weakly ompa t in L1 ( ). For a modied

Helly prin iple then see [688, Prop. 6℄.

3.4

A lass of onvex

 - ompa ti ations of L p -spa es

In this se tion we will join the results from Se tion 3.3 with the theory of onvex

- ompa ti ations

of normed linear spa es as presented in Se tion 2.3 in order

- ompa ti ations of the Lebesgue L p ( ; S) with S a separable Bana h spa e. These onvex - ompa ti ations

to onstru t a su iently ri h lass of onvex spa es

will be sometimes also lo ally ompa t,

B - oer ive, and norm- onsistent.

Through-

out this se tion, we will onsider

U # L p ( ; S) endowed by the norm bornology

(3.132)

B.

We will onsider again the spa e

Carp ( ; S) from Se tion 3.3, but here we endow

it by the olle tion of seminorms { -  % } % òN dened again by



whi h makes

!!

!!

h% # sup !!!!X h(x ; u(x)) dx!!!! ; uòB % !

!

(3.133)

Carp ( ; S) a lo ally onvex spa e. We will refer to this topology as the

natural one. Obviously, it is the oarsest topology whi h makes all the identities

Ë

192

3 Young Measures and Their Generalizations

Carp ( ; S) Ù Carp% ( ; S) with % ò N ontinuous; note that (3.133) oin ides with (3.82). p p The mapping : Car ( ; S) Ù C ( U B ) and the embedding i : U Ù Car ( ; S) are dened respe tively by (natural extension of) (3.83) and (3.84); i.e. < i ( u ) ; h > # [ h ℄( u ) # P h ( x ; u ( x )) d x . Let us note that  h  % ¢  h  % % for any % ò N. Also note that

p  h  % #  h  B % # h C 0 B % so that is a homeomorphi al embedding of Car ( ; S) p into C ( U B ) if one onsiders an appropriate fa tor spa e, namely Car ( ; S)/Ker ; *

1

(

re all that

)

C(UB ) was endowed with the olle tion of seminorms † - B % ‡%òN .

For a linear subspa e

H ’ Carp ( ; S)

we dene

FH

’ C ( U B )

again by an

extension of (3.85), i.e.

# (H) % { onstants on U}:

FH

(3.134)

e H : U Ù C(UB ) and i H : U Ù H are dened as in Se tion 3.3, i.e. e H (u) # e(u)FH and i H (u) # i(u)H . Eventually, we put *

Also

p

YH

;

#

% ( ; S)

i

l H *  H (

B % ) ;

(3.135)

and

p

YH ( ; S) #

p ℄ YH ; % ( ; S) % òN

p # b lB H  i H ( L ( ; S)) :

(3.136)

*

Convention 3.65 (Generalized Young fun tionals).

p

The elements of YH (

; S) will be ad-

dressed as generalized Young fun tionals. Let us note that this onvention agrees with the previous Convention 3.38 be ause

p

H endowed with the topology of the seminormed spa e Car% ( ; S) is a subp spa e of the dual of H endowed with the (relativized) topology of Car ( ; S) (or by any p ner lo ally onvex topology), and it is easy to see that Y H % ( ; S) from Se tion 3.3 an p p be a tually onsidered as a subset of Y ( ; S), oin iding obviously with YH % ( ; S) H

the dual of

;

;

dened by (3.135). This justies our notation.

Theorem 3.66.

Let

H be a linear subspa e of Carp ( ; S), p ò [1; %℄, U

B the norm bornology. Then: The linear subspa e F H of C ( U B ) is B - onvexifying ( f. * (M(F H B ) ; F ; e H ) is a onvex  - ompa ti ation of ( U; B ). H

and

FH

given

by (3.132) and (3.134), and (i)

(2.16)) and thus

Carp ( ; S) su h that H ’ H , then M(FH B ) ³ M(FH B ), and if H has the same losure in Carp ( ; S) as H , then M(FH B ) lo Ê

(ii) If

H







is another linear subspa e of





FH B ).

M(

H is endowed with a lo ally onvex topology ner than the natural topolCarp ( ; S), then (YHp ( ; S); H ; i H ) forms a onvex - ompa ti ation of ( U; B ) whi h is equivalent with (M(F H B ) ; F ; e H ) via the adjoint mapH ping .

(iii) Moreover, if

*

ogy indu ed from

*

*

Proof. By Theorem 3.39, every Therefore,

FH

is

FH B %

#

FH ; %

is a onvexifying subspa e of

C0 (B % ).

B - onvexifying with respe t to the anoni al norm bornology base.

3.4 A lass of onvex

By (3.134), (

FH



- ompa ti ations of

ontains onstants, so that M(F H B ) is a onvex

U; B ) by Theorem 2.22, whi h proves (i). As for the point (ii), obviously F H

’ FH



provided

H’H



Lp

-spa es

Ë 193

- ompa ti ation of

, and therefore M(F H B )

² M(FH B ) again by Theorem 2.22. p As the natural topology of Car ( ; S) is proje tively indu ed from C B ( U ) via , the fa t that lCarp S H # lCarp S H implies l C B U ( H ) # l C B U ( H ), whi h implies lo

l C B U F H # l C B U F H , whi h eventually implies M(F H B ) Ê M(F B ) by TheoH 

;

(

(

)

)

(

(



;

(

)

)

(



)





)

rem 2.22. Let us go on to the point (iii). As the topology on

H

is ner than the topology in-

Carp ( ; S) proje tively via from C(UB ), the linear operator : H Ù FH is ontinuous. The fa t that the adjoint operator : FH Ù H restri ted on M # { ò FH ; # 1} is inje tive and has a weakly* ontinuous inverse an be demonstrated exa tly as in the proof of Theorem 3.39. As M(F H B ) ’ M by the very denition of M(F H B ) and Ž e H # i H by (3.87), we an on lude that realizes p Å the ane homeomorphism between M(F H B ) and Y ( ; S). H

du ed on

*

*

*

*

*

*

Proposition 3.67 (Lo al ompa tness, onsisten y). Let H be a linear subspa e of Carp ( ; S) endowed with a lo ally onvex topology ner than the topology indu ed from Carp ( ; S). Then: m (i) If S # R and H “ L p ( ) ‹ (Rm ) ; (3.137) 

then (ii) If

*

YH ( ; Rm ) is sequentially B - oer ive. p

p   % and H ontains a oer ive integrand h in the sense H ó h ;

where

p

h (x ; s) £ s S ;

(3.138)

p

YH ( ; S) is B - oer ive and lo ally ompa t. Moreover, if there is an equality in p m m (3.138) and if also p ¡ 1, S # R , and (3.137) is fullled, then YH ( ; R ) is norm onsistent; i.e. the embedding i H is (strong,weak*)-homeomorphi al. then

Proof. Supposing (3.137) and taking a sequen e { u k } k òN su h that { i H ( u k )} k òN on-

, we obtain in parti ular that {< i H ( u k ) ; g ‹ v >} k òN onverges in m

) and v(s) # m l #1 v l s l for some ( v 1 ; :::; v m ) ò R . As < i H ( u k ) ; g ‹ v > # p m P g ( x ) u k ( x ) d x with g ò L ( ; R ) given by [ g ( x )℄ l # g ( x ) v l , we an see that { u k } k òN

p m m

onverges weakly* in L ( ; R ) if p ¡ 1 or in L ( ; R ) Ê vba( ; Rm ) if p # 1. p m In any ase, we have a weak* onvergen e in a dual to the Bana h spa e L ( ; R ) and, by the Bana h-Steinhaus prin iple, the sequen e { u k } k òN must be bounded in L p ( ; Rm ); for p # 1 we used also of the oin iden e on L1 ( ; Rm ) of the norms of L ( ; Rm ) and of L1 ( ; Rm ). Thus (i) is proved. Let us take f # h ò F H with h from (3.138). Then

verges weakly* in

R

p for any g ò L

H

*



(





*





*

f (u) #

X h ( x ; u ( x )) d x

£

p X u ( x ) S d x

# u pLp S : (

;

)

Ë

194

3 Young Measures and Their Generalizations

Therefore we have the oer ivity

inf uòU

\

rem 2.22 we an on lude that M(F H B ) is

p

p

£ % p Ù % for % Ù %. By Theo-

B % f (u)

B - oer ive and lo ally ompa t, hen e so

YH ( ; S) be ause M(FH B ) Ê YH ( ; S) by Theorem 3.66 and be ause B - oer ivity and lo al ompa tness are invariant under the equivalen e of onvex  - ompa ti ations, f. Proposition 2.20. The inverse ontinuity of the embedding i H was shown in

is

Å

Theorem 3.39(iii).

N 2

Corollary 3.68.

Let p ò [1 ; %). There are at least 2 lo ally ompa t B - oer ive on- ompa ti ations of L p ( ; S) whi h are even norm- onsistent provided additionm m N lo ally ally S # R and p ¡ 1. Besides, if S # R with m ¡ 1, there are at least 2 p m

ompa t B - oer ive onvex  - ompa ti ations of L ( ; R ) whi h are even sequenvex

tially lo ally ompa t. Proof. It su es to take

p [Ô v ℄( s ) that

p

YH

s

H # H(s) # C( ) ‹ Ôp (R (s)) from (3.88) with Ôp v dened by

# v(s)(1 % s Sp ); f. (3.89). Sin e always 1 ò R (s), we have (3.138) fullled, so p ( ; S) is a lo ally ompa t B - oer ive onvex  - ompa ti ations of L ( ; S)

( )

by Proposition 3.67. If

S#

Rm and p ¡ 1, then these onvex - ompa ti ations are

norm- onsistent by Proposition 3.67(ii). For dierent

s ò S m"1

- ompa ti ations ( f. again the m ¡ 1 the sphere S m"1 ontains at least 2N ele-

we get dierent onvex

proof of Theorem 3.41. However, for

ments. As the above subrings are separable, the respe tive ompa ti ations of

Rm

are metrizable, and the sequential lo al ompa tness follows by the arguments of Ex-

Å

ample 3.70 below.

The reader an easily verify that, having a linear subspa e

L



(

G su h that C( ) ’ G ’

), the property of H to be G-invariant an be dened again by (3.91) and Proposip YH ( ; S) need not be bounded in

tion 3.43 is still relevant. The only dieren e is that

H

*

be ause, instead of (3.95), we have at our disposal only the estimate



i H (u) H ¢ sup *



h H ¢1

[

h℄(u) ¢ C % sup 

h % ¢1

[

h℄(u) ¢ C %

(3.139)

u ò B % , where C % denotes here the onstant from the assumed estimate  h  % ¢ C % h H . Of ourse, a blow-up C % Ù  with % Ù  is not ex luded, see Exp ample 3.76 where C % # % . Therefore, the joint ontinuity of ( h ;  ) ÜÙ h DZ  stated in p Proposition 3.43 is relevant for  ranging only Y H % ( ; S), whi h is ertainly a bounded subset of H . provided

;

*

Let us now have a look how the examples from Se tion 3.3 an be modied.

Example 3.69 (L p -Young measures).

Let us take

p ò [1; %), S #

Rm , and H from

(3.97) endowed with the norm (3.98). This strong topology is ner than the (relativized) natural topology indu ed of

Carp ( ; Rm ), whi h an be seen from the estimate (3.99)

valid for any

is

% ò N. Again H

C( )-invariant, satisfying also (3.93). By Example 3.46

3.4 A lass of onvex



Lp

- ompa ti ations of

Ë 195

-spa es

and Proposition 3.22,

Y H ( ; Rm ) lo Ê Y L1 p

p

(

; C 0 (Rm ))

(

; Rm )

Ê Yp ( ; Rm ) #

†

ò Y( ; Rm );

X

Rm

s

 

p

x (d s ) ò L

1

(

) ‡ :

(3.140)

p ¡ 1, then also Yp ( ; Rm ) lo Ê Y H1 ( ; Rm ) with H1 # H % L p ( ) ‹ m (R ) . By Theorem 3.66 it su es to show that H is dense (in the natural topology of Carp ( ; Rm )) in H1 . Indeed, having some h # g ‹ idò L p ( ; Rm ) ‹ L(Rm ; Rm ), we an m take always a sequen e h k # g k ‹ id with g k ò C ( ; R ) onverging to g in the norm of p m m p L ( ; R ) be ause the embedding C( ; R ) ’ L ( ; Rm ) is dense if p ¡ 1, and then p

Moreover, if



*











by the Hölder inequality 

h k " h% #

!! !!X h ( x ; u ( x )) k !! u L p ;R m ¢ % !

sup (

¢

!

)

sup



u L p ;Rm



u L p ;Rm

¢

!!

" h(x ; u(x)) dx!!!!

(

X

)

¢%

)

¢%

sup (





g k (x) " g(x) - u(x)dx

g k " g L p



(

;R m )



u L p

(

Ù 0:

;Rm )

h k Ù h in Carp ( ; Rm ). On the other hand, ea h h k lives in H from (3.97) provided p ¡ 1 be ause we an ertainly write h k ( x ; s ) # g k ( x ) - s # m  l #1 [ g k ( x )℄ l v l ( s )(1%  s  p ) with v l ( s ) # s l /(1% s  p ), and obviously v l ò C 0 (R m ) provided p ¡ 1. As (3.137) is obviously fullled for H 1 , the onvex  - ompa ti ation p Y H1 ( ; Rm ) is sequentially B - oer ive. However, Yp ( ; Rm ) itself is not sequentially This just shows that

B - oer ive.75

Example 3.70 (The generalization by DiPerna and Majda).

Let us take

p ò [1; %), S #

Rm , R a omplete subring of C Rm , and H from (3.102). Again we an endow H by 0

(

h

)

the norm H dened by (3.98), whi h satises (3.93) and indu es a ner topology than the topology indu ed from

Carp ( ; Rm ) be ause of (3.99). Then the respe tive

R R

R

p m ) is equivalent with the subset DM p ( ; m ) #

onvex  - ompa ti ation Y ( ; H R p U % òN DMR; % ( ; m ) of r a( , R m ) dened by (3.51). Let us note that (3.138) is fulp m ) forms a B - oer ive lled, so that the set of all DiPerna-Majda measures DM ( ; R p m lo ally ompa t onvex  - ompa ti ation of L ( ; ). If 1   p , this onvex  - om-

R







R

R

pa ti ation is even norm- onsistent thanks to Theorem 3.39(iii).

p m m DMR ( ; R ) is lo ally sequentially ompa t provided R R is metrizm m m able. Indeed, if R R is metrizable, so is , R R , and then C ( , R R ) ontains a m

ountable dense subset,76 and therefore the weak* topology of r a ( , R R ) is metrizm able on subsets whi h are bounded with respe t to the dual norm on C ( , RR ) . Moreover,









*



p One an easily see that, for any sequen e { u k }kòN unbounded in L ( ; k"1 , Æ(u k ) onverges weakly* to Æ(0) in Yp ( ; m ).

75

76

R

See, e.g., Bourbaki [144, X.3.3, Theorem 1 and IX.2.8, Proposition 12℄.

Rm

supp(u k ) ¢

) su h that 

Ë

196

3 Young Measures and Their Generalizations

#   % u pLp Rm guarantees that lr a , Rm i(B % ) is R m ) for every % ò N. However, for every bounded (and thus metrizable) in C ( , R R p  ò DM ( ; R m ), there is % ò N large enough for l  R r a , Rm i(B % ) to be a (sequeniu

The identity ( )

C ( ,

R

Rm )

*

(

;

)

(

)

*





(

R



)

tially ompa t) neighbourhood of  .

Example 3.71 (A renement of DiPerna-Majda measures). For a ring G su h that C( ) ’ G ’ L ( ), we an again dene a G-invariant subspa e H by (3.104) and en



h

dow it by the norm H dened by (3.98). We get thus a lo ally ompa t

R

G #Ö C( ),

p m ). If

onvex  - ompa ti ation of L ( ; p m DiPerna-Majda measures DM ( ; ). R

R



B - oer ive

it is stri tly ner than the

Example 3.72 (Fonse a's extension of L -spa es). For p # 1 one an take H from m m (3.105) and then obtain Y ( ; R ) a onvex  - ompa ti ation of L ( ; R ) whi h is H m equivalent with the set F( ; R ) of all Fonse a's measures. It is not B - oer ive, howm " ) with C ( , S m " ) ever. A slight enlargement of H from (3.105) by repla ing C ( , S m yields Y ( ; R ) a B - oer ive, lo ally (sequentially) ompa t onvex  - ompa ti aH m m tion of L ( ; R ) equivalent with the set F( ; R ) of Fonse a's measures on ; f. 1

1

1

1

0

1



1

1





also Summary 3.54.

Example 3.73 (Coarser onvex - ompa ti ations).

Let us take

p ò [1; %℄, S #

R

Rm ,

p m ) is equivalent for from (3.107) with the norm dened by (3.108). Then Y ( ; H   p ¢ % with L p ( ; m ) endowed by the weak* topology77 while for p # 1

H

1

R

; Rm ) Ê vba( ; Rm ) endowed by the weak* topology, 1 m i.e. the bi-dual spa e of L ( ; R ). The ane homeomorphism is via the mapping p m H Ù L ( ; R ) adjoint to the operator g ÜÙ g ‹ id : L p ( ; Rm ) Ù H . These spa es serve as examples of homogeneous sequentially B - oer ive onvex  - omit is equivalent with

*



L



(

*



*

pa ti ations whi h are not

B - oer ive; f. also Examples 2.25 and 2.26. Analogously,

Y H1 ( ; Rm ) Ê r a( ; Rm ) while Y H ( ; Rm ) Ê L p ( ; Rm ) for p ¡ 1. Likewise, the hoi e (3.110) yields Y H1 ( ; Rm ) Ê r a( ; Rm ) and (3.111) yields Y H1 ( ; Rm ) Ê L1%" ( ; Rm ).

the hoi e (3.109) yields

p



Remark 3.74 (Metri ompletion of L p ( ; S)).

d on U # L p ( ; S), the most natural extension is its ompletion with respe t to the metri d , whi h is a omplete metri spa e ( U ; d ) su h that U is embedded densely into U and d  U , U # d . p Negle ting the onvex stru ture, Y ( ; S) an a tually be identied with a suitable H p

ompletion of L ( ; S) provided H is separable and satises (3.138). Then an appropriate metri on d an be: d(u1 ; u2 ) # d h (u1 ; u2 ) %

77

For

Having a metri



H2 k #1

"k

d h k (u1 ; u2 ) ; 1 % d h k (u1 ; u2 )

1   p   % it is merely the weak topology be ause of the reexivity of L p ( ; Rm ).

3.4 A lass of onvex

where, for

h ò H , d h (u1 ; u2 ) #



P ( h ( x ; u 1 ( x ))

h



- ompa ti ations of

Lp

-spa es

Ë 197

" h(x ; u (x))) dx and h is the oer ive 2

H . It is an easy exer ise d if and only if it is bounded and weakly* onvergent when embedded into H via i H . Let us note also p m that d indu es just the strong topology on L ( ; S) provided S # R , p ¡ 1, and H 

integrand from (3.138) and the olle tion { k } k #1 is dense in to show78 that a sequen e in

L p ( ; S)

is Cau hy with respe t to *

satises (3.137) and (3.138) with an equality; f. Proposition 3.67(ii).

Summary 3.75.

Some properties of onvex

- ompa ti ations from Examples 3.69

3.73 are summarized in the following table:

Convex  - ompa ti ation: L p ( ; Rm ), p ¡ 1 r a( ; Rm ), p # 1 r a( ; Rm ), p # 1 vba( ; Rm ), p # 1 L1%" ( ; Rm ), p # 1 Yp ( ; Rm ), p £ 1 F( ; Rm ), p # 1 p m DMR ( ; R ) DiPerna-Majda, rened

B - oer ive

sequentially B - oer ive yes yes yes yes yes no yes yes yes

no no no no no no yes yes yes

Table 3.1. Properties of on rete onvex

Properties: lo ally

ompa t no no no no no no yes yes yes

norm

onsistent no no no no no no no yes yes

linear manifold yes yes yes yes yes no no no no

- ompa ti ations.

p Let us note that, in fa t, the L -Young measures

Yp ( ; Rm )

have the worst geomet-

ri al/topologi al properties,79 no matter how useful they are and how inspiring role they histori ally played. Let us still add that all onvex and Hausdor. Besides, for DM

p R ( ;

Rm

sures if both

) if the ring

R’C

0

R

(

- ompa ti ations in Table 3.1 are homogeneous

F( ; Rm ) is lo ally sequentially ompa t, whi h holds also R is separable and also for the rened DiPerna-Majda mea-

m ) and

G’L



(

) are separable.

Example 3.76 (A norm on Carp ( ; S)). We want to show that, ex ept the ase p # %, p even the whole lo ally onvex spa e Car ( ; S) an be normed in su h a way that (3.93) is satised, although parti ular subspa es may admit stronger norms whi h are some-

N

u i }iòN is Cau hy means: :" ¡ 0 ;i0 ò :i1 ; i2 £ i0 : d(u i1 ; u i2 ) ¢ ". For su h d h (u i ; 0)}iòN is bounded in so that, by (3.138), {u i }iòN is bounded in L p ( ; S). The weak* onvergen e of { i H ( u i )}iòN is standard; see, e.g., Bishop and Bridges [121, Se t. 7.6℄, Holmes [392, 78

A sequen e {

R

a sequen e {

Se t. 15℄ or Warga [791, Thm. I.3.11℄.

Rm

m # 1 for simpli ity) is not lo ally ompa t an be shown by taking, 1 * N of Æ(0) ò L w* ( ; r a( )) Ê L ( ; C 0 ( )) , a sequen e { Æ ( k A )}kòN ’ p Y ( ; ) whi h lies in N whenever A ’ has a su iently small positive measure depending on N p ). but not on k ; su h sequen e weakly* onverges but its limit, being zero on A , does not live in Y ( ;

79

The fa t that

Yp ( ;

) (with

for any weak* neighbourhood

R

Note also that this sequen e is not tight.

R

R

R

198

Ë

3 Young Measures and Their Generalizations

times easier to be handled or inevitable for rate-of-error estimates, f. the ondition (3.154 ) below. Considering

h Carp

(

p ò [1; %), we put

#

;S)

inf

:(x ; s)ò ,S: h(x ; s)¢a(x)%b s Sp



a L1

(

%b:

)

(3.141)

Carp ( ; S), it will be a norm. The positive homogeneity h Car p S #   h Car p S for any ò R is obvious. Let us prove the triangle inequality h % h Carp S ¢ h Car p S % h Carp S . For l # 1 ; 2 and for every " ¡ 0, there are a l " ò L ( ) and b l " ò R su h that h l Carp S £ a l " L 1 % p b l " " " and h l (x ; s) ¢ a l " (x) % b l " s S . Realizing that h (x ; s) % h (x ; s) ¢ [a " % p a " ℄(x) % (b " % b " ) s S , we an estimate

Making possibly a suitable equivalen e on (

;

)

(

1

2

;

1;

2

)

;

(

)

(

;

1

;

h1 %h2 Carp

(

;S)

¢ a

1;

¢  a ¢ h " ¡ 0

;

(

;

)

(

2

)

1;

2;



As

)

;

;

2;

;

1

)

1

;

;

(

%a

"

2;

" L1 ( )

1;

% (b

" L1 ( )

%b

1;

"

%b

2;

")

"  %  a 2 ; " L 1 ( ) %b 2 ; " 

% h

Carp ( ;S)

1

1;

2

Carp ( ;S)

% 2" :

has been arbitrary, the triangle inequality is proved. The fa t that (3.93) is

satised in this ase is plain. The topology generated by the norm (3.141) is ner than

% ò N. Indeed, for every " ¡ 0 there are again a " ò L1 ( ) and b " ò R su h that h Car p ;S £ a " L1 % b " " " p and  h ( x ; s ) ¢ a " ( x ) % b " s . Obviously, both a " and b " must be non-negative. Then S for every % ò N we an estimate

the lo ally onvex topology indu ed by the seminorms - % with



h% ¢

sup

X  h ( x ; u ( x )) d x

u L p ;S ¢ % (

)

¢

X a " (x)

u L p ;S ¢ % (

¢ a Passing with

sup

(

)

(

)

% b " u(x) Sp dx

)

" L1 ( )

% b " % p ¢ % p  h Car p S % ": (

" to zero, we get the estimate h% ¢ % p h Carp

(

;

(3.142)

)

;S) , whi h shows that the

norm from (3.141) generates the ner80 topology than the natural one. It is ertainly useful to have at our disposal a pro edure how to onstru t larger linear subspa es of

Carp ( ; S) from original ones, without deteriorating signi ant proper-

ties of the original spa es. This is handled by the following assertion.81

Proposition 3.77. Let G be a linear subspa e C( ) ’ G ’ L ( ) and H; H ; H be p subspa es of Car ( ; S) equipped with some norms generating ner topologies than the 



1

2

In fa t, (3.141) indu es even a stri tly ner topology be ause h Car p ( ;S) ¡  h % # 0 for any h ( x ; s ) # a(x) with a #Ö 0 but P a(x) dx # 0. Anyhow, one an show that h Carp ( ;S) ¢ h% % % h" % with % # 1, h% # max(0; h) and h" # max(0; "h) if one use the al ulations dedu ing the estimate (3.81) from p 1 the boundedness of N h : L ( ; S) Ù L ( ), f. Lu

hetti and Patrone [499, proof of Thm. 3.1(2)℄, uniformly with respe t to h . 81 The reader is en ouraged to prove that similar assertion holds also for the subspa e H 1  H 2 endowed with the norm h H H 1 2 # h H1 % h H2 .

80

3.4 A lass of onvex

(relativized) natural topology on (i)



- ompa ti ations of

Lp

Ë 199

-spa es

Carp ( ; S). Then:

H1 % H2 endowed with the norm

The subspa e

h H1 %H2 #



inf



h#h1%h2 h1 òH1 ; h2 òH2

h1 H1 % h2 H2

(3.143)

has a ner topology than the (relativized) natural topology on

Carp ( ; S). Moreover,

H1 and H2 are G-invariant and satisfy (3.93), then H1 %H2 is G-invariant and H1 and H2 are separable, then H1 %H2 is separable,

if both

satisfy (3.93), too. Also, if both too. (ii) If

G is a ring (i.e. G - G # G), then the linear hull k – H gl l #1

span(G - H) #

-

h l ; k ò N; g l ò G; h l ò H —

(3.144)

G-invariant linear subspa e of Carp ( ; S) ontaining H . Moreover, if both G and H are separable, then span( G - H ) is separable if equipped with the norm

is the smallest

(3.141). Proof. The fa t that (3.143) denes a norm is quite obvious,82 let us only show the trian-

h ; h ò H1 % H2 . By the denition (3.143), for any " ¡ 0 there are h 1 " ; h 1 " ò H 1 and h 2 " ; h 2 " ò H 2 su h that h # h 1 " % h 2 " , h # h 1 " % h 2 " ,

gle inequality. Let us take ;

;



;



and





;

;

;





;



;

h1 " H1 % h2 " H2 " " ¢ h H1 %H2 ¢ h1 " H1 % h2 " H2 ; ;

;

;

(3.145)

;

h 1 " H1 % h 2 " H2 " " ¢ h H1 %H2 ¢ h 1 " H1 % h 2 " H2 : Now we an estimate ;



;







;



;

h % h H1 %H2 ¢ h1 " % h 1 " H1 % h2 " % h 2 " H2 ;



¢ „ h

1;

# „ h

1;

As



;

;

;



" H1 %



h 1 " H1 … % „ h2 " H2 % h 2 " H2 …

" H1 %



h2 " H2 … % „ h 1 " H1 % h 2 " H2 … ¢ „ h H1 %H2 % " … % „ h H1 %H2 % " … :

;



;

;



;



;



;



" ¡ 0 was taken arbitrarily, the triangle inequality for the norm - H1 %H2

has been

proved.

h ¢ C1 % h H1 and h% ¢ C2 % h H2 valid for any % ò N. Taking the de omposition h # h 1 " % h 2 " satisfying (3.145), we an estimate Let us now suppose   %

;

;



h% ¢



h1 " % % h2 " % ¢ C1 % h1 " H1 % C2 % h2 " H2 ;

;

¢ max(C with

;

;

1;

;

% ; C 2 ; % ) h 1 ; " H 1

;

% h

;

2;

" H2 

;

¢ C %  h H1 H2 % "

C % # max(C1 % ; C2 % ). Letting " ÿ 0, we an see that the norm (3.143) generates a Carp ( ; S). ;

;

ner topology than

82

The impli ation

h H1 %H2

# 0 âá h # 0 (in the sense h(x ; -) # 0 for a.a. x ò ) follows from the

assumption that the normed spa es

onvex spa e

H1 and H2 are ontinuously embedded into the Hausdor lo ally

Carp ( ; S); f. Gajewski et al. [342, Chap. I, Rem. 5.13℄.

Ë

200

3 Young Measures and Their Generalizations

The fa t that H 1 % H 2 is G -invariant is obvious. Let us now suppose g - h H 1 ¢ C1 g L h H1 and g - h H2 ¢ C2 g L h H2 valid for any g ò G and h belonging to H 1 and H 2 , respe tively. Taking the de omposition h # h 1 " % h 2 " satisfying (3.145), 

(



)

(

)

;

;

we an estimate

g - h H1 %H2 ¢ g - h1 " H1 %H2 % g - h2 " H1 %H2 ;

¢ g - h

1;

;

" H1

¢ max(C ; C 1

% g - h 2 )

g L



2;

(

" H2

¢ C g L 1



(

) h1; " H1

)  h 1 ; " H 1 % h 2 ; " H 2 

% C g L 2

¢ C g L



(



(

) h2; " H2

)  h H 1 % H 2 % " 

C # max(C1 ; C2 ). Letting " ÿ 0, we obtain (3.93) valid for H1 % H2 . The separability of H 1 % H 2 follows from the separability of H 1 and H 2 by the separate (strong,strong,strong)- ontinuity83 of the mapping ( h 1 ; h 2 ) ÜÙ h 1 % h 2 : H 1 , H2 Ù H1 % H2 . The point (i) has been thus proved. p The fa t that span( G - H ) “ H is a G -invariant linear subspa e of Car ( ; S) is obvious. Also, span( G - H ), being a linear hull of the set G - H # { g - h ; g ò G ; h ò H }, is the smallest G -invariant subspa e greater than H . Sin e the mapping ( g; h ) ÜÙ g - h : G , H Ù G - H is separately (strong,strong,strong)- ontinuous,84 G - H and thus also span(G - H) is separable provided G and H are so. This proved (ii). Å

with

Let us now turn our attention to an important property of generalized Young fun -

p-non on entrating DiPerna-Majda measure. In p analogy with this, we will say that a generalized Young fun tional  ò Y H % ( ; S) is p-non on entrating if it an be attained by a net {u  } ò in L p ( ; S) (in the sense that lim ò i H (u  ) #  weakly* in H ) su h that the set { u  Sp ;  ò } is relatively weakly 1

ompa t in L ( ). We saw already one impa t of this property in Proposition 3.43(ii). m Conning ourselves to the ase S # R , let us now state another important onse-

tionals. In Se tion 3.2. we dened a

;

*

quen e of this property:

Proposition 3.78 (Young-measure representation). (i)

H be separable. Then: p  ò YH ( ; Rm ), there exp m ists a (not ne essarily uniquely determined) Young measure ò Y ( ; R ) su h that For any

p-non on entrating

: hòH :

Let

generalized Young fun tional

 ; h

#

X X

Rm

h(x ; s)

x (d s ) d x :

(3.146)

ò Yp ( ; Rm ) determines by the formula (3.146) a p-non on entrap m ting generalized Young fun tional  ò YH ( ; R ).

(ii) Conversely, any

ò Y( ; Rm ) is a onsequen e of the Ball lemma 3.20 if one

Proof. The existen e of realizes that, sin e

H is separable and thus the weak* topology on bounded subsets of

h1;1 % h2 " (h1;2 % h2 ) H1 %H2 # h1;1 " h1;2 H1 %H2 ¢ h1;1 " h1;2 H1 . g - h1 " g - h2 Carp ( ;S) ¢ g L ( ) h1 " h2 Carp ( ;S) and

83

This follows from the estimate

84

This follows from the obvious estimates



g 1 - h " g 2 - h ¢ g 1 " g 2 L



(

) h Carp ( ;S) .



3.4 A lass of onvex



- ompa ti ations of

Lp

Ë 201

-spa es

 is weakly* attainable by a bounded sequen e {u k }kòN ’ L p ( ; Rm ) su h that { u k k ò N} is relatively weakly ompa t in L1 ( ), and that also the set 1 { h Ž u k ; k ò N} is relatively weakly ompa t in L ( ) be ause h has at most p -growth. p m Then it is also lear that ò Y ( ; R ). Let us go on to (ii). It is lear that (3.146) determines a linear fun tional on H . Let us take a sequen e { u k } k òN su h that, with some % ò R, u k L p ;Rm ¢ % and that generp m ates ò Y ( ; R ) in the sense that lim k Ù P h ( x ; u k ( x )) d x # P P m h ( x ; s ) x (d s ) d x

R 1 m for any h ò L ( ; C 0 (R )). As H is separable, we an even suppose this onvergen e to hold for any h ò H ; f. the proof of Proposition 3.22 and Remark 3.23. Then we an H

*

is metrizable,

p  ;

(

)



estimate

!! !!X X !! ! Rm

h(x ; s)

!! ! x (d s ) d x !!! !

!!

!!

¢ sup !!!!X h(x ; u k (x)) dx!!!! k òN !

¢

! !! !! sup !!!!X h(x ; u(x)) dx!!!! ! u L p ;R m ¢ % !

(

where  -  % is the seminorm of fun tional

# h% ;

)

Carp ( ; Rm ), see (3.82). This shows the ontinuity of the

 determined by (3.146); re all that the topology on H

is always supposed

to be ner than the topology indu ed by  -  % . Altogether, we proved that

u

 ò YH ( ; Rm ). p

In view of the biting lemma 1.29, the sequen e { k } k òN an be (if ne essary) modied

u k }kòN again generates and the set { u k p ; k ò N} is L ( ); f. the proof of Proposition 3.81 below. It shows

so that the modied sequen e { relatively weakly ompa t in that





1

Å

 is p-non on entrating. The following assertion shows, in parti ular, that every sequen e in

whi h attains a

ò Y Hp ( ;

p-non on entrating  - ompa ti ation in

vided the onvex

R

L p ( ; Rm )

m ) does not on entrate energy pro-

H

question is ne enough, i.e. if

is large

enough.

Proposition 3.79 (Non on etration of sequen es).85 Let {u k }kòN be a bounded sep m quen e in L ( ; R ) su h that ea h weak* luster point of { i H ( u k )} k òN in H is p -nonp

on entrating and let H be su iently ri h,86 e.g. let H ontain H # C ( ) ‹ Ô (R ) *

0

85 For

It should be emphasized that this assertion does not hold on oarser onvex

p

p ¡ 1 and Y H ( ;

Rm

)

u k p ; k

0

- ompa ti ation.

Ê L p ( ; Rm ) (i.e. H is from (3.107)) one an take any sequen e {u k }kòN from

Figures 3.77-9; then obviously

{



i H (u k ) onverges to i H (0) whi h is ertainly p-non on entrating while

ò N} is not relatively weakly ompa t in L ( ). For p # 1 a similar example was already 1

onstru ted for the Fonse a measures in Remark 3.36.

86

H 's. For example, we an ompa tify not the origL p ( ; m ) but the spa e of energies L1 ( ) by taking H0 # C( ) ‹ V with V # {v ò m ); ; v ò C ([0 ; %℄) : v ( s ) # v ( s )(1% s p )}, whi h makes Y p ( ; m ) equivalent with 0 0 H0

In fa t, the assertion holds also for a bit smaller

inal spa e

R

Cp (

R



R

DM1R0 ( ; R); H0 ; i) with R0 the smallest omplete subring in C0 (R) and [i(u)℄(x) # u(x)p so that

(

*



we an use Lemma 3.27(ii) on this oarser ompa ti ation.

Ë

202 with

R0

set { u k

3 Young Measures and Their Generalizations

C0 (Rm ) ontaining onstants. Then the k ò N} is relatively weakly ompa t in L1 ( ).

being the smallest omplete subring of

p  ;

Proof. Suppose that the assertion does not hold, i.e. { u k  weakly ompa t in

L ( ). 1

p;

k ò N} u

is not relatively

Then we an sele t a subsequen e { k } k ò N 1 , every subse-

u

quen e of whi h does on entrate energy, i.e. { k 

p;

k ò N2 } is not relatively weakly

L ( ) whenever N2 ’ N1 is innite.87  of {i H (u k )}kòN1 and a ner net {i H (u k  )} ò onverging to ; p of ourse k  ò N 1 for any  ò  and therefore the set { u k   ;  ò  } is inevitably not 1 relatively weakly ompa t in L ( ). As  is also a luster point of { i H ( u k )} k òN , it must be p -non on entrating. We have also w*-lim  ò  i H 0 ( u k  ) #  #   H 0 , where  is p -non on entrating, too. Sin e H 0 is separable, we an now onsider  ’ N 1 dire ted by the standard ordering p indu ed from N. By Lemma 3.27(ii), we an see that { u k   ;  ò  } is relatively weakly 1

ompa t in L ( ), a ontradi tion. Å 1

ompa t in

Take a luster point





Let us introdu e another important notion, whi h will serve as a powerful tool

Ž p Ž Ž  ;  ò YH ( ; Rm ), we say that  is a p-non on entrating modi ation of  if  is Ž p-non on entrating and # holds for any h ò H su h that h(x ; s) ¢ a(x) % o(sp ) with some a ò L1 ( ) and o : R% Ù R satisfying limrÙ o(r)/r # 0. Let us note that the on ept of the p -non on entrating modi ation is sensible only if H ontains integrands whi h have (in absolute value) the growth pre isely p be ause otherwise every generalized Young measure is, by the very denition, automati ally the p -nonlater. For



on entrating modi ation of itself. Let us also remind that an example of a on rete

Ž  ÜÙ  was demonstrated in Proposition 3.30 for the

pro edure realizing the mapping

ase of the DiPerna-Majda measures. The following assertion justies our notation, showing that mined uniquely by

Ž , if exists, is deter-

 in question.

Proposition 3.80 (Uniqueness of p-non on entrating modi ation). Every Ž p p  ò YH ( ; Rm ) admits at most one p-non on entrating modi ation  ò YH ( ; Rm ). Ž Ž p 1 ; 2 ò YH ( ; Rm ) are two p-non on entrating modi ations Let us take h ò H and put h r ( x ; s ) # h ( x ; s ) v r ( s ) with the  ut-

Proof. Let us suppose

 ò YH ( ; Rm ). o  fun tion v r given again by (3.38). Without loss of generality we an suppose 1 m that L ( ; C 0 (R )) ’ H . More in detail, if it is not the ase, we an repla e H by Ž Ž 1 H # H % L ( ; C0 (Rm )) and extend  ; 1 ; 2 on this enlarged spa e so that again p

of

p Indeed, by Dunford-Pettis theorem 1.28(ii), { u k  ; k ò N} is not uniformly integrable, whi h means ;" ¡ 0 :n ò N ;k n ò N: P xò uk x p ¢n u kn (x)p dx £ ". Putting N # {k n ; n ò N}, we have, for n any N ’ N innite, :K ò R ;n ò N (e.g. n # min(N  [K ; %℄)) P xò un x p £K u n (x)p dx £ p p P  u n ( x ) d x £ " , whi h shows that the set { u k  ; k ò N } is not uniformly integrable. xò u n x p £n

87

{

2

{

; 

1

( )

; 

( )

}

1

}

2

2

{

2

; 

( )

}

3.4 A lass of onvex

Ž Ž p  ; 1 ; 2 ò YH ( ; Rm ).



- ompa ti ations of

Ž 1

Lp

-spa es

Ë 203

Ž 2

remain p -nonŽ Ž

on entrating as well. If one shows  1 #  2 in the sense of H , then it is obvious that it holds for the original fun tionals on H as well. Thus, adopting the agreement 1 m that H ontains L ( ; C 0 (R )), we may and will suppose h r ò H be ause always 1 m h r ò L ( ; C0 (R )). Ž Ž As h r has a growth less than p and both  1 and  2 are p -non on entrating modi ations of  , we have Besides, the extended fun tionals

and *

Ž

 1 ; h r

Ž

#  ; h r #  ; h r :

(3.147)

2

Now we want to show that

Ž

Ž

lim  ; h r #  ; h :

r Ù

1

(3.148)

1

Ž 1 is p-non on entrating, there is a net {u  } ò bounded in L p ( ; Rm ) su h that w*Ž lim ò i H (u  ) # 1 and the set {u  p ;  ò } is relatively weakly ompa t in L1 ( ) and

As

therefore, by the Dunford-Pettis theorem 1.28(ii), this set is also uniformly integrable. This means that, for any

" ¡ 0, one an nd r "

sup  ò

As In

X {



x ò ;  u  ( x )p £ r " }

su iently large so that

u  (x)p dx ¢ " :

h ò H ’ Carp ( ; Rm ), we have h(x ; s) ¢ a(x) % bsp for some a ò L1 ( ) and b ò R. parti ular, a is absolutely ontinuous in the sense that, for any " ¡ 0, there is

m " ¡ 0 small enough so that

sup X a(x) dx A ’ measurable A  A ¢ m "

¢ ":

x ò ; u  (x) £ r} ¢ (C/r)p with C # "1 p 1 p r £ max(Cm " 2 ; r " 2b ) and every  ò , we an

Let us noti e that it ertainly holds {

sup ò u  L p Rm (

;

).

Then, for every

/

/

/

/

estimate 

!!

!!

i H (u  ); h r " h  # !!!!X h(x ; u  (x))(v r (u  (x)) " 1) dx!!!! !

!

¢

X {

¢

X {

a ( x ) x ò ;  u  ( x )£ r }

a(x) dx % X

x ò ;  u  ( x )£ r }

Passing to the limit with

% bu  (x) {

p



dx bu  (x)p dx ¢

x ò ;  u  ( x )£ r }

"

2

%

"

2

# ":

Ž  ò , we obtain  # . As this holds for any h ò H with the growth less Ž that p , we have shown that  is the p -non on entrating modi ation of  . Å so that we showed that
# <  " P  ; h > # <  ; h " Pd h >. By the estimate d <  "  d ; h > ¢  H h " Pd h H together with (3.154b) we an see that  d Ù  weakly*. It remains to prove (3.150). Of ourse, we put again  d # P  ò K d . Then d Let us go on to (3.149b). For a given

*

*

*

*



 "  d p H # sup *



h pH ¢1



"  d ; h # sup  ; h " Pd h

h pH ¢1

¢ sup  H h " Pd h H ¢ sup Cd   H h pH # Cd   H :

*

h pH ¢1



h p H ¢1

*

Statement (iv) follows dire tly from (3.149b) be ause always thanks to (3.152) and (3.153); re all that

*

p

K d ’ YH ( ; S)

B - oer ivity implies losedness due to Propo-

Å

sition 2.20(i).

Though the abstra t onstru tion introdu ed above is quite simple, the proper task onsists in a hoi e of

H and a onstru tion of the parti ular proje tors Pd and of

the norms - H and - p H whi h t with a treated on rete problem, an be easily implemented, and satisfy the above required onditions. Let us remark here that (3.154b) is not ne essary for (3.149b) and a tually sometimes (3.149b) must employ another

onstru tion than

 d # Pd ; see, e.g., (5.95b). *

P

Obviously there is a great amount of possibilities how to onstru t d , but we mention now only some (hopefully quite representative) examples whi h will be used also in the following hapters.

Ë 209

3.5 Approximation theory

We will use two parameters

d1

and

d2

for dis retisation of

and S, respe tively,

and onstru t our proje tor always as a omposition

Pd $ P where the parti ular proje tors variable

(

d1 ; d2 )

# Pd 1 PdS2 # PdS2 Pd 1 ;

Pd 1 and PdS2

(3.155)

are responsible for the dis retisation in the

x ò and s ò S, respe tively. Possibly either Pd 1 or PdS2

may be the identity.

The following assertion is useful if one wants to verify (3.154 ) for (3.155) from the knowledge of (3.154 ) for the parti ular proje tors

h

Pd

in the form

and

p ’ H and two norms - p H ;H 2 H1 # h pH1 % h pH2 .

suppose that we have given two subspa es p1 The norm - p H 1 p H 2 is dened by p H1 p H2

Pd 1

PdS2 . Let us and - p H2 .

Proposition 3.84. Let there be C , C , C ,  ;  ¡ 0 su h that, for all h ò H and d # d ; d ) ¡ 0, the following estimates hold: 0

(

1

1

2

1

2

2

" " " h " "

"

" Pd 1 h""""H ¢ C d 1 h pH1 1

" " " h " "

and

1

"

" PdS2 h""""H ¢ C d 2 h pH2 ; 2

(3.156a)

2

and

" " " " P h""" p " " d1 " H2

¢ C h pH2 0

" " " " PS h""" p " " d2 " H1

or

¢ C h pH1 :

(3.156b)

0

Then the approximation property (3.154 ) is valid. More spe i ally, for and for

C # max(C1 ; C2 ; C0 C1 ; C0 C2 ), it holds " " " "h

Pd

from (3.155)

" Pd h""""H ¢ C(d 1 % d 2 ) h pH1 pH2 : 1

(3.157)

2

Proof. Let us suppose that, for example, the rst part of (3.156b) is satised. Then we

an estimate:

" " " "h

"

"

"

"

"

"

" Pd h""""H # """"h " PdS2 Pd 1 h""""H ¢ """"h " Pd 1 h""""H % """" Pd 1 h " PdS2 Pd 1 h""""H

¢ C d 1 h pH1 % C d 2 Pd 1 h pH2 ¢ C d 1 h pH1 % C C d 2 h pH2 ¢ (C d 1 % C C d 2 ) h pH1 pH2 ¢ C(d 1 % d 2 ) h pH1 pH2 1

1

with with

2

1

1

0

2

2

2

1

0

1

1

2

2

2

C # max(C1 ; C0 C2 ). If the se ond part of (3.156b) is valid, we get su h estimate C # max(C2 ; C0 C1 ). Å

Remark 3.85 (Approximations of Type III).

Though approximations of Type III may

seem a bit less natural, they are used most often mainly be ause some of them an be implemented by the same way as original, non-relaxed problems. We have in mind the situation when simply

K d # i H (U d ) ; U d ’ L p ( ; Rm ) nite-dimensional:

(3.158)

U d is a onvex subset (or a linear subspa e) of L p ( ; Rm ), but the embedp m ding i H : L ( ; R ) Ù H is not ane provided H ontains at least one non-ane integrand, whi h makes eventually K d from (3.158) non onvex.

Typi ally,

*

Ë

210

3 Young Measures and Their Generalizations

U d ontains the L p ( ; Rm ) on a nite-element triangulation Td

1 of like in (3.160) for d # d 1 . For the ase m # 1 and ’ R , the approximation is outlined on Figure 3.11 where an equi-distant partition of onto sub-intervals of the length d is used. Let us illustrate this kind of approximation in the ase that

element-wise onstant fun tions from

S

S

PSfrag repla ements

d

Fig. 3.11:



The onventional non onvex approximation of a Young measure.

We would like to noti e that, in fa t, su h kind of approximation has been already

onstru ted in the Step 2 of the proof of Theorem 3.6. Let us only remark that error estimates an be also obtained for this ase; e.g. if

n # 1 and H # L1 ( ; C(S)) with S ’ Rm ompa t, one an derive90 the estimate

 "  d C 0 1 ;

[

(

; C ( S )) L



(

¢ Cd 1 2

; C 0 2 ( S ))℄ ;

*

/(

1 2 %1 m%2 )

 d # i H (u d ), where u d is pie e-wise onstant

’ R1 onto the sub-intervals of the length d # d1 .

with



 L1

(

; C ( S ))

*

(3.159)

on the equi-distant partition of

To ompare (3.159) with (3.169), one should estimate the dimensionality of the resulting problems. For this it is essential that, to over ments of the diameter less than

d1

and

’

Rn and S ’ Rm by ele-

d2 , one needs minimally O(d"1 n ) and O(d"2 m )

mesh points (=variables), respe tively. Therefore, to realize (3.169) the number of mesh points

D

n # 1

as only

must be proportional to

d"1 n d"2 m ,

while for (3.159) we have

D È d "1

is admitted in this ase. In the ase of (3.169), we fa e the question

d1 and d2 to get the highest rate of onverd11 /d22 È onst., whi h yields the rate of error Thus for n # 1 the estimate (3.169) yields the

of an optimal syn hronization between gen e. This optimal ratio is obviously

D as O(D"1 2 1 m%2 n ). "  1 m%2 ) while error O( D 1 2 /(

in terms of rate of

/(

O(D"1 2 /(1 m%2 %1 2 ) ).

)

)

the estimate (3.159) gives a slightly worse rate

Su h omparison with the semi-dis retisation of Type II, reated e.g. by the proje tor

Pd

*

with

Pd # Pd 1 , is not possible for the ase of (generalized) Young fun tionals

without any spe ial properties. On the other hand, one is often interested only in (generalized) Young fun tionals exhibiting some spe ial properties (like being solutions of optimization or variational problems). Then a semi-dis retisation of Type II may

90

C in the error estimate depends linearly on the Hölder ';2 , f. [661℄ for details.

We refer to [661, Lemma 3.1℄, realizing that

ontinuity onstants

';1

and

Ë 211

3.5 Approximation theory

appear even far more e ient than the full dis retisation, as we will see in Se tions 4.3.e and 6.6.

An approximation over

3.5.b

The simplest approximation over spa e (or time) is by a dis retisation of



and by

an element-wise homogenization.91 Supposing the reader to be roughly familiar with basi ideas of the nite-element method (FEM), we dis retise the domain

’

Rn by

a nite-element mesh, say a triangulation. For simpli ity, we will suppose that



is

polyhedral and, for any d 1 ¡ 0, T d 1 is a triangulation of onsisting of elements of

the diameter not ex eeding d 1 . Ea h element E ò T d 1 is therefore a simplex with n % 1 



verti es. For d 1 £ d 1 ¡ 0, we suppose that T d 1 ’ Td 1 , this means Td 1 is a renement



* of T . Our aim is to onstru t P so that ( P )  will be element-wise homogeneous d1 d1 d1 



generalized Young fun tionals. This will be done ( f. Proposition 3.86 below) if the proje tor

Pd 1

makes spatial averages within ea h element, so that the result

Pd 1 h will

be an element-wise onstant Carathéodory integrand dened by

Pd 1 h
(x ; s) # Equivalently, [

1 

E

X h( x ; s) d x E 

if 

x ò E ò Td 1 :

(3.160)

1 Pd 1 h℄(x ; s) # [ Pd 1 h(-; s)℄(x) where the average operator P

d : L ( ) Ù 



L ( ) is dened by 1

P

d g
( x ) #

1





E

X g( x ) d x E 

x ò E ò Td :

if 

Let us illustrate the interpretation of the operator ( Young measures, i.e.

#{

91

x } x ò via the mapping

from Lemma 3.4,

*

Indeed, for every

#

p

Pd 1 )  an be identied with an element-wise homogeneous (= onstant d1 x

1

on the ase of the lassi al

H # L ( ; C(S)) and S ’ S ompa t, f. Se tion 3.1. If  ò YH ( ; S)

d1

on ea h element) Young measure

* ´( Pd )  ; h µ

*

1

is identied with the Young measure we laim that (

Pd 1 )

(3.161)

#

1



E

#{

X E

x

d1 x } x ò dened by

dx 

x ò E ò Td 1 :

if

(3.162)

h ò H # L1 ( ; C(S)), we an write, using Fubini's theorem, that

# ´ ; Pd 1 hµ # 1

1

X h( x ; s) d x  H X X œ E S E E E òTd

1 

H X œ X X h ( x ; s ) x (d s ) d x  E E E S E òTd

1 

dx



x (d s ) d x



For numeri al approximation of Young measures by an element-wise homogenization see also Pe-

dregal [598, 599℄, or also [671℄.

Ë

212

3 Young Measures and Their Generalizations

d1 H X X h ( x ; s ) x (d s ) d x E S E òTd

1

#



where



d1 X h ( x ; s ) x (d s ) d x

S

#X

#



(

d1

)

; h ;

was dened in Lemma 3.4. This proves (3.162).

The element-wise homogenization pro edure is illustrated on Figure 3.12, whi h uses

S ’

R

1

a one-dimensional domain (=an interval) dis retised by an equid1 .

and

distant partition onto the sub-intervals of the length

S

S (

PSfrag repla ements

Pd 1 )*



d1 Fig. 3.12:

The element-wise homogeneous approximation of a Young measure.

We would like to noti e that su h onstru tion has been already used in the Step 2A of the proof of Theorem 3.6, f. (3.18). Let us investigate some approximation properties of the proposed proje tor (3.160). The requirement (3.151) as well as (3.154a) are ertainly satised for

L ( ; C(S)) 1

h # P supsòS h(x ; s) dx. Also both (3.154b) H # W  1 ( ; C(S)). This follows from the estimates



S % Pd 1 h L 1 ; C S ¢ 2 h L 1 ; C S and h " Pd 1 h L 1 ; C S

with the standard norm H

and (3.154 ) are satised for p

h " P h L 1 ( ; C ( S )) ¢ h L 1 ( ; C ( d1

¢ Cd h W 1 1 C S ;

1



H #

(

;

h " Pd 1 h L1

(

(

)) .

;

))

(

(

))

(

(

))

(

(

))

Then by interpolation92 we obtain

; C ( S ))

¢2

" C  d  h

1

1

B 1 1 ( ; C ( S )) ;

¢2

" C  d  h

1

1

W  1 ( ; C ( S )) ;

(3.163)

B pq (-) denotes a Besov spa e. p Furthermore, let us investigate H # G ‹ Ô (R ) from (3.104) used to rene the DiPerna-Majda measures. For G # C ( ) ( f. (3.102)) we get the standard DiPerna-Majda measures but then Pd H ’Ö H so that (3.151) is not fullled; to approximate the DiPerna

Majda measures, we would have had to hoose another P d 1 than (3.160), e.g. a on-

where



tinuous, element-wise ane interpolation instead of the element-wise onstant averaging as in (3.160). For our hoi e (3.160), the requirement (3.151) will be fullled

G # L ( ), in whi h ase H is not separable, however. Yet there exist subspa es C( ) ’ G ’ L ( ) whi h satisfy (3.151) and yield H separable provided the ring R is separable. For example, we an take for G the spa e 

if





G0 #

92

℄ d 1 ¡0

G d1

with

G d1 # †g ò L



(

); :E ò Td 1 ; gE ò C(E )‡ ;

We refer to, e.g., Bergh and Löfström [109℄ for details.

(3.164)

3.5 Approximation theory

or also the losure of

G0 in L



(

). Su h G is separable be ause ea h G d1

and the olle tion of triangulations {T

Ë 213

is separable

d 1 } d 1 ¡0 is supposed ountable. We an easily see

that (3.154a) is fullled, but (3.154b) is not! Nevertheless, (3.149b) an be ensured by another way than via Proposition 3.83, namely by a dire t onstru tion of appropriate

u d1 element-wise onstant on Td 1 su h that i H (u d1 ) onverges weakly* to a given  ò p p ’ H on erns, we an take, e.g., p YH ( ; S). As far as the subspa e H H # C0  ( )‹ V , enm m dowed with the norm h p H # Ô p h C 0  ; C S , where Ô p : Car( ; R ) Ù Car( ; R ) p is dened by [Ô p h ℄( x ; s ) # h ( x ; s )/(1%  s  ). Then (3.154 ) is satised with C # 1. p p

The very nontrivial fa t that ( P ) : Y ( ; S) Ù Y ( ; S), needed for (3.152), is H H d1 ;

;

(

(

))

*

obvious for the ase of lassi al Young measures93 while for the general ase, this will

be proved later in Proposition 3.86. Let us still investigate some theoreti al properties of the proje tor

P

d from (3.160),

whi h we will be frequently used in what follows. In parti ular, we want to show a quite nontrivial fa t that it satises the hypothesis (3.152).

Proposition 3.86 (Properties of the proje tor Pd ).94 Let a linear subspa e H ’ Carp ( ; S) ontains densely95 some G ‹ V with a subspa e G su h that G ’ G ’ L ( ), where G is from (3.164). Then, for every d ¡ 0: p p

(i) ( P ) maps Y ( ; S) into Y ( ; S). H H d p



(ii) For any  ò Y ( ; S), ( P )  is element-wise homogeneous, i.e. (1 ‹ v ) DZ ( P )  is H d d element-wise onstant for any v ò V . p (iii) If  ò Y ( ; S) is p -non on entrating and v ò V , then H 

0

0

*

*

" lim """ (1 d Ù0 "

*

"

‹ v) DZ (P d )  " (1 ‹ v) DZ  """"L1 # 0 : *

(

)

R

G ‹ V by the norm h G‹V # Ôp h L ( ,S) ; for S # m it is just

(3.98). Note that P maps G ‹ V into G ‹ V and is ontinuous with respe t to this norm. d p p lo Sin e we suppose G ‹ V ’ H densely, we have Y G ‹ V ( ; S) Ê YH ( ; S) so that we an Proof.96 Let us endow



onne ourselves to test-integrands from

G ‹ V.

* * * P

d ) : (G ‹ V) Ù (G ‹ V) p p p * maps Y G ‹ V ( ; S) ’ ( G ‹ V ) into YG ‹ V ( ; S). Let  ò YG ‹ V ( ; S). By the denition of p YG‹V ( ; S), there is a bounded net {u  } ò ’ L p ( ; S) su h that i G‹V (u  ) Ù  weakly* p p *  in ( G ‹ V ) . As L ( ; S) is dense in L ( ; S) (with respe t to the L -norm topology

In view of Theorem 3.66, it su es to show that (

in whi h the embedding

93 94

i : L p ( ; S) Ù ( G ‹ V )

*

is ontinuous), we may and will

This follows simply from the expression (3.162) together with Theorem 3.6.

L p -Young measures. The onverp  ò YH ( ; S) provided v ò V  C p (S). p It refers to the natural topology of Car ( ; S) but, of ourse, it su es to have the density in any The point (iii) generalizes the result by Pedregal [598℄ stated for

gen e (iii) holds, in fa t, even for arbitrary

95

ner (e.g. a strong) topology.

96

We use basi ally the te hnique by Kinderlehrer and Pedregal used in [424, 426℄ for the ase of

Young measures. Here it is a bit modied be ause we do not require any non on entration of separability of

H (hen e metrizability of bounded sets in H *).

, nor

Ë

214

3 Young Measures and Their Generalizations

u ò L

; S) although, of ourse, the net {u  } ò is generally unbounded  ranges the universal index set  # N , {nite subset of G ‹ V } dire ted by the relation assume that

in

L



(

; S).



(

Without any loss of generality, we an always assume that the index

¢ , ’, f. also Example 1.4.

Let us now make our onstru tion only for an (arbitrary) element

E ò Td . For every

k ò N, we take a overing (up to a set of zero measure) of E by a ountable (or possibly Pk of pairwise disjoint subsets of the form x kj % " kj E with some x kj ò E and

nite) family

0   " kj ¢ 1/k. The existen e of su h overing follows by the Vitali argument97 [780℄

E has; f. also Figure 6.3 on p. 445. Besides, we an always suppose k%1)th overing is a renement of the kth overing. Then, for  # (k ; {h l }),

whatever shape that the ( we put

u  (x)

#

. 6 > 6 F

u Œ

0

x"x kj " kj



for

x ò x kj % " kj E ; j ò N ;

elsewhere

Making this onstru tion on every element

(3.165)

:

E ò Td , we get eventually u  ò L p ( ; S). As p

u  } ò is bounded in L p ( ; S), so is {u  }   ò, . Sin e YG‹V % ( ; S) are ompa t for % any % ò R and the universal index set  is ri h enough, its image via i must onverge in ( G ‹ V ) (possibly only as a ner net but indexed again by  , f. Example 1.4) to p

some element in Y G ‹ V ( ; S). We want to show that it is just ( P d )  provided the ner {

( ;

)

;

*

*

net is sele ted arefully. First, we take

 ò  xed. The net {i G‹V (u  )}ò must onverge (possibly as a ner p

  ò YG‹V ( ; S). We want to show that   # (P

d ) i G ‹ V ( u  ). Thus we are to



show # <   ; g ‹ v > # # d d

for any g ò G and v ò V . Note that P , dened by (3.161), maps G into itself be ause d



we supposed G “ G 0 with G 0 from (3.164), and that P ( g ‹ v ) # ( P g ) ‹ v , hen e d d



# . Let us again lo alize our onsidd d erations on E and take g k ò G  E pie ewise onstant on the partition P k . Then net) to some

*

*

*





*



 X g k ( x ) v ( u  ( x )) d x E

whenever

#

x"x kj

H X g k (x)vŒ u   k  dx k k "j j òN x j % " j E

#

k n k H (" j ) X g k (x j E j òN

% " kj x)v(u  (x)) dx #

#

k n k X v ( u  ( x )) d x H ( " j ) g k ( x j ) E j òN

#

1

X g k (x) dx X v ( u  ( x )) d x E E E

(3.166)

 ò  is su iently large, namely  # (k ; {h l }) with k £ k. Altogether, this

gives

97

k n k H ( " j ) X g k ( x j ) v ( u  ( x )) d x E j òN

See also e.g. Dunford and S hwartz [275, Se t. III.12.2℄.





Ë 215

3.5 Approximation theory

 i G ‹ V ( u  ) ; g k

‹ v #

 H X g k ( x ) v ( u  ( x )) d x E E òT

d

1

#

X g k (x) dx H X v ( u  ( x )) d x E E E E òT

#

* ( P d ) i G ‹ V ( u  ) ; g k

# i G‹V (u  ); P d g k ‹ v 

d

‹ v #

(1

‹ v) DZ (P d ) i G‹V (u  ); g k : *

u  ; g k ‹ v> Ù #

i

1 ‹ v) DZ   ; g k >. # for every g k ò G pie ewise

Simultaneously, we know that < G ‹ V ( ) Therefore, / E .

1 As to (iii), it is an easy exer ise to show that limd Ù0 P g # g strongly in L ( ) for d 1 any g ò L ( ), in parti ular for g # (1 ‹ v ) DZ  , as well. Å

As this holds for an arbitrary ontinuous

g

supported on

*



3.5.

An approximation over S PdS2 in the simplest ase where, instead of a separable m ompa t polyhedral. For every d ¡ 0, take only S ’ R 2

Now we give an example for Bana h spa e

S,

we take

Ë

216

3 Young Measures and Their Generalizations

than





TdS2 . Then denote by PdS2 

ea h

TdS2

of S onsisting from elements of the diameter less d2 . For d2 £ d2 ¡ 0, we suppose that TdS2 ’ TdS , this means TdS is a renement of

a nite-element triangulation



2

2

: C(S) Ù C(S) the linear ontinuous proje tor whi h assigns

v ò C(S) the element-wise ane interpolation whi h oin ides with v at all mesh TdS2 . Then we dene PdS2 by

points of the triangulation

PdS2 h
(x ; s) # PdS2 (h(x ; -))
(s) :

(3.167)



Let us illustrate the interpretation of ( measures, i.e.

H # L ( ; C(S)). 1

PdS2 )

*

again for the ase of the lassi al Young

We will see that this proje tor makes an aggrega-

tion of Young measures so that the resulting Young measures are omposed of a -

x ò . The Ld PdS2 C(S) ’ C(S) possesses the base, denoted by {v ld2 }l#12 , su h

nite number of atoms (=Dira measures) at xed supports independent of nite-element subspa e



L d2 l l that ea h v d 2 ò C ( S ) is non-negative and  l #1 v d 2 ( s ) # 1 for any s ò S . Moreover, we L d2 S l l l

an write [ P h ℄( x ; s ) #  d2 l #1 h ( x ; s d 2 ) v d 2 ( s ), where s d 2 ò S denotes the mesh points. p If  ò Y ( ; S) is identied with the Young measure # { x }xò via the mapping H S * from Lemma 3.4, we laim that ( P )  an be identied with an aggregated Young d2 d2 # { d2 } dened by measure x ò

x d2 x

#

Æ s denotes the L1 ( ; C(S)) we have

where

S ´( Pd

2

*

)

L d2

l H a d (x)Æ s l 2 d2 l #1

with

a ld2 (x) # X v ld2 (s) S

Dira measure supported at

s ò S.

x (d s ) ;

Indeed, for every

(3.168)

hòH #

L d2 l l X X ” H h ( x ; s d ) v d ( s )• x (d s ) d x 2 2

S l #1 L d2 L d2 l l l l H X h ( x ; s d )X v d ( s ) x (d s ) d x # H X h ( x ; s d ) a d ( x ) d x 2 2 2 2 S l #1

l #1

 ; hµ # ´ ; PdS2 hµ #

#

d2 X h ( x ; s ) x (d s ) d x

S

#X

#



(

d2

)

; h

d 2 ò Y( ; S ) is taken as in (3.168). Let us note that a l ( x ) £ 0 and  L d2 a l ( x ) d2 l #1 d 2 L # PS l#d12 v ld2 (s) x (ds) # PS x (ds) # 1 so that (3.168) a tually determines a Young mea-

provided

sure.

a oneS ’ R1 dis retised by an equi-distant partition

The aggregation pro edure is illustrated on Figure 3.13, whi h uses dimensional domain (=an interval) and onto the sub-intervals of the length

d2 .

Ë 217

3.5 Approximation theory

S

S

d2

repla ements (

PdS2 )*



Fig. 3.13:

The aggregation of a Young measure.

We would like to noti e that su h onstru tion has been already used in the Step 2B of the proof of Theorem 3.6.

L1 ( ; C(S)), the requirement (3.151) as well as (3.154a,b) p # L 1 ( ; C 0  ( S )). The is fullled. Besides, also (3.154 ) is satised with C # 1 for H p p S fa t that ( P ) : Y ( ; S) Ù Y ( ; S), needed for (3.152), follows from the obtained H H d2 For the standard norm of

;

*

representation (3.168) together with Theorem 3.6.

PdS2

Of ourse, we ould also think about another onstru tion of

by means of a

suitable higher-order interpolation or another approximation method. When proje tor

S ’

Rm is unbounded, in parti ular if S # Rm , the onstru tion of the

PdS2 be omes te hni ally more di ult but the previous ideas an be straight-

forwardly modied, for example, in the ase of the DiPerna-Majda measures using the subring

R

from (3.45), where the metrizable ompa ti ation

homeomorphi with a ompa t polyhedral domain in

Remark 3.87.

R

m.

Rm Ê Rm € S m"

1

R

The onstru tion presented here ertainly reminds that one from Se -

Pd : H Ù H determines proje tors FH Ù FH used 2.4 provided Ker ’Ker( Pd ). If this ondition were supposed, we ould *

tion 2.4. A tually, the proje tors in Se tion

is

*

shorten the proofs of Lemma 3.82 and Proposition 3.83.

Remark 3.88 (Approximations of Type I).

If

S #

Rm and H has the form G ‹ V as in

Pd with Pd # Pd 1 from Se t. 3.5.b yields dire tly a full dis retisation (Type I) provided V is nite-dimensional as in (3.107), while if V is innite-dimensional it yields only a semi-dis retisation (Type p m p m II). In parti ular, for H from (3.107) where Y ( ; R ) Ê L ( ; R ), this proje tor H

(3.97), (3.101), (3.102), (3.104), or (3.107), then the proje tor

*

makes nothing else than the element-wise onstant approximation of fun tions from

L p ( ; Rm ).

It is easy to see that

PdS2

from (3.167) ommutes with

Pd 1

from (3.160). Therefore,

having a Young measure as in the left-hand part of Figure 3.12 (or 3.13), we an apply the omposed proje tor (

Pd 1 PdS2 )

*

(whi h is the same as (

PdS2 Pd 1 )

*

), whi h gives an

element-wise homogeneous aggregated Young measure. This omposed approximation is outlined on Figure 3.14, f. also Figures 7.1 and 7.3 on pp. 505507.

218

Ë

3 Young Measures and Their Generalizations

S

S

d2

PSfrag repla ements (

Pd 1 PdS2 )*



d1 Fig. 3.14:

The element-wise homogeneous aggregation of a Young measure.

It is lear that this omposed pro edure yields a tually an approximation of Type I, this means onvex and nite-dimensional. Also note that (3.156b) is satised. In parti ular, for the ase of the lassi al Young measures with

H # L1 ( ; C(S)) endowed with the standard norm, by Propositions 3.83

and 3.84 one gets the error estimate

valid for

 "  d W 1 1 ;

[

d $ 

(

d1 ; d2 )

(

; C ( S )) L 1( ; C 0 2 ( S ))℄ ;

*

¢ Cd 1 %d 2   L1 C S 1

2

L

; ;{a l }l#d2 ; a l £ 0 ; 1

H

L d2 l #1 a l

H

*

U

#

L d2  l #1

dependent on

and, for any

L d2 ò N, dene

# 1;

:h ò H : # X

In their Young-measure representations,

s ld2

(3.169)

*

as



or altrnatively also

))

Inspired by Corollary 3.10, one an

u

the nite-dimensional onvex subset in

by having

(

*

onsider a xed ountable olle tion { l } l òN dense in

*

;

# [Pd 1 PdS2 ℄ .

Remark 3.89 (Another approximations of Type I).

Ž ò H

(

H

L d2 l #1 a l h ( x ; u l ( x )) d x DZ :

's from (3.170) will read as

a l Æ u l ò r a%1 (U). It diers from the

x as s ld2 # u l (x) while a ld2

x

(3.170)

L

# l#d2 a l Æ u l 1

(

x) ,

approximation (3.168)

are now independent of

x. This

sort of approximation is supported by arguments that the set of Young measures is onvex and ompa t, and that ea h element of onvex ompa t sets an be approximated by a onvex ombination of extreme points due to the Kren-Milman theorem 1.14, and that the extreme points of the set of the Young measures are Dira measures a.e.,

f. Proposition 3.24. This approximation, devised by V.M. Tikhomirov [769℄ and used e.g. in [39, 40℄ under the name mix of ontrols, does not seem to be indu ed by any proje tor as in Lemma 3.82. Of ourse, ea h su h

 is attainable from U ; for this, one

an take the Young-measure representation and onstru t a fast os illating sequen e

L d2 Ù , p Y H ( ; Rm ) due to the density of

like in Step 2 of the proof of Theorem 3.6 on p. 128. Moreover, passing the sets (3.170) in rease and their union is dense in {

i H (u l )}lòN . This allows for the onvergen e proof behind this sort of onvex approxi-

mation.

3.5 Approximation theory

3.5.d

Ë 219

Higher-order onstru tions by quasi-interpolation

The onstru tions from Se tions 3.5.b and 3.5. falls into more general s heme involv-

x- or/and in s-variables, let us denote it by Cark l p ( ; S) :#

; S   %} with k and l referring to order of dierentiability ; ;

ing higher smoothness in

h òCar ( ; S); h Cark l p in x ò and s ò S , respe tively. A natural hoi e seems p

{

; ;



h Cark l p ; ;

(

(

)

:# h Carp

;S)

(

" k " " " h" " " " " k " x " "Carp ( ; S ) " "

% """"

;S)

" l " " " h" " " " " l " s " "Carp"l ( ; S ) " "

% """"

p £ l, with the onvention that, if n ¡ 1,  k /x k means all kth-order derival l tives and analogously, if m ¡ 1,  / s means all l th-order derivatives. Let us note k l p k 1 ( ; C ( S ))  L 1 ( ; C l ( )). We will rely on that, if S bounded, then Car ( ; S) Ê W provided

; ;

;

(3.156a) in ombination with the former property in (3.156b) and present more general

onstru tions devised in [520℄. As for the proje tor

Pd 1 , we may onsider a olle tion of ansatz fun tions {g i }Ii#d11 ’ (

I(d

)

L1 ( ), using also a dual olle tion {g i }i#11 ’ L ( ). Then we dene the operator Pd1 by a quasi-interpolation with respe t to these bases, i.e. *

[

Pd1 h℄(x ; s) :#

I(d1)

 i (s)g i (x)

H i #1

)



 i (s) :#

with

P

g i (x)h(x ; s) dx *

P

g i (x) dx *

:

(3.171)

The former desired approximation property in (3.156a) will now read as

Pd1 h L1

(

¢ C d k h W k 1

; C ( S ))

1

1

;

(

; C ( S ))

:

(3.172)

I(d1 ) * I(d1) In fa t, this property links the olle tions { i } i #1 and { i } i #1 with ea h other to some

g

g

extent. Sometimes, these olle tions will be orthogonal with respe t to the natural

L2 -

type s alar produ t in the sense

X g i (x)g j

*

(

x) dx 

Proposition 3.90 (The proje tor Pd 1 ). for

Pd1 *

: [

Moreover,

Pd1 *

olle tions { g i }

Pd1 *

℄x

#

For

I(d1 ) H i #1

#0 ¡0

Pd 1

g i (x) *

i #Ö j for i # j : for

from (3.171), the following formula holds

P

g i ()

P

{

I(d

g i }i#11 *

)

 d

g i ()d *

is a Young measure or, in other words,

I(d1 ) i #1 and

(3.173)

:

(3.174)

Pd1 Y( ; S) ’ Y( ; S), provided the *

satisfy

g i £ 0; g i £ 0; i # 1; :::; I(d1 ); *

I(d1) H i #1

g i (x) # 1 *

X g i (  )d 

for a.a.

xò

(3.175a)

and

(3.175b)

# X g i ()d ; i # 1; :::; I(d ) : *



1

(3.175 )

220

Ë

3 Young Measures and Their Generalizations

Pd1 *

Proof. The on rete form of adjoint operator

an be obtained straightforwardly

from denitions if one uses the Fubini theorem:

Pd

*

1

g i ()h( ; s)d

I(d1) P

; h # ; Pd1 h # X

*

X H

S i #1

#

I(d1 )

P

1

H i #1 P

g i ()d *

g i ()d *

X X gi

, S

*

I(d1 )

#X Obviously, [

X [ Pd 1 S *

Pd1 *

x (d s )d( x ;  )

 h( ; s)g i (x)

( )

g i (x) *

X h ( x ; s )¤ H

S i #1 P

x (d s )d x

g i (x)

g i () d *

X g i (  )  d  ¥(d s )d x :

℄ x is a positive measure be ause of (3.175a), and moreover

ds) #

I(d1 )

℄x (

H i #1

g i (x) *

P

g i ()X

S

P

 (d s )d 

g i () d *

#

I(d1) H i #1

g i (x) *

P

P

g i ()d g i () d *

#1

be ause of (3.175b, ). Hen e we proved that this approximation is onformal. For the parti ular onstru tion from Se tion 3.5.b using P0-nite elements, the formula (3.171) is the Clément quasi-interpolation [226℄ of 0-order and

Pd 1 is indeed a pro-

je tor. Now we have still few other options more:

Example 3.91 (Pd 1 by P1/Q1-nite elements). Consider a simpli ial mesh and resulting dis retisation Td 1 , and put g i # g i #the hat element-wise ane, ontinuous fun tion orresponding to i th node. Then (3.173) is not satised, but (3.172) with k # 2, *

and (3.175) are satised. The formula (3.171) is the Clément quasi-interpolation of 1order and

Pd 1

is again a proje tor. A dis ontinuous P1-variant arises when putting

g i # g i # the ane fun tion supported however only on the parti ular simplex from Td1 , and vanishing at all its nodal points ex ept one. Then we get all properties as in *

the pre eding ontinuous P1- ase. Considering a Cartesian mesh, one an onstru t

gi # gi

*

as tensorial produ t of the P1-fun tions, onsidered in 1-dimensional vari-

ant (possibly ombined in various dire tions). Again, all properties of these examples

Pd 1 . Yet, it should be emphasized that for quadrati (or higher-order) nite elements, (3.172) with k £ 3 an be satised, but (3.175) annot

hold and thus the resulting approximation [ P ℄ Y( ; S ) is not onformal. d1 are inherited by the resulting

*

As to the proje tor

PdS2 , we dene it again as a quasi-interpolation with respe t to

J(d2) some ansatz fun tions { j } j #1

v

’ C(S) and {v j }Jj#d1 ’ r a(S) Ê C(S) *

(

)

*

1

by the formula

analogous to (3.171), i.e.

[

PdS2 h℄(x ; s) :#

J(d2) H j #1

 j (x)v j (s)

with

 j (x) :#

P S

h(x ; s)v j (ds) *

P S

v j (ds) *

:

(3.176)

The latter desired approximation property in (3.156a) will now read as

h " PdS2 h L1

(

; C ( S ))

¢ C d l h L1 C l S 2

2

(

;

(

))

(3.177)

3.5 Approximation theory

Ë 221

J ( d 2) * J(d1) and again, in fa t, it links the olle tions { j } j #1 and { j } j #1 with ea h other to some

v

v

extent. Sometimes, these olle tions are so- alled B-dual, i.e. orthonormal with respe t to the natural

L2 -type s alar produ t in the sense X v i (s)v j S

*

Proposition 3.92 (The proje tor PdS2 ). S * for ( P ) : d2

(

S d2

Moreover, ( P )

*

PdS2 )

*


x

0 1

ds) # 

(

For

#

PdS2

for for

i #Ö j; i # j:

(3.178)

from (3.176), the following formula holds

J ( d 2 ) P v (  ) (d  ) x S j H v*j : * P v (d  ) j #1 S j

(3.179)

Y( ; S) ’ Y( ; S) whenever

v j £ 0; v j £ 0; j # 1; :::; J(d2 ) *

J(d2) H j #1

v j (s) # 1

for all

and

(3.180a)

sòS :

(3.180b)

S * d 2 ) an be obtained straightforwardly from

Proof. Con rete form of adjoint operator ( P

denitions if one again uses the Fubini theorem:

S * ( Pd ) 2

J(d2) P S

; h # ; PdS2 h # X

X H

S j #1

#X

J(d2 )

H

j #1 P S

1

v j (d) *

*

P S

v j (d) *

X h(x ; )v j (s)[v j S,S

*

J(d2) P S

#X

h(x ; )v j (d)

X h ( x ; s )¤ H

S j #1

v j () P S

x (d  )

v j (d) *

,

v j (s)

x (d s )d x

x ℄d(  ; s )d x

v j (ds)¥ dx ; *

(3.181)

whi h yields the formula (3.179). More pre isely, Fubini's argument holds lassi ally if

x and all

v j , j # 1; :::; J(d2 ), *

are absolutely ontinuous. In the oppo-

site ase, one an he k (3.181) by a ontinuous extension of the absolutely on-

, 1 ; 2 ) ÜÙ f(s1 ; s2 )2 (ds2 )℄1 (ds1 ) and (1 ; 2 ) ÜÙ PS [PS f(s1 ; s2 )1 (ds1 )℄2 (ds2 ) for any f ò C(S, S). Indeed, onning ourselves (e.g.) on the former ase, for any sequen es 1k Ù 1 and 2k Ù 2 weakly* in r a(S), one has F k (s1 ) :# PS f(s1 ; s2 )2k (ds2 ) Ù P f ( s 1 ; s 2 )  2 (d s 2 ) #: F ( s 1 ) for any s 1 ò S , hen e also F k Ù F uniformly on S be ause S S is assumed ompa t, and thus eventually PS F k (s1 )1k (ds1 ) Ê Ù Ê P F ( s 1 )  1 (d s 1 ) # P [P f ( s 1 ; s 2 )  2 (d s 2 )℄  1 (d s 1 ). S S S S ℄ x is non-negative due to (3.180a), and moreover Obviously, [( P ) d2 tinuous ase, relying on the joint (w* w*)- ontinuity of the mappings (

P [P S S

*

S * X [( Pd ) 2 S

s ds #

℄x ( )

J ( d 2 ) P v (  ) (d  ) x S j * H X v j (d s ) * S P v (d  ) j #1 S j

Ë

222

3 Young Measures and Their Generalizations

J(d2 )

#X

” H S j #1

x (d  )

v j ()•

#X

x (d  )

S

#1

due to (3.180b). Hen e we proved that this approximation is again onformal. In addition to the aggregation approximation indu ed by (3.167), we have now other options:

Example 3.93 (PdS2 by P1/Q1-nite elements). Consider a simpli ial mesh and result¡ 0 a mesh parameter, and put v j #the hat ing dis retisation Td 2 of S with d element-wise ane, ontinuous fun tion orresponding to j th node s j ò S , and v # j Æ s j =Dira 's distribution at s j . Then (3.177) with l # 2, (3.178), and (3.180) are satised, 2

*

PdS2 is a proje tor. Likewise, for a Cartesian mesh, one an onstru t v j as tensorial

and

produ t of the fun tions from Examples 3.6 onsidered in 1-dimensional variant, and

vj

*

as Dira 's distributions in parti ular nodal points. Again, all properties of the previ-

PdS2 . Let us remark that this operator

ous P1- onstru tion are inherited by the resulting

has been proposed by Tartar [749℄. Let us point out that, for quadrati (or higher-order)

l ¢ 3 an be satised but (3.180) annot hold and thus the PdS2 ) Y( ; S) is not onformal.

nite elements, (3.177) with resulting approximation (

The operators

*

Pd 1 and PdS2 in the form (3.171) and (3.176) always ommute with ea h

other. This follow easily from Fubini's theorem (possibly extended by ontinuity) by the dire t al ulation:

Pd 1 PdS2 h
(x ; s) #   Pd 1 ”( x ; s ) ÜÙ 





I ( d 1 ) J ( d 2 ) P g * ( x ) P h ( x ; s ) v * (d s )d x j

i S g i (x)v j (s) H H * * P g i ( x )d x P v j (d s ) i #1 j #1

S

#

I(d1 ) J(d2) P g* ( x )h( x ; s )[L , v*j ℄d( x ;

,S i H H * * P g i ( x )d x P v j (d s ) i #1 j #1

S







# ¨PdS2 ”( x ; s ) ÜÙ 













3.6



#



where

J ( d 2 ) P h ( x ;  ) v * (d  ) j S H v j ( s )•¡(x ; s) * P v (d  ) j #1 S j 









s)

g i (x)v j (s)





I ( d 1 ) P h (  ; s ) g * (  )d  j

H * P g i (  )d  i #1 S 

g i ( x )•© (x ; s) # [PdS2 Pd 1 h℄(x ; s) ; 

L stands for the Lebesgue measure on .

Extensions of Nemytski mappings

An important lass of nonlinear mappings from one Lebesgue spa e into another one is formed by the Nemytski mappings. Here we want to study a (perhaps somewhat surprising) feature that they may admit an ane ontinuous extension on appropriate onvex

- ompa ti ations. Let us realize that these mappings are (ex ept trivial

Ë 223

3.6 Extensions of Nemytski  mappings

ases) nonlinear but with respe t to the original linear stru ture of Lebesgue spa es whi h may be (and mostly is) deformed. This deformation makes possible that the extended mappings are ane in some (or all) arguments. Let us begin with the Nemytski mappings

3.6.a

N'

of one argument only.

One-argument mappings: ane extensions p; q ò [1; %), two separable Bana h spa es S1 and S2 ,

We will onsider

U 2 # L q ( ; S2 )

U 1 # L p ( ; S1 ) ;

endowed respe tively by the norm bornologies tion 2.5) and a Carathéodory mapping

B1

B2

and

(3.182)

( f. the notation from Se -

' : , S1 Ù S2 satisfying the growth ondition

;a ' ò L q ( ) ;b ' ò R% :(x ; s) ò , S : 1



p/ q

'(x ; s) S2 ¢ a ' (x) % b ' s S1 :

Re all that (3.183) guarantees the Nemytski mapping

(3.183)

N ' : L p ( ; S1 ) Ù L q ( ; S2 )

dened by

N' (u)℄(x) # '(x ; u(x))

(3.184)

[

to be ontinuous and bounded; see Theorem 1.24. For

h òCarq ( ; S2 ), we dene S ' h by [

In other words, For any

S'

B ò B1 ,

S ' h℄(x ; s) # h(x ; '(x ; s)) :

substitutes the fun tion

'

into the Carathéodory integrand

S hB ¢ hN'

we have the estimate  '

(3.185)

(

h.

B ) ; it holds even as equality. q

Car ( ; S2 ) ' fulls (3.183) whi h ensures N' (B) ò B2 for every B ò B1 . p Again, we dene here two linear homeomorphi al embeddings 1 : Car ( ; S1 ) Ù q CB1 (U1 ) and 2 : Car ( ; S2 ) Ù CB2 (U2 ) by [ l h℄(u) # P h(x ; u(x)) dx with l # 1; 2. Furthermore, let us onsider two linear subspa es H1 ’ Carp ( ; S1 ) and H2 ’ Carq ( ; S2 ). Of ourse, FH l will mean l (H l ) % { onstants on U l }, l # 1; 2. It turns out

Therefore, it is easy to see that into

Carp ( ; S

S'

is a linear ontinuous mapping from

1 ) provided

that the ondition

S ' (H2 ) ’ H1 is essential98 for

N'

to admit an ane ontinuous extension

(3.186)

N ' :

M(

F1 B1 )

Ù

F2 B2 ).

M(

p

- ompa ti ation Y H1 ( ; S1 ) is B1 - oer ive (in parti ular if (3.138) is satS ' (H2 ) ’ H 1 # lCarp ( ;S1 ) H1 p p be ause eventually Y ( ; S1 ) Ê Y ( ; S1 ) by Theorem 3.66 and by Proposition 2.20. On the other H1 H 1 p hand, a pre ise knowledge of the losure of H 1 in the natural topology of Car ( ; S1 ) is usually not at 98

In fa t, if the onvex

ised), then the ondition (3.186) an be weakened by requiring only

our disposal in on rete ases.

224

Ë

3 Young Measures and Their Generalizations

Proposition 3.94 (Ane extensions of Nemytski mappings). N'

be valid. Then the Nemytski mapping

p

q

Let (3.183) and (3.186)

admits an ane ontinuous extension

p

YH1 ( ; S1 ) Ù YH2 ( ; S2 ). This extension oin ides on YH1 ( ; S1 ) with the adjoint operator S ' : H 1 Ù H 2 . Alternatively, we an say that N ' possesses a ontinuous ane *

*

*

extension M(F1 B1 ) ping to

Ù M(F

2

B2 ) whi h oin ides on M(F1 B1 ) with the adjoint map-

f2 ÜÙ f2 Ž N' : FH2 Ù FH1 . Altogether, the following diagram ommutes: S' *

p YH1 ( ; S1 )

✲ Y q ( ; S2 ) H2

✒ ■ i H1 ✻❅ i H2 ✻ N' ❅ q p 1 2 L ( ; S1 ) ✲ L ( ; S2 ) e H2 ❅ ❄ ✠ e H1 ❘ ❄ ❅ N ' ✲ M(F2 B2 ) M(F1 B1 ) *

*

 # N' . Then (3.183) implies  bounded f2 Ž  ò FH1 provided f2 ò FH2 be ause of the obvious identity q ( 2 h ) Ž N ' # 1 ( S ' h ) valid for any h ò Car ( ; S2 ). Therefore, by Proposition 2.32, the ontinuous ane extension  # N ' : M(F1 B1 ) Ù M(F2 B2 ) does exist and

oin ide with the adjoint mapping to Q : F H 2 Ù F H 1 : f ÜÙ f Ž N ' . On the other hand, it is obvious that S ' : H 1 Ù H 2 is ontinuous and S ' Ž i H 1 # i H2 Ž N' be ause of the identity Proof. Let us employ Proposition 2.32 with and (3.186) implies





*

S ' *

Ž i H 1 ( u ) ; h # i H 1 ( u ) ; S ' h #

*

*

X h ( x ; ' ( x ; u ( x ))) d x

# i H2 Ž N' (u); h

u ò U1 . Therefore, the restri tion of S ' the ane ontinuous extension of N ' .

valid for any

h ò H2

*

*

and

on

p

YH1 ( ; S1 ) realizes

Å

By applying Theorem 3.66 twi e, we obtain eventually the above diagram.

It is now lear that various hoi es of

H1 and H2 give various on rete representa' when H1 and H2 are onsid-

tion of (3.186), whi h is, in fa t, a ertain ondition on

ered as xed. We will mention only a few examples. For some of them it will be very di ult to hara terize (3.186) in terms of

' pre isely, so that mostly we will be able

to pose only su ient onditions.

Example 3.95 (The largest H makes (3.186) void). Let us take H # Carp ( ; S ) and H ’ Carq ( ; S ) arbitrary and suppose (3.183). Then (3.186) is always fullled. Indeed, q q any h ò Car ( ; S ) satises  h ( x ; s ) ¢ a h 2 ( x ) % b h 2 s S2 with a h 2 ò L ( ), whi h 1

2

1

1

2

2

2

1

2

enables us to estimate

!! ! !![ S ' ( h 2 )℄( x ; s )!!!

¢ a h2 (x) % b h2 '(x ; s)q ¢ a h2 (x) % b h2 Œa ' (x) % b ' s Sp 1q  /

¢ a h2 (x) % 2q" b h2 a q' (x) % 2q" b h2 b q' s Sp 1 : 1

1

q

Ë 225

3.6 Extensions of Nemytski  mappings

In view of (3.183),

q

a h2 % 2q"1 b h2 a ' ò L1 ( ), and therefore S ' (h2 ) ò Carp ( ; S1 ) # H1 ,

hen e (3.186) is a tually satised. Let us observe that the weakest mode of the ondition (3.183) is for

q # 1.

Example 3.96 (The substitution h DZ ). Let us put S # R, q # 1, and H # G ‹ R p with some subspa e C ( ) ’ G ’ L ( ). Then (3.183) means pre isely ' òCar ( ; S ). Taking some linear subspa e H ’ Carp ( ; S ), one an easily verify that (3.186) is valid for any ' ò H if and only if H is G -invariant; see (3.91). If H is G -invariant, p we an extend the Nemytski mapping N h : L ( ; S ) Ù L ( ) with h ò H arbitrary, 2

*

2



1



1

1

1

1

1

1

1

1

obtaining the identity

:h ò H : ò YHp1 ( ; S ) : 1

when one identies99

H2

*

with

Sh  # h DZ  *

1

G

*

; for

G#L



(

(3.187)

) or G # C( ) f. Example 3.50. Also

note that we did not use here the ondition (3.93) but, on the other hand, we have got only a ontinuous mapping

p

 ÜÙ h DZ  : YH1 ( ; S1 ) Ù G

*

with

h

xed, whi h is a

weaker result than that obtained in Proposition 3.43. Eventually, note that for

q ¡ 1,

the ondition (3.183) oin ides with (3.94). This allows us to extend Nemytski map-

Nh : L p ( ; S1 ) Ù L q ( ).

pings

Example 3.97 (Mappings between DiPerna-Majda measures). Let us take H # C( )‹ p C (S )) and H # C ( ) ‹ Ô q (R ) with R arbitrary omplete subring of C (S ); f. 1

Ô (

0

1



0

2



2

S1 and S2 are nite-dimensional. We will onsider an autonomous ase, i.e. a Nemytski mapping N ' with ' ( x ; s ) # # ( s ) for some # : S1 Ù S2 ontinuous and satisfying the growth ondition

Example 3.47 for the ase when



Then

N'

p/ q

#(s) S2 # O( s S1

)

for

s

S

1

Ù :

(3.188)

- omh2 ò H2 in the form g l ò C( ), and v l ò C0 (S2 ), we have

possesses an ane ontinuous extension on the respe tive onvex

pa ti ations. Indeed, (3.186) is fullled be ause, for every

h2 (x ; s) #

L  l #1

q

g l (x)v l (s)(1 % s S2 ) with s ò S2 ,

obviously [

S ' h℄(x ; s) #

# q

L H g l ( x ) v l ( # ( s ))(1 l #1



% #(s) Sq 2 )

L 1 % #(s) Sq 2 ­ (1 H g l ( x ) ¬ v l ( # ( s )) 1 % s Sp 1 l #1

% s Sp 1 ):

p

v l (#(s))(1 % #(s) S2 )(1 % s S1 )"1 ò C0 (S1 ) provided # satises the growth ondition (3.188), we have surely S ' h ò H 1 , hen e (3.186) is a tually valid. In parti ular, if S1 and S2 are nite-dimensional, we an substitute any DiPerna-Majda measure p from DM 0 C S1 ( ; S1 ) into the fun tion # with the growth (3.188), the result being some q DiPerna-Majda measure from DM ( ; S2 ). R

As

(

99

)

In terms of (3.185),

S h : G Ù H1 in (3.187) is dened as g

ÜÙ g - h with [g - h℄(x ; s) # g(x)h(x ; s).

226

Ë

3 Young Measures and Their Generalizations

Example 3.98 (Mappings from Young to DiPerna-Majda measures). Let us take H as in Example 3.97 and H # C( ) ‹ C p (S ); f. Example 3.46 for the ase when S is nite-dimensional. We will onsider the Nemytski mapping N ' with ' ( x ; s ) # # ( s ) for some # : S Ù S ontinuous and satisfying the growth ondition 2

1

1

N'

1

2



Then

1



p/ q

#(s) S2 # o s S1



for

s

S1

Ù :

(3.189)

possesses an ane ontinuous extension on the respe tive onvex

- om-

pa ti ations. Indeed, (3.186) an be veried analogously as in the pre eding example

p

q

v l (#(s))(1 % #(s) S2 )(1 % s S1 )"1 ò C0 (S1 ). Let us note that, for S1 # S2 # m R and p # q, this Nemytski mapping operates from the Young measures Yp ( ; S1 ) p into the DiPerna-Majda measures DM ( ; S2 ), although the former one is a stri tly R

oarser onvex  - ompa ti ation than the later one. Of ourse, this is possible thanks to the growth restri tion (3.189). Likewise, if p ¡ q and ' ( x ; s ) # # ( s ) # s , then S ' q p m m embeds Y ( ; R ) into DM ( ; R ); for p # q f. also Remark 3.29. R be ause now

*

Example 3.99 (Canoni al surje tion). A very spe ial ase appears for S # S and '(x ; s) # s, whi h fulls (3.183) with p # q. Then N' is just the identity on L p ( ; S ), and (3.186) means pre isely H ’ H . The ontinuous ane extension of N' (if 1

2

1

2

1

it exists) is just the anoni al surje tion from the ner onvex

p

p

YH1 ( ; S1 ) onto the oarser onvex - ompa ti ation YH2 ( ; S1 ).

3.6.b

- ompa ti ation

Two-argument mappings: semi-ane extensions

Let us pro eed this se tion to some universal and often used assertions about spe ial extensions of the two-argument Nemytski mapping

L  ( ; S

3 ),

dened by

N ' : L q ( ; S1 ) , L p ( ; S2 ) Ù

N ' (y; u)
(x) # '(x ; y(x); u(x)) ;

' : ,S1 ,S2 Ù S3 a Carathéodory mapping, S1 ; S2 ; S3 separable Bana h spa es, q; p;  ò [1; %℄. Let us agree that, for notational simpli ity, we will write likewise in the nite-dimensional ase s - s in pla e of < s ; s > with s ò Si and s ò Si . We will start with an extension only in the se ond argument, the rst

with

S3

being reexive, and

*

*

*

*

argument remaining in the original Lebesgue spa e. As this extension will be ane only in the se ond argument, we will speak about a semi-ane extension. Parallel to Corollary 2.33 together with Proposition 3.94, we now have:

Lemma 3.100 (Semi-ane extensions of Nemytski mappings). Let q; p ò [1; %),  ò [1 ; %), let C ( )-invariant linear spa es H ’ Carp ( ; S ) and H ’ Car ( ; S ) be given, and, for ' Ž y dened by [ ' Ž y ℄( x ; s ) # ' ( x ; y ( x ) ; s ), let 2



S 'Žy (H3 ) ’ H2

and

y ÜÙ S 'Žy : L q ( ; S1 ) Ù Carp ( ; S2 )

with respe t to the norm on

'.

2

Carp ( ; S

Then the Nemytski mapping

N'

2 ),

with

3

be ontinuous

3

(3.190)

S 'Žy dened by (3.185) with 'Žy instead of ' u)℄ (x) # '(x ; y(x); u(x)) admits

dened by [N ( y;

Ë 227

3.6 Extensions of Nemytski  mappings

an extension

'

p

N : L q ( ; S1 ) , YH2 ( ; S2 ) Ù YH3 ( ; S3 ) dened by

'

N (y; ) # S*'Žy  :

(3.191)



If restri ted on

,

p

L q ( ; S1 ) , YH

;

% ( ; S2 ) with '

(strong weak*,weak)- ontinuous and

% ò

R%

arbitrary, this extension is

N (y; -) is ane.

y, we use Proposition 3.94 when realizing that S 'Žy (H3 ) ’ H2 is just S 'Žy i H2 (u) # N'Žy (u) # N ' (y; u). For nets   Ù  weakly* in H2 and for y  Ù y strongly in L q ( ; S1 ), we have Proof. For xed

(3.186) up to notational modi ations. In parti ular we have

*

*

¼N

'



(

'

y  ;  )"N (y; ); h½ # ´S 'Žy    " S 'Žy  ; hµ *

*





# ´  "  ; S 'Žy hµ % ´  ; (S 'Žy  "S 'Žy )hµ Ù 0 for any

h ò H3 . Here we used the ontinuity of y ÜÙ S 'Žy assumed in (3.190).

We will o

asionally write g - ( ' Ž y ) instead of < g; ' Ž y >, meaning a fun tion

,S2 Ù R : (x ; s) ÜÙ means <  ; g - ( ' Žy )>.

100

Of ourse, we use here an



with

228

Ë

3 Young Measures and Their Generalizations

H is C( )-invariant, (3.192a) ensures101 that ' Ž y DZ  òr a( ; S3 ). By (3.192b), q  p  ' Ž y℄(x ; s) S3 ¢ a(x) % 1 s S2 with a # a1 % b1 y S1 ò L  ( ),  whi h yields ' Ž y DZ  ò L ( ; S3 ) by Proposition 3.43(iii), modied for the S3 -valued '

ase. Obviously, for  # i H ( u ), we get ' Ž y DZ  # N ' Ž y ( u ) # N ( y; u ) so that (3.193) ' a tually determines the extension of the original mapping N . Proof. As





/

/

we have ensured [

It remains to show the ontinuity of the extended mapping. Let us take a net

y in the strong topology of L q ( ; S1 ) (then su h net must be p % q eventually bounded in L ( ; S1 )), and a net {   }  ò  ’ Y H % ( ; S2 ) with some % ò R

onverging weakly* in H to  ; we an use the ommon dire ted index set  without  any loss of generality. We want to show that ' Ž y  DZ   Ù ' Ž y DZ  weakly in L ( ; S3 ).  By (3.192b), we an see that the net { ' Ž y  DZ   }  ò  is eventually bounded in L ( ; S3 ) and therefore it su es to show ' Ž y  DZ   Ù ' Ž y DZ  weakly* in r a( ; S3 ). For every g ò C( ; S3 ) we an write {

y  } ò

onverging to

;

*



*



'

Ž y  DZ   " ' Ž y DZ  ; g # ' Ž y DZ (   " ) ; g % (' Ž y  " ' Ž y) DZ   ; g #: I  % I  : (1)

(2)

I  #   "  ; g - (' Ž y) , so that I  Ù 0 be ause   Ù  weakly* in H and be ause g - (' Ž y) ò H by (3.192a).

p

Let h # Ô (1), whi h means h ( x ; s ) # s . Obviously, h p q òCar ( ; S2 ). MoreS2 over, we may suppose that H ontains h p q so that h p q DZ  has a good sense. If possibly h p q ò Ö H , we an repla e H (just for the purpose of this proof ) by H % H p q where H p q # L q ( ) - {h p q } òCarp ( ; S2 ). Iterating this tri k, we may and will also suppose 1 that L ( ) ‹ 1 ò H . By the ontinuity arguments, the following three general relations p are at our disposal for any  ò Y ( ; S2 ): H

As for the rst term, we have

(1)

(1)

*

/

/

/



/







;

;



/

:h ; h ò H : h ¢ h âá h DZ  ¢ h DZ  ; :g ò L ( ) : (g ‹ 1) DZ  # g ; 1

2

1

2

1

(3.194a)

2

1

:h ò H; h £ 0 :

 ; h

#

X



(3.194b)

h DZ  dx # h DZ  r a : (

)

(3.194 )

Then the se ond term an be estimated by means of (3.192 ) and the Hölder inequality as:

I  #   ; g - (' Ž y  " ' Ž y) (2)

¢ ´  ; g S3 (a ‹ 1 % b y  Sq"1 ‹ 1 % b y Sq"1 ‹ 1 % h p q ) y  " y S1 µ *

¢

101

" " " " " g S3 ( a 2 " *

1

1

2

2

2

/



"

% b y  Sq"1 % b y Sq"1 % h p q DZ   ) y  " y S1 """"" 1

2

1

2

In fa t, we suppose here, for a moment, that

fullled if

2

H is normed appropriately.

2

/



g - (' Ž y) H

¢ C y g C S 3 (

;

*

)

r a( )

, whi h an be always

Ë 229

3.6 Extensions of Nemytski  mappings

"

"

¢ g C S """" y  " y""""L q S1 """"a % b y  Sq"1 % b y Sq"1 % h p q DZ   """" q " "L 3 ;

(

Therefore

*

;

(

)

1

2

)

1

2

2

2

/





(

)

:

I  Ù 0 as well.

Å

(2)

For the ane extension of a given mapping, the spa e of test fun tions

H1

is to

be su iently large. On the other hand, for the sequentially on ept whi h is onventional in omparison with the on ept of nets,

H1

should be separable. The proof of

the following assertion will also be an interesting illustration of the usage of the norm (3.141) in addition to the proof of Proposition 3.77(ii).

Proposition 3.102 (Separability of H ).

Let

(3.192b, ) is fullled. Then the linear

subspa e

H # span †g - ('Žy); g ò C( ; S3 ); y ò L q ( ; S1 ) ‡ *

(3.195)



Carp ( ; S ) is separable with respe t to the norm (3.141), and thus in the natural topolp ogy of Car ( ; S ), too. of

2

2

Proof. Let us prove the ontinuity of the mapping

y ÜÙ ' Ž y : L q ( ; S1 ) Ù Carp ( ; S2 )

with respe t to the norm (3.141). By (3.192 ) and by Young's inequality, we have

'(x ; y1 (x); s) " '(x ; y2 (x); s) S3

¢ a(x) % b y (x) Sq"1 % b y (x) Sq"1 % s Sp 2q  y (x) " y (x) S1 1

1

1

¢ „a % b y

1

/



2

q "1 S1

% b y

1

q "1 S1 …  y 1

2

"y %

Æ q q

2





2

"q q p Æ s % y1 "y2 S1 1

q

# a Æ (x)%b Æ s Sp 2

q "1

q "1

q

Æ ¡ 0 provided a Æ :# (a % b y1 S1 % b y2 S1 ) y1 " y2 S1 % Æ1"q y1 " y2 S1 /q q and b Æ :# Æ ( q "1) / q . By (3.141), we have for any " ¡ 0 for any



" " " " 'Žy1

" 'Žy

2

" " " "Carp

#

inf



p

a L1

a ( x )% b s S2 £ ' ( x ; y 1 ; s )" ' ( x ; y 2 ; s ) S3

(

)

% b ¢ inf a Æ L1 % b Æ ¢ (

Æ ¡0

)

"

2

"

%

2

Æ ¡ 0 is hosen so small that b Æ ¢ "/2 and then y1 " y2 L q ;S1 is so small ¢ "/2; note that a Æ L1 # O((1 % Æ1"q ) y1 " y2 qL q ;S1 ). This shows p q that the mapping y ÜÙ ' Ž y is even uniformly ontinuous from L ( ; S1 ) to Car ( ; S2 ). Moreover, we an dedu e that also the mapping ( g; y ) ÜÙ g - ( ' Ž y ) : C ( ; S3 ) , L q ( ; S1 ) Ù Carp ( ; S2 ) is ontinuous (even uniformly on bounded sets) be ause of provided

)

(

a

that Æ L 1 ( )

(

)

(

)

*



the obvious estimate

" " " " g 1 -( ' Ž y 1 )

" g -('Ž y 2

" " " "Carp

2)

¢ """"(g "g ) - (' Ž y )""""Carp % """" g - (' Ž y " ' Ž y )""""Carp ¢ """"g "g """" C S """" 'Žy """"Carp % g C """"'Žy "'Žy """"Carp : 1

1

As both

2

2

1

(

;

*

3)

2

1

1

2

(

)

2

1

2

C( ; S3 ) and L q ( ; S1 ) are separable, the spa e (3.195) is also separable *



if equipped with the norm (3.141).

Ë

230

3 Young Measures and Their Generalizations

For 1st-order optimality onditions, we will also need a smoothness property of the extended Nemytski mapping

H

pose that

is normed so that

H

N

'

. We will use Convention 1.55. Also we will sup-



*

H

is a Bana h spa e; re all that

always admits a

p

norm generating a topology ner than the natural topology oming from Car (

; S2 );

f. the universal norm from Example 3.76.

Lemma 3.103 (Dierentiability of semi-ane extensions).102 Let H be a C( )-invariant p linear subspa e of Car ( ; S ), q ò [2  ; %), p ò [1 ; %),  ò (1 ; %), let ' satisfy

2

(3.192b) and

:g ò L  ( ; S 

and let

*

3

:y ò L q ( ; S ) :

)

g - ('Ž y) ò H ;

1

(3.196)

'(x ; -; s) : S1 Ù S3 be ontinuously dierentiable su h that

:g ò L  ( ; S ) :y; y ò L q ( ; S ) : 

g - (' r Žy) - y ò H ;

*

3

;a ò L q

/(

3

q")

(

/(

4

q "2  )

' r (x ; r; s) L S1

¢ a (x) % b r Sq1"  % s Sp 2q" (



(

(

;

S3 )

3

) ;b4 ; 4 ò R% : q "2  )/  b4 r1 S1 %

¢ a (x) %

(

4

(3.197a)



) ;b3 ; 3 ò R% :



;a ò L q



1





(

)/

)/

q

3

3

' r (x ; r1 ; s) " ' r (x ; r2 ; s) L S1 

;

(3.197b)



(

q "2  )/  b4 r2 S1 % (

;

S3 )

p ( q "2  )/ q

4 s S2  r 1 " r 2 S1 :

(3.197 )

'

: L q ( ; S ) , Y Hp ( ; S ) Ù L  ( ; S ) is sepap q rately103 Fré het dierentiable at any ( y;  ) ò L ( ; S ) , Y H ( ; S ) with the dierential ' q  › N ( y;  ) ò L( L ( ; S ) , H ; L ( ; S )) given by

Then the extended mapping

N

1



2

3

1

*

1



ˆ› N

'



'

(

2

3

y; )‰ ( y ;  ) # (' r Ž y DZ ) - y % ' Ž y DZ  







(3.198)



) : H Ù L  ( ; S3 ) is (weak*,weak)- ontinuous. Moreover, ›y N ' (-; ) : L q ( ; S1 ) Ù L(L q ( ; S1 ); L  ( ; S3 )) is lo ally Lips hitz ontinuous uniformly with re' p q %  spe t to  ò YH % ( ; S2 ) for any % ò R and [›  N (- ; -)℄(  ) : L ( ; S1 ) , H Ù L ( ; S3 ) p % is lo ally Lips hitz ontinuous uniformly with respe t to  ò YH % ( ; S2 ) for any % ò R . and ›  N ( y;

*



*



;





;

Furthermore, if

sup



g L  ;S 3 

*

(



)

¢1

sup

g - ('Ž y1 "'Ž y2 ) H ¢ C y1 L q

y L q ;S1 ¢1 g  L ;S3 ¢1

(



102

If

(

% y





;S1 )

L q ( ;S1 )  y 1 "y 2 L q ( ;S1 )

g - (' r Ž y) - y H ¢ C y L q

(

;S1 ) 

;

(3.199a) (3.199b)

)

*

)

' does not depend on s, i.e. '(x ; r; s)

tiability properties of the Nemytski  mapping also Krasnoselski  at al. [442℄.

103

2

(

'

This means that both

N (y; -) and N



' (-

# #(x ; r), then Lemma 3.103 speaks about the dierenN# : L q ( ; S1 ) Ù L  ( ; S3 ). For su h sort of results see

; ) are Fré het dierentiable.

3.6 Extensions of Nemytski  mappings

R% Ù R% ontinuous in reasing, then

'

Ë 231

p

N : L q ( ; S1 ) , Y H ( ; S2 ) Ù q *  L(L ( ; S1 ) , H ; L ( ; S3 )) is lo ally Lips hitz ontinuous. Finally, if  # 1, all these * *   statements remain valid with L ( ; S3 ) and L ( ; S3 ) repla ed respe tively by C ( ; S3 ) and r a( ; S3 ) as far as the partial dierential ›  N on erns.104 with some

C:

›











'

)℄(  ) # ' Ž y DZ  is obvious be ause ' N (y; -) is linear. Also it is obviously the Fré het dierential. Besides, the mapping  ÜÙ ' Ž y DZ  is (weak*,weak)- ontinuous be ause, for   Ù  weakly* in H * and for *  every g ò L ( ; S3 ), we have Proof. The expression for the omponent [›  N ( y;

















'

Ž y DZ   ; g #   ; g - (' Ž y) Ù  ; g - (' Ž y) # ' Ž y DZ  ; g ; 





where also (3.196) has been used.



'

y; ). Let us note that, by (3.192b) and p  (3.197b), both ' Ž y DZ  and (( ' r Ž y )- y ) DZ  live in L ( ; S3 ) for any  ò Y ( ; S2 ); f. PropoH sition 3.43(iii) modied for the S3 -valued ase. First, let us noti e that, by (3.197 ), we Let us al ulate the omponent › y N (





have

" " ' " " " " " " "

¢

Ž (y% " y ) " ' Ž y



"

1 " ""  X "' " 0 "" r "

" "S3

0









;

(

% b y Sq1"

2

(

 )/ 

4

2

(



4









" "S3





% b y % " y Sq1"

2

(

 )/ 

4



)

4

¢ 2" "(a (x) % b y Sq1" 1

" "

Ž (y% " y )… - y d " " (' r Ž y) - y """"

"

Ž (y % " y ) " ' r Ž y""""L S1 S3 y S1 d "

" X (a4 (x)

for any

" " 1 "  " " X „' r " " " "" 0



1

¢

" " "

" (' r Ž y) - y """" #





% b y Sq1" (



4

% s Sp 2q"

 )/ 

(

2

 )/ 

 )/ q

% s Sp 2q" (

2

 )/ q

4



0 ¢ " ¢ 1, where b # b (1 % max(1; 2q 

4

4

2

4



/

)

)

" y S1 y S1 d " 







y 2S1 : 

 "1 )). Using (3.194) together with

the onvention from the proof of Lemma 3.101 (this means here that we an suppose

h p q"2 (

" " ' " " " " " " "

¢

)/

q ò H ), one an estimate

Ž (y % " y ) " ' Ž y 

"

"

 

2

¢

"

2





" "L  ( ;S3 )

" " " " "( a 4 "

% b y Sq1"

" " " a4 " " "

% b y Sq1"

 

" " "

DZ  " ((' r Ž y) - y ) DZ """"



2

(

 )/ 



2

(

% b y Sq1"

 )/ 

4

(



4

4

2

(





% h p q" 4

% b y Sq1" 4

 )/ 



2

 )/ 

2

(

% h p q" 4

" 2 " "  )/ q DZ  ) y S1 " "  ( ;S ) "L 3

2

(



" " " " "2 q - " " " y"  )/ q DZ  " " "L q q"2 ( ) " " L ( ;S1 ) 

/(

)

" Ù 0 thanks to the assumption (3.197 ). By the denition of q the dierential, for any y ò L ( ; S1 ), one gets

whi h tends to zero for



'

# 1, the partial dierential ›y N remains valued in L ( ; S ) be ause '  ) # (' r Ž y DZ ) -  always q £ 2 ¡ 1; indeed, ›y N ( y;  )‘ ( y;  y ò L ( ; S ) be ause y ò L q ( ; S ) and

104

Let us note that, for



1





1



'r Ž y DZ  ò L q ( ; L(S1 ; S3 )) thanks to (3.197b) and Proposition 3.43(iii). 

3

3

1

Ë

232

3 Young Measures and Their Generalizations

ˆ› y N

'



(

'

N (y % " y ; ) " N ' (y; ) " Ù0 " ' Ž (y % " y ) " ' Ž y DZ # # lim " Ù0 "

y; )‰ ( y ) # lim 







((

' r Ž y) - y ) DZ  ; 



whi h is just the orresponding omponent in (3.198) after a trivial re-arrangement, proving that › y N

'



is the Gâteaux dierential.

By (3.197 ), one an also estimate

" ' " " ›y N (y1 ; ) " " "

#

sup

y L q ;S1



¢

(

)

¢1

" " ›y N ' (y ; )""""

"L( L q ( ;S1 ); L  ( ;S3 )) " ' r Ž y2 ) DZ ) - y """"L  ( ;S ) 3

2

"  " " (( ' r " "

Žy " 1





" ( q "2  )/  " " % b4 y2 S(q1"2)/ % 4 h p(q"2)/q DZ  " " Œa 4 % b 4 y 1 S1  y L q ;S1 ¢1 " " " - y 1 " y 2 S y S " 1 1" " "L  ( )

sup (

)



¢

" " ( q "2  )/  " " % b4 y2 (Sq1"2)/ % 4 h p(q"2)/q DZ """""L q q"2 y1 " y2 L q ( ;S1 ) " " a 4 % b 4 y 1 S1 " ( ) /(

)

;

' ; ) already follows. In parti -

from whi h the lo al equi-Lips hitz ontinuity of › y N (ular, we showed that › y N

'



is even the Fré het dierential.

By (3.197b) one an strengthen (at least if



" "

1

"

0

 ¡ 1) the hypothesis (3.192 ) as follows " "

'(x ; r1 ; s) " '(x ; r2 ; s) S3 # """"X ' r (x ; r1 % a(r2 " r1 ); s)(r2 " r1 ) da """"

¢ Œa (x) % 3

q "  )/  b3 r1 S1 % (



q "  )/  b3 r2 S1 %

From this we an estimate, for any

(

" S3 p ( q "  )/ q

3 s S2  r 1 " r 2 S1 :

y1 ; y2 ò L q ( ; S1 ), 1 ; 2 ò H

*

(3.200)

p

 ò Y H ( ; S2 ),

, and



" " ' ' " " " " # """(' Ž y1 " ' Ž y2 )DZ  """"L  ( ;S3 ) " " "[›  N ( y 1 ;  1 )℄(  ) " [›  N ( y 2 ;  2 )℄(  )" "L  ( ;S3 ) " " " " ¢ """"(a3 % b3 y1 S(q1")/ % b3 y2 S(q1")/ % 3 h p(q")/q DZ  ) y1 " y2 S1 """"  " "L ( ) " " ( q "  )/  ( q "  )/  " " " " % b3 y2 S1 % 3 h p(q")/q DZ  """L q q" y1 "y2 L q ( ;S3 ) ¢ """a3 % b3 y1 S1 ( )













/(

whi h shows the mapping (

)

;

y; ) ÜÙ [› N ' (y; )℄(  ) to be lo ally Lips hitzian. 

Furthermore, let us estimate

" ' " " ›y N (y1 ; 1 ) " " "

¢ % The term

I1

" ›y N ' (y ; 

" " " " " "L( L q ( ;S

2)

 1 ); L ( ;S3 )) " " ' " " ' " › y N ( y 1 ;  1 ) " › y N ( y 2 ;  1 )" " " " " " "L( L q ( ;S1 ); L  ( ;S3 )) " " ' " " ' " " " "L( L q ( ;S ); L  ( ;S )) "› y N ( y 2 ;  1 ) " › y N ( y 2 ;  2 )" " " 2





1

3

#: I % I :

was already estimated above by means of (3.197 ), while

estimated by means of (3.199b) as follows

1

I2

2

an be now

Ë 233

3.6 Extensions of Nemytski  mappings

I2 #

"  " " "( ' r

sup

y L q ;S1



¢

(

)

¢1 "

sup

´ g; ( ' r

*

(

(



-

Ž y DZ ( "  2

1

"

y """"L  

2 ))

(

;S3 )

yµ ¢

-



sup

´ 1 y L q ;S1 ¢1 g  L ;S3 ¢1

"

" " " 2 """H """"g -('r Ž y2 )- y """"H *

¢ C( y



"  ; g - (' r Žy 

2

2)

-

yµ 

)

(



)

sup



2 ))



" " " " 1 y L q ;S1 ¢1 g  L ;S3 ¢1

¢

1

)

(



2



y L q ;S1 ¢1 g  L ;S3 ¢1

Ž y DZ ( " 

*

(

2

)

" " L q ( ;S1 ) ) " " 1

"

2

" " " "H

*

;

)

*

(

)

'

proving thus the lo al Lips hitz ontinuity of › y N .

'

Eventually, let us prove the ontinuous dependen e of ›  N . For

L q ( ; S1 ) and 1 ; 2 ò H

*

" ' " " " "›  N ( y 1 ;  1 ) "

" › N ' (y ; 

#

Ž y " ' Ž y ) DZ  ; g #



sup ('  H ¢1 g  L ;S3 ¢1

¢

2

1

" " " " " "L( H ; L  ( ;S3 ))

2)

2

*

(

sup





 H

¢1





" " " "( ' Ž y 1

- (

" 'Žy ) DZ  """"L  S3 2

(



;

)

' Ž y1 " ' Ž y2 )

*



*

(

)

 H g - ('Ž y1 " 'Žy2 ) H ¢ C y1 L q 

*

sup  ; g  H ¢1 g  L ;S3 ¢1



)

 H ¢1 g  L ;S3 ¢1

# sup

*

*



y1 ; y2 ò

, we an estimate by means of (3.199a):

*

(

;S1 ) % y 2 L q ( ;S1 )  y 1 " y 2 L q ( ;S1 ) ;

*



*

(

)

whi h shows that ›  N

'

Å

is lo ally Lips hitz ontinuous.

Remark 3.104 (The ase q # %).

The pre eding two lemmas hold also for

q # %

but the assumptions (3.192b), (3.192 ), (3.197b), and (3.197 ) must be suitably modied;

b r

b r



namely the resulted terms of the type 

R ÙR %

%

b:

S1  must be repla ed by ( S1 ) with arbitrary in reasing ontinuous fun tion. Also the proofs must be suitably

 # 1, the norms of L q ( ) and L q q"2 ( ) must be ' repla ed by the norm in r a( ) and, likewise, › y N may be valued in G , supposing additionally that H is G -invariant. 

modied; for example, if also

/(

)

*





Notation 3.105 (A shorthand onvention: CAR- lasses). For the notational simpli ity, H , the mapping ' : , S , S Ù S belongs to the lass

we will say that, for a given

1

CARqH p  ( ,S ,S ; S ;

;

1

if

'

2

2

3

3)

is a Carathéodory mapping satisfying the quali ation hypothesis (3.192). If

additionally

' satises also (3.196), (3.197), and (3.199), then we say that ' belongs to

the lass

CARqH pdi ( ,S ,S ; S ) : ;

;

If

S1 # S11 , S12

and, instead of

;

1

2

3

(3.201)

L q ( ; S11 , S12 ) Ê L q ( ; S11 ) , L q ( ; S12 ), , L q2 ( ; S12 ), we will work with

q we need an anisotropi spa e L 1 ( ; S11 )

Ë

234

CARqH1 q2 ;

;;

3 Young Measures and Their Generalizations

p; 

(

,S11 ,S12 ,S2 ; S3 )

q2 CARqH1 di ;

or

;;

p; 

;

(

,S11 ,S12 ,S2 ; S3 )

for whi h the

onditions (3.192) or (3.196), (3.197), and (3.199) are modied straightforwardly; details are omitted. On the other hand, if

' # '(x ; r) does not depend on s-variable, H

q; p;  q; p;  be omes irrelevant and, instead of CAR ( ,S1 ,S2 ; S3 ) or CAR H H ; di ( ,S1 ,S2 ; S3 ), q ;  q; 

CAR ( ,S ; S ) or CARdi ( ,S ; S ), respe tively. Let us note that, CARp ( ,S; R) is just Carp ( ; S) dened in (3.81).

we will write just in parti ular,

1

3

1

3

;1

The last onditions, namely (3.199), involve expli itly the norm of

H . The following

example demonstrates that, in fa t, neither (3.199a) nor (3.199b) represent any further restri tion on

' if one hooses a su iently oarse norm on H , as always possible.

Example 3.106 (A universal approa h). We shown in Example 3.76 that every p spa e H of Car ( ; S ) an ertainly be normed by the universal (semi)norm

sub-

2



h H #

inf



:(x ; s)ò ,S2: h(x ; s)¢a(x)%b s Sp 2

a L1

(

%b:

)

(3.202)

H #Carp ( ; S2 ), whi h obviously re-

In parti ular, this hoi e enables to norm also

- ompa ti ation from the investigated lass. show that (3.199a) will be fullled whenever ' satises

ates the nest onvex We want to

(3.196) and

(3.197b). Indeed, by (3.197b) one gets the estimate (3.200), from whi h we an further estimate

!! ! !! g -( ' Ž y 1 " ' Ž y 2 )!!!

¢ g S3 Œa % b y 3

*

¢ C y " y 0

1

 L q ( ;S1 ) ” g S

*

3

3

¢ C y " 1

0

q "  )/  S1

q "  )/  S1 (

2

% b y

(

1



2

% b y 

3

3

% (a % b y 3

3

% s Sp 1q" (

)/

3

y2 L q ( ;S1 ) ” g S 3 

*

%

q "  )/  S1 (

2

q "  )/  S1

q q /( q "  ) )

q

y " y2 S1 q

y "y 2 L q

(

and

C0 

depending on

q

 y 1 " y 2 S1

q

y " y2 S1

1

%

% a q

;S1 )

3

C0

)/

q

y "y2 L q

1

/(

(

•

;S1 )

q")

3

% b y

for suitable onstants

(

3

(

1

1 1

% s Sp 2q"

1

q S1

% b y 3

2

q S1

% s Sp 2 • 3

p, q, and  only. Taking into a

ount

our hoi e (3.202), we get immediately the estimate

g - (' Ž y1 " ' Ž y2 ) H ¢ C0 y1 " y2 L q 

" " " "

-”" "

¢ C y " y 

0

1

g S % 

*

3

q y1 " y2 S1 q y 1 "y 2 q L ( ;S1 )

 L q ( ;S1 ) Πg L  ( ;S ) 3 

2



(

;S1 )

% a q

/(

q")

3

% 1 % a

3

% b y 3

1

q S1

% b y 3

2

q " " " " S1 " "L 1 ( I )

% • 3

q /( q "  ) L q q" ( ) /(

%b y 3

)

1

q L q ( ;S1 )

% b y 3

2

q L q ( ;S1 )

%

3

…

;

from whi h we obtain already the assumption (3.199a) with

q /( q "  ) q" ( )

C(r) # C0 Œ2 % a3 L q 

/(

)

% b rq % 3

3



:

(3.203)

Ë 235

3.6 Extensions of Nemytski  mappings

Furthermore, (3.197b) together with (3.197a) also guarantees the assumption (3.199b). Indeed, likewise previously one an estimate

!! !! g

- (

' r Ž y) - y !!!! ¢ g S

*

3



¢ C ¢ C

0

(

)/

 Πg S

% y Sq 1 % a q

(

3

*

3

q

)/

/(

q")

3

 y S1 

p ( q "  )/ q q /( q "  )

3 s S 

3



3

)/

3

% Œa % b y Sq1"  %

*





(

3

 ” g S



0

% b y Sq1"  % s Sp 2q"

Œa3



% y Sq 1 •

2



% b y Sq 1 % s Sp 2  : 3

3

Then one gets



"







*

"



¢C

3

"

q /( q "  )

q

g - (' r Ž y) - y H ¢ C0 ”"""" g S % y S1 % a3 

 Πg  % y qL q ( ;S1 ) % L ( ;S3 )

% b y Sq 1 """""L1 3





0







(

)

q /( q "  ) q" ( ) %

a3 L q

/(

)

%

3

• q

b3 y L q

(

;S1 ) %

3  ;

and therefore

sup

y L q ;S1 ¢1 g  L ;S3 ¢1

q /( q "  ) q" ( )

g - (' r Ž y) - y H ¢ C0 Œ2 % a3 L q 





/(

)

% b y qL q 3

(

;S1 )

%

;

3

)

(





*

(

)

whi h veries already the assumption (3.199b) with

C given again by (3.203).

Analogously, one an also show the estimate

sup



y L q ;S1 ¢1 g  L ;S3 ¢1

(



(

g - (' r Ž y1 " ' r Ž y2 ) - y H 



)

*



¢ C y

)

1

% y

L q ( ;S1 )

2

L q ( ;S1 )  y 1 " y 2 L q ( ;S1 )

;

(3.204)

whi h will be found useful later; f. Example 4.56. Indeed, by (3.197 ) one gets



g - (' r Ž y1 " ' r Ž y2 ) - y  





¢ g S3a % b y 4

*

¢ C y "y 1

1

4

 L q ( ;S1 ) ” g S3 

2

% b y 4

¢

q "2  )/  S1

(

1

*

q "2  )/  S1

(

2

for suitable onstants



q "2  )/  S1 (

2

% (a % b y 4

4

2

(

4



*

%a

4

% s Sp 1q"

C1 y1 "y2 L q ( ;S1 ) ” g S 3 

% b y

%

q y S1

q /( q "2  ) 4



%

%

% s Sp 2q" (

2

 )/ q

3

 y 1 " y 2 S

1

q "2  )/  S1



y S1 

(

1

 )/ q q /( q "2  ) )

%

q

y "y2 S1

1

q

y "y2 L q

1

% y Sq 1 • 

(

;S1 )

q

y "y 2 L q

1

q b4 y1 S1

q

y "y2 S1

1

%

;S1 ) q b4 y2 S1 (

% s Sp 2 • 4

C1 and C1 depending on p, q, and  only. This results to 

q 

;S1 ) Πg L  ( ;S3 ) % y L q ( ;S1 ) % 1 /( q "2  ) % a4 q % b4 y1 qL q ( ;S1 ) % b4 y2 qL q ( ;S1 ) L q q"2 ( )

g - (' Ž y1 " ' Ž y2 ) H ¢ C1 y1 "y2 L q 

/(





(

)



%

4

;

Ë

236

3 Young Measures and Their Generalizations

whi h yields (3.204) with

q /( q "2  ) q"2 ( )

C(r) # C1 Œ3 % a4 L q 

/(

)

% b rq % 4

4

:

q # %, all these estimates go through with the hypotheses (3.197b, ) modied

If

in the spirit of Remark 3.104.

Remark 3.107.

p

One an observe that the spa e Car (

First, its norm (3.202) ensures the mappings

y :

L q ( ; S



1)

Ù H

y ÜÙ g

; S2 ) is very natural, indeed. ' Ž y) and y Ù Ü g - (' r Ž y) 

- (

to be (strong,strong)- ontinuous; f. (3.199a) and (3.204), re-

spe tively. Se ondly, the assumptions (3.192a), (3.196), and (3.197a) are void provided

H #Carp ( ; S2 ). Indeed, for any y ò L q ( ; S1 ) and g ò L  ( ; S3 ), the growth ondition p  p (3.192b) ensures ' Ž y S3 òCar ( ; S2 ) so that ertainly g -( ' Ž y ) òCar ( ; S2 ), and thereq fore both (3.192a) and (3.196) are satised. Likewise, for any y ò L ( ; S1 ), the growth q q "  p 

ondition (3.197b) ensures ' r Ž y L S1 S3 ò Car ( ; S2 ) so that, for any g ò L ( ; S3 ) q p and y ò L ( ; S1 ), we have g - ( ' r Ž y ) - y òCar ( ; S2 ) and (3.197a) is satised, as well. 

/(



)

;

(



)







Remark 3.108 (Counterexamples for smoothness).

Smoothness

of

Nemytski map-

pings is, in fa t, quite strong property and the relaxation whi h makes them partly ane (hen e smooth) is thus worthy also from this analyti al reason. For example,

f ò C (R; R) with the growth at most linear but not ane, the superposition p p operator N f : u ÜÙ f Ž u : L ( ) Ù L ( ) is not ontinuously dierentiable. Indeed,

for any



one an see that

" "   " " " "L( L p ( ); L p ( )) "Nf ( u )"Nf ( u )" 

(

whi h an be pushed to 0 for

3.6.

1

p "  !  !p "   " ”X !!!( f ( u )" f ( u )) w !!! d x • # " " " f ( u )" f ( u )" "L

w p L ¢1

# sup







(

) ;

)

u Ù u only if f # onstant, hen e f 



is ane.

Two-argument mappings: bi-ane extensions

Let us end this se tion with a bi-ane extension of the Nemytski mapping

L q ( ; S1 ) , L p ( ; S2 ) Ù L  ( ; S3 ) with ' : , S1 , S2 Ù S3

N' :

a Carathéodory map-

ping satisfying again the growth ondition (3.192b). For this reason, let us have two

C( )-invariant subspa es H1 ’ Carq ( ; S1 ) and H2 ’ Carp ( ; S2 ) and assume

:s ò S :s ò S :g ò C( ; S ) : g - '(-; -; s ) ò H and g - '(-; s ; -) ò H ; *

1

1

2

2

2

1



3

1

2

(3.205)

s1 and s2 in pla e of r and s '(-; -; s2 ) DZ 1 ò L  ( ; S3 ) and '(-; s1 ; -) DZ 2 ò q p L  ( ; S3 ) for any 1 ò YH1 ( ; S3 ) and 2 ò YH2 ( ; S3 ) provided  ¡ 1. For  # 1 the

in a

ord with the notation from Chapter 7, we will use in the rest of this se tion. Then one has

Ë 237

3.6 Extensions of Nemytski  mappings

same holds true with

r a( ; S

in pla e of

3)

L  ( ; S3 ). The ondition (3.205) is trivially

satised105 in the following situation:

Lemma 3.109. If S is reexive, ' satises (3.192b) and has the form '(x ; s ; s ) # h (x ; s ) % h (x ; s ) with h ò H and h ò H , then the Nemytski mapping N ' admits a ' q p  bi-ane jointly ontinuous extension N from YH ( ; S ) , YH ( ; S ) to L ( ; S ) (or 1 2 to r a( ; S ) if  # 1) given obviously by the formula 3

1

1

2

1

2

1

1

2

1





2

2

2

3

3

'

N (1 ; 2 ) # h1 DZ 1 % h2 DZ 2 :

(3.206)



To extend non-additively oupled Nemytski mappings, we must onne ourselves to

 ¡ 1; f. also Example 3.115 below. Putting ' 1 (x ; s2 ) # ['(-; -; s2 ) DZ 1 ℄(x)

and

' 2 (x ; s1 ) # ['(-; s1 ; -) DZ 2 ℄(x) ;

(3.207)

we further suppose

: ò YHq1 ( ; S ) : ò YHp2 ( ; S ) :g ò C( ; S ) : g - ' 1 ò H and g - ' 2 ò H : *

1

1

2

2

2



3

(3.208)

1

Lemma 3.110 (Bi-ane extensions of Nemytski mappings). Let S and S be nitedimensional, S be reexive, H and H be separable,  ¡ 1, and ' satisfy (3.192b), 1

3

1

2

2

(3.205), and (3.208). Then the following ommutativity property holds

' 1 DZ 2 # ' 2 DZ 1 and the Nemytski mapping

ontinuous extension

'

N :

N'

in

L  ( ; S3 )

(3.209)

,

has a bi-ane separately (weak* weak*,weak)-

p q YH1 ( ; S1 ) , YH2 ( ; S2 )

Ù L  ( ; S

3)

dened by the formula

'

N (1 ; 2 ) # ' 1 DZ 2 :

(3.210)



 ¡ 1, the growth ondition (3.192b) ensures ' to have a lesser growth q and p in the variables s1 and s2 , respe tively. Then we an repla e 1 and 2 Ž Ž by their q - and p -non on entrating modi ations  1 and  2 whi h do exist by Proposition 3.81 be ause H 1 and H 2 are supposed separable and S1 and S2 nite-dimensional. Ž Ž 1 By Proposition 3.78,  1 and  2 admit Young-measure representations ò Y q ( ; S1 ) 2 p ò Y ( ; S2 ), respe tively. and From (3.207) we obtain by Proposition 3.78 for a.a. x ò :

Proof. Sin e than

' 1 (x ; s2 ) # X '(x ; s1 ; s2 ) S1

105

x (d s 1 )

1

Stri tly speaking, this is true if both

however.

and

' 2 (x ; s1 ) # X '(x ; s1 ; s2 ) S2

x (d s 2 ) :

2

H1 and H2 ontain L1 ( ) ‹ 1, whi h an be always supposed,

Ë

238

3 Young Measures and Their Generalizations

It is obvious that

' 1

and

' 2

' 1 DZ 2
(x) # ' 2 DZ 1
(x) #

p/   p and q/   q, respe tively. Then

has the growth

(3.208) yields the formulae for a.a.

x ò :

x (d s 1 )

X œX S2 S1

'(x ; s1 ; s2 )

1

X œX S1 S2

'(x ; s1 ; s2 )

2

x (d s 2 )

2

x (d s 2 )

and

(3.211a)

x (d s 1 ) :

1

(3.211b)

By (3.192b) we an estimate

X œX ' ( x ; s 1 ; s 2 ) S x (d s 2 ) 3 S1 S2

x (d s 1 )

2

¢

X œX „ a 1 ( x ) S1 S2

# a (x) % b 1

for a.a.

1

1

% b s 1

X s1  S1

1

q/

x

,

2

1

x (d s 1 )

q/

1

%

1

2

p/ 

x (d s 2 )

2

…

X s2  S2

x (d s 1 )

1

x (d s 2 )

p/ 

2

  %

x ò . Then we are authorized to use the Fubini theorem, whi h ensures the '(x ; s1 ; s2 )( 1x , 2x )(ds1 ds2 ), with S ,S

both right-hand sides in (3.211) to be equal to P 1

% s

1

2

x denoting the standard produ t of the measures

1

2

x and

x . Thus (3.209) has

been proved. Then by (3.208) and by Proposition 3.43 (generalized for the

1 ÜÙ

' 2

DZ 1 :

q YH1 ( ; S1 )

Ù

L  ( ; S

3)

and

2 ÜÙ

' 1

DZ 2 :

S3 -valued ase) both Ù L  ( ; S3 )

p YH2 ( ; S2 )

are ane and (weak*,weak)- ontinuous.

1 # iH1 (u1 ) with u1 ò L q ( ; S1 ) and 2 # i H2 (u2 ) with u2 ò L p ( ; S2 ), one has for any g ò C ( ; S3 ): Moreover, for

*



¼ g; N

'



(

1 ; 2 )½ #

 g; ' 1

DZ 2 # 2 ; g - ' 1

# ´ i H2 (u ); g - ' 1 µ # 2

# whi h shows that sion of

 X g ( x ) ' 1 ( x ; u 2 ( x )) d x

X g ( x ) ' ( x ; u 1 ( x ) ; u 2 ( x )) d x

# g; N' (u ; u 1

'

N (i H1 (u1 ); i H2 (u2 )) # N ' (u1 ; u2 ), so that N

'



2)

;

is a tually an exten-

Å

N ' , as laimed.

q # % or p # %, the respe tive terms of the type  - %  should % % be repla ed by b ( - ) with an arbitrary nonde reasing ontinuous b : R Ù R ; f.

Remark 3.111.



If

also Remark 3.104.

Notation 3.112.

In ase

N'

possesses a bi-ane extension, let us agree to write

'

N (1 ; 2 ) # ' DZ 1 DZ 2 :

(3.212)



This notation is to indi ate that the mapping (

'

'; 1 ; 2 ) ÜÙ N (1 ; 2 ) is, in fa t, tri , S1 , S2 Ù S3

linear. Also let us abbreviate the lass of Carathéodory mappings satisfying (3.192b), (3.205), and (3.208) by

CAR qH1p H2 ( , S , S ; S ) : ;

;

;

1

2

3

(3.213)

3.6 Extensions of Nemytski  mappings

Remark 3.113.

Lemmas 3.109 and 3.110 an be ombined together. Thus we an get a

bi-ane separately ontinuous extension of

N'

for any

'(x ; s1 ; s2 ) # '0 (x ; s1 ; s2 ) % h1 (x ; s1 ) % h2 (x ; s2 ) q; p;  '0 òCAR H1 ; H2 (

Although

'

Ë 239

, S , S ; S ); h ò H ; h ò H : 1

2

3

1

1

2

(3.214)

2

CAR qH1p H2 ( , S , S ; S

1 2 3 ) for  ¡ 1 be ause it may q and p in the variables s1 and s2 , respe tively, the bi-ane

need not belong to

have the growth pre isely

with

;

;

;

separately ontinuous extension does exist, and is obviously given by the formula

' DZ 1 DZ 2 # '0 DZ 1 DZ 2 % h1 DZ 1 % h2 DZ 2 :

(3.215)

'Ž y instead of ' as we did in Se tion 3.6.b. Thus we obtain semi-bi-ane extension ' Ž y DZ  1 DZ  2 . Moreover, we an also use it for

Example 3.114 (Failure of the joint ontinuity).106 An intera tion of os illations generally prevents the extended Nemytski mapping to be jointly ontinuous. Let us demon-

# (0; 1), S1 # S2 # S3 # R, q"1 % p"1 ¢ "1 , H1 q , H 2 ontaining L (0 ; 1) ‹ R , and ' given by

strate it on a simple example, using

p

ontaining L (0 ; 1)

‹R

*

*

'(x ; s1 ; s2 ) # s1 s2 :

(3.216)

Let us note that (3.192b) is satised by the Hölder inequality, (3.205) holds trivially,

' 1 (x ; s2 ) # ['(-; -; s2 ) DZ 1 ℄(x) # [(1 ‹ id) DZ 1 ℄(x)s2 # g(x)s2 for some g ò L q (0; 1) so that obviously ' 1 ò H2 and analogous onsiderations qp  yield also ' 2 ò H 1 . Altogether, we an see that ' òCAR H 1 H 2 ((0 ; 1) , R , R; R). Let us k q now take u 1 ò L (0 ; 1) dened by

and also (3.208) is valid be ause

;

;

;

u1k (x) # 

1 "1

if

; x ò (0; 1/2) ; l ò N : x # 2 xl/k ; 



otherwise

(3.217)

;

see the left-hand part of Figure 3.15. Furthermore, let us dene a shifted fun tion

L q (0; 1) by

u 1 (x) # u1 (x % 1/k) and u2 k

k

ò L p (0; 1) simply by

k



u 1k ò 

u1 # u2 . It is left as an easy k

k

exer ise to verify that

i H1 (u1k ) #

w*-lim

i H2 (u2k )

k Ù

k Ù

If

N'

1 1 i (1) % i H1 ("1) #:  # 2 H1 2 1 1 # i H2 (1) % i H2 ("1) #:  : 2 2

w*-lim

1

'

'

lim N (i H1 (u k ); i H2 (u k )) # N ( ; 

106

k Ù

i H1 ( u 1k ) ;

(3.218a)



(3.218b)

2

would have the jointly ontinuous extension

k Ù

w*-lim





1

2

1

2)

N ' , then inevitably '

# lim N (i H1 ( u k ); i H2 (u k )) ; k Ù





1

2

In onne tion with game theory, this lassi al example an be found basi ally also in Balder

[55, Example 2.6℄, Krasovski  and Subbotin [443, Se t. 9.1.2℄, Subbotin and Chentsov [737, Se t. VI.1℄, Warga [791, Se tions IX.2 and X.0.1℄, et .

240

Ë

3 Young Measures and Their Generalizations

N ' (u1k ; u2k ) # u1k u2k # 1 while # u 1 u2 # "1 for any k ò N.

whi h does not hold be ause the left-hand side equals to

k ' k the left-hand side equals to N ( u 1 ; u 2 ) 

k k



Example 3.115 (Failure of the separate ontinuity).

The



assumption

¡

1

in

Lemma 3.110 is really ne essary be ause otherwise the os illation ee ts in one variable, say

u1 , may intera t with

the on entration ones in the other variable, i.e.

u2 , to prevent even the separate ontinuity. To demonstrate it, let us take # (0; 1), S1 # S2 # S3 # R, and q # p # 2,  # 1, and ' given by '(x ; s1 ; s2 ) # PSfrag repla ements Furthermore, let us take the sequen es {

u1k

S

s1 s22 : 1%s1 

(3.219)

u1k }kòN and {u2k }kòN a

ording to Figure 3.15. u2k

$k

1/k 1

0



0

1

1/k

"1 Fig. 3.15:

Os illating and on entrating sequen es that intera t via Nemytski  mapping.

More pre isely,

u1k ò L2 (0; 1)

is dened again by (3.217) while

u2k ò L2 (0; 1)

is

dened now by

u2k (x) # 

$k

if

0

0   x   1/k ;

otherwise

(3.220)

:

!! s

!

s22 /(1%s1 )!!!! ¢ s22 , ' satises (3.192b) with  # 1, a 1 # b 1 # 0, and 1 # 1. Taking H # H 1 # H 2 # C ([0 ; 1℄) ‹ 0 2 Ô (R ) with R being the smallest omplete subring of C (R), we an identify ea h 2  ò Y H (0; 1; R) with a DiPerna-Majda measure ( ;  ) ò DM2R (0; 1; R); f. Se tion 3.2. Then  1 from (3.218a) has the representation (  ;  ) with Let us note that, thanks to the obvious estimate !! !

 # 2;

x

1

1 1 # Æ % Æ" 2 2 1

1

2 # w*-limkÙ i H (u2k ) (note that this limit does exist be ause every v ò R has a limit at innity) has the representation ( ;  ) with

f. (3.56), while



 # 1 % Æ0 ;

x

#

Æ0 Æ



f.

(3.57).

Let

us

note

s1 (1%s1 )"1 Æ0 ò Ö L1 (0; 1)

that

(3.205)

is

if if

x #Ö 0 ; x # 0;

trivially satised,

but

'(-; s1 ; -) DZ 2

#

so that (3.207) looses a sense and thus the existen e of a

separately ontinuous extension

N

'

is not guaranteed by Lemma 3.110.

3.6 Extensions of Nemytski  mappings

Ë 241

In fa t, su h extension does not exist, otherwise it would have to hold 1

'

1

lim lim X '(x ; u k ; u l ) dx # X N ( ;  ) dx # lim lim

k Ù l Ù

1



1

0

2

2

1

l Ù k Ù

0

X 0

'(x ; u1k ; u2l ) dx ;

0; 1℄. However,

where the entral integral is understood in the sense of measures on [ this is not true be ause the left-hand side an be evaluated as follows 1/

lim

¬ lim X k Ù l Ù 0

u1k (x)l

l

1%u (x) k

dx­ # lim

k Ù

1

1 1 # ; 2 2

while for the right-hand side one has

lim ¬ lim

l Ù

k Ù

1/

l

X 0

u1k (x)l

1%u k (x)

dx­ # lim 0 # 0 : l Ù

1

Remark 3.116 (Relations with Se tion 2.5). Let us onsider U # L q ( ; S ), U # L p ( ; S ), Y # L  ( ; S ),  # N ' , and Fl # FH l # l (H l ) % { onstants on U l } with l : H l Ù CBl (U l ) dened by [ l h l ℄(u l ) # P h l (x ; u l (x)) dx, l # 1; 2. Then we an 1

2

1

2

3

relate the extension stated in Lemma 3.110 with the abstra t situation stated in Proposition 2.36: indeed,

  1 # (N ' )  1

y òY *

where

*

from (2.35) take here the form

N ' )1 y* ℄(u2 ) #

X g(x)

-

' 1 (x ; u2 (x)) dx ;

(3.221a)

N ' )2 y* ℄(u1 ) #

X g(x)

-

' 2 (x ; u1 (x)) dx ;

(3.221b)

[(

[(

  2 # (N ' )  2

and

is now denoted by

g ò L  ( ; S3 ) 

*

and

1 # 1 1 , 2 # 2 2 . *

*

For

example, (3.221a) follows from the hain:

N ' )  1 y *
( u 2 ) #

Ž (-; u ) #  ; (g - (' Ž u )) # #  ; g - (' Ž u ) # g; (' Ž u ) DZ  # g; ' 1 Ž u

(

1 ; y 1

*

2

1

1

2

2

2

1

2

:

2 # i2 (u2 ) implies g - (' Ž u2 ) ò H1 , so that 1 (g - (' Ž u2 )) ò 2 (g - (' Ž u1 )) ò FH2 , whi h veries (2.33). Moreover,  1 (3.208) also implies g - ' ò H2 , hen e  1 y ò FH2 . The linearity of the mapping  1 g # y ÜÙ  y is obvious, while its ontinuity as a mapping L  ( ; S3 ) # Y Ù FH2 Let us note that (3.208) for

FH1 .

Similarly, we have also

*

*

with



*

FH2

*

*

endowed with the natural seminorms follows immediately from (3.192b) by

the estimate

!! !!X [ g !! !

- (

!!

' 1 Ž u2 )℄(x) dx!!!! ¢ g L  !



(

" " " a1

;S3 ) " " " *

% b % q  % u /

1

1

p/  " " " " S2 " "L  ( ;S

2

q

3)

1 ò YH1 % ( ; S1 ), with a1 , b1 , and 1 oming from (3.192b). Altogether we shown  1 ò L(Y ; FH2 ). Likewise,  2 ò L(Y ; FH1 ), hen e (2.36) has been veried. Eventu-

for

;

*

*

ally, the ommutativity (2.37) follows from (3.209) proved in Lemma 3.110.

242

Ë

3 Young Measures and Their Generalizations

Remark 3.117.

From the proof of Lemma 3.110 one an see that the separately ontin-

uous bi-ane extension for

N ' : YH1 ( ; S1 ) , YH2 ( ; S2 ) Ù L  ( ; S3 ) p

q

 # 1 but one must restri t N

'

on

does exist even

q- and p-non on entrating generalized Young 2 was not

fun tionals only. This also ex ludes the situation in Example 3.115 where

p-non on entrating.

4 Relaxation in Optimization Theory ...

I

observed

that

the

maximum

prin iple

in

ontrol theory is equivalent to the onditions of Euler-Lagrange

and

Weierstrass

in

the

lassi al

theory. [384, p. viii℄

Magnus Rudolph Hestenes

(1906-1991)

I had a proof of the Maximum Prin iple. Not as a su ient ondition, but as a ne essary ondition... [126℄

Vladimir Grigorevi h Boltyansky In

addition

onsider

to

`original'

"approximate"

solutions solutions

(1925-2019)

..., that

we are

also se-

quen es ... and `relaxed' solutions that are a form of weak, or extended, solutions. ... we study relaxed solutions for a number of reasons: they yield a

omplete theory that en ompasses both existen e theorems and ne essary onditions; they provide the means for onstru ting optimal approximate solutions;

they

properly

model

ertain

physi al

situations [791, pp. xi-xii℄

Ja k Warga (1922-2011)

This hapter begins with a relaxation theory for abstra t optimization (esp. optimal ontrol) problems: Se tion 4.1 studies basi relations between the original and the relaxed problems and also a omparison of various relaxations is performed there. The exposition in this se tion pro eeds on the most general level, using extensions of the original problems by the onvex ompa ti ation theory from Chapter 2. This enables us also to formulate the rst-order optimality onditions, whi h takes the form of abstra t maximum prin iples if one works with onvex ompa ti ations in their

anoni al forms. The remaining part deals with more on rete optimal ontrol problems with ontrols ranging some Lebesgue spa e. Therefore, the relaxation is performed in terms of the generalized Young fun tionals developed in Chapter 3. Two general important prin iples are treated separately in Se tion 4.2: the rst one on erns Pontryagin-type maximum prin iples reated by a lo alization of the integral maximum prin iples resulted straightforwardly from the abstra t maximum prin iples derived in Se tion 4.1, and the se ond one establishes a ertain non on entration regularity of generalized Young fun tionals whi h satisfy these integral maximum prin iples. This se tion further dis usses various onsequen es of the maximum prin iple. All this enables us to treat a wide lass of on rete problems with ontrols ranging Lebesgue spa es by a routine way, assembling already prefabri ated tools and results. Usage of these prefabri ated tools is demonstrated on quite on rete optimal ontrol problems in Se tions 4.36. They on ern su

essively the nonlinear dynami al https://doi.org/10.1515/9783110590852-004

Ë

244

4 Relaxation in Optimization Theory

systems (i.e. initial value problems for ordinary dierential or dierential-algebrai equations), partial dierential equations of the ellipti and the paraboli types, and the Hammerstein integral equation. The rst ase is handled quite omprehensively to demonstrate all possible appli ations of the abstra t results, i.e. impa ts of results for the relaxed problem to the original problem itself, relations between the original and the relaxed problems, existen e of solutions to the relaxed problems and their stability, rst-order optimality onditions, qualitative properties of optimal relaxed ontrols and their numeri al approximation. The resting ases are exposed more briey rather to show various pe uliarities onne ted with the parti ular distributed parameter systems; nevertheless, always the well-posed relaxed problem is onstru ted and a Pontryagin-type maximum prin iple is derived. The on rete form of su h maximum prin iples is losely related with the hosen onvex ompa ti ation used to relax the original problem, the Hamiltonians involved in these prin iples having a very denite meaning, namely Gâteaux dierentials in anoni al forms. In all ases, we will admit

ontrols that need not be bounded in the

L



-norm; in parti ular we will onsider a

polynomial growth (of possibly dierent orders

p and q) of both the ontrols and the

states. Being usual in on rete problems and not essentially onning appli ability, ex ept Se t. 4.5.a we onne ourselves on the more onventional on ept of sequen es rather than general nets (and thus e.g. subsequen es instead of ner nets used in Chapters 2 and 3). To this goal, essentially without loss of generality, we will assume here separability of spa e of test fun tions.

4.1

Abstra t optimization problems

In this se tion we will develop a relaxation theory of the abstra t optimization problem in the form (PO )



Minimize subje t to

(u) for u ò U ; R(u) ¢ 0 ;

:UÙR a  ost fun tion, R : U Ù  a mapping, and  an ordered Bana h spa e with D ’ 

where

U

is a set (say a topologi al spa e) endowed with a bornology

B, 

 by a  "  ò D.

a losed onvex one with the vertex at the origin. Re all that the ordering of

one

D

is dened as follows: for

;  ò  

we write

 ¢ 



if and only if



D has a non-empty interior int (D), we will write     if and only if  "  òint  (D). Besides, we will onsider a lo ally onvex topology  on  ner than the weak topology (so that D remains losed with respe t to  ); f. also Se tion 1.2.d. 

Moreover, if 

Without any further data quali ation, (PO ) need not have any solution or the set of solutions

Argmin(PO ), even if nonempty, need not be stable with respe t to data

perturbations. The on ept, more natural than the solutions in the above lassi al sense, relies on asymptoti ally admissible and minimizing sequen es, invented basi-

u

ally by Levitin and Polyak [489℄. A sequen e { k } k òN will be alled

-asymptoti ally

Ë 245

4.1 Abstra t optimization problems

"D in the topology , whi h means that for any -neighbourhood N ’  of 0 there is k N ò N su h that R(u k ) ò N " D whenever k £ k N . For example, if  is the strong topology, then {u k }kòN is -asymptoti ally admissible if lim k Ù inf ¢0 R(u k ) "   # 0. If  is the weak topology, then {u k }kòN is  -asymptoti ally admissible if, for any  ò  , lim k Ù inf ¢0 £ 0 provided h ò H su h that h(x ; s) £ a0 (x) for some a0 ò L1 ( ), Ž Ž p <  "  ; h > ¡ 0 provided  #Ö  and h ò H is oer ive in the sense h ( x ; s ) £ a 0 ( x ) % b  s  1 with some a 0 ò L ( ) and b ¡ 0.


£ 0. The point (i) is proved. Ž Let us suppose that (ii) does not hold, so that <  "  ; h > # 0 for some h ò H

from whi h we obtain in the limit
£ 0. Taking into a

ount also our assumption Ž <  "  ; h > # 0, we obtain bsp with b p



























Ž

Ž

"  ; h ¢  "  ; h # 0 :

(4.40)





Ž " h , we an see that ¢

Å

Let us now turn our attention to onsequen es whi h the pointwise maximum prin iple may have in on rete situations. Let us mention two typi al examples.

Example 4.25 (Bang-bang ontrols).

If the ontrol a ts linearly both in the ontrolled

p

H # L p ( ; S ) so that Y H ( ; S) # H Ê   p ¢ %; for S # Rm f. also Example 3.73. Supposing S measurable with losed onvex values, we have U ad # Uad . For a given p p m Hamiltonian h ( x ; s ) # < g ( x ) ; s > with some g ò L ( ; S ) and an optimal  ò Y ( ; R ) H 

system and in the ost fun tional, then we an take25 *



23

*

L p ( ; S) provided S is reexive and 1

*

Similar kind of results for spe ial lass of minimization problems has been obtained also by Kinder-

lehrer and Pedregal [425℄. For optimal ontrol problems see Berlio

hi and Lasry [114℄ where this was only supposed as a hypothesis, however.

"h required in Lemma 4.22(ii). L p ( ; S* ) with a subspa e of Carp ( ; S) via the mapping g

24

Note that (4.36) ensures just the oer ivity of

25

Here we identify naturally



h(x ; s) # .

ÜÙ h with

4.2 Optimization problems on Lebesgue spa es

identied with some

Ë 269

u ò L p ( ; S), the maximum prin iple (4.34) results in

:a.a. x ò :

g ( x ) ; u ( x )

# max g(x); s :

(4.42)

sòS(x)

Let us note that the maximum in (4.42) is a tually attained, for example, just at the point

s # u(x). Furthermore, (4.42) an be rewritten into the form

:a.a. x ò : g(x) ò N S x (u(x)) ;

(4.43)

( )

whi h implies parti ularly that

:a.a. x ò : u(x) òbd(S(x))

or

g(x) # 0 :

(4.44)

The phenomenon that some optimal ontrol tends to follow the boundary of

S(x)

is

alled a bang-bang prin iple.

Example 4.26 (Chattering ontrols). If p # , S # Rm , S(x) is ompa t, ahd H # Car ( ; Rm ), then the orresponding onvex - ompa ti ation of L ( ; Rm ) is m equivalent with the set of the Young measures Y ( ; R ). Taking a Hamiltonian p m m # h(x ; s) ò Car ( ; R ) and identifying  ò YH ( ; R ) with a Young measure 







x } x ò , then (4.35) results in

{

:a.a. x ò :

h(x ; s) x (ds) X S(x)

# max h(x ; s) :

(4.45)

sòS(x)

S(x) is ompa t x ò , h(x ; -) attains its maximum on S ( x ) at a nite number of points, say s l ( x ) ò S ( x ), l # 1 ; :::; k . Then (4.45) Let us note that the maximum in (4.45) is a tually attained be ause

h(x ; -)

and

is ontinuous. It is an often ase that, for a.a.

says that, in parti ular, the optimal relaxed ontrol must be ne essarily a onvex

ombination of the Dira measures:

x

k

k H a l (x)Æ u l (x) l #1

(4.46)

) and u l ò L ( ; Rm ) su h that 0 ¢ a l (x) ¢ 1, u l (x) ò S(x), and a l (x) # 1 for a.a. x ò . In other words, the Young measure is omposed from

with some

k  l #1

#

al ò L





(

atoms. Relaxed ontrols of this type are alled hattering ontrols.26 Su h ontrols

are espe ially important if they are pie ewise onstant (resulting, for example, as optimal ontrols for approximate problems reated by the approximation from Se t. 3.5.b be ause then they an be readily implemented on omputers. Paraphrasing this denition for the general ase, a generalized Young fun tional

 ò YH ( ; Rm ) will be alled hattering if it admits the following representation p

 #

26

H

k a i (u ) ; l #1 l H l

with

u l ò L p ( ; Rm )

and

al ò L



(

)

(4.47)

Chattering ontrols are also o

asionally alled, e.g. in [267, 343℄, sliding modes or regimes, or

o

asionally Gamkrelidze's ontrols. Also, sometimes dierent meaning of hattering ontrols an o

ur; f. Zelikin and Borisov [814℄.

Ë

270

4 Relaxation in Optimization Theory

a l (x) £ 0 and kl#1 a l (x) # 1 for a.a. x ò , where the expression kl#1 a l  l with a l ò G and  l ò H is dened for H being G -invariant by27 su h that

*

:h ò H :

¼H

k a l #1 l l

; h½ #

H

k h DZ  l ; a l : l #1

(4.48)

Let us note that this extended denition has the previous meaning of a linear ombi-

 l provided a l are onstants on .  in the form (4.47) will be also said k-atomi with k ò N referring to the number k in (4.47). As every i H ( u l ) is p -non on entrating, it an be seen that  from (4.47) is p -non on entrating, as well. Therefore, by Proposition 3.78, every hattering p  ò YH ( ; Rm ) admits a Young-measure representation ò Yp ( ; Rm ), and obviously nation of

Every

takes the form (4.46). The following assertion treats situations when every optimal solution is hattering:

Proposition 4.27 (Chattering ontrols I). Let  ò U ad ’ YHp ( ; Rm ) satisfy the maximum prin iple (4.32), H be separable, U ad #Ö  take the form (4.33) with a measurable losedm valued mapping S : ± R , and h satisfy the des ent ondition (4.36). Then: (i) If, for a.a. x ò , h ( x ; -) attains its maximum on S ( x ) at no more than k ò N points, 

k-atomi . h(x ; -) is stri tly on ave for a.a. x ò and S is onvex-valued, then  is 1-atomi . In other words, the relaxed ontrol  , being of the form i H ( u ) with some u ò U ad , is, is inevitably

(ii) If

in fa t, an original ontrol. Proof. By Theorem 4.24,



is

p-non on entrating, and therefore it admits a YoungRm ) su h that # P PRm h(x; s) x (ds) dx for

ò Yp ( ;

measure representation





h ò H ; see Proposition 3.78. Moreover28 supp( x ) ’ S(x) for a.a. x ò . By Theorem 4.21,  satises also the pointwise maximum prin iple (4.35), whi h results here in P h(x ; s) x (ds) # [h DZ ℄(x) # h S (x) :# supsòS x h(x ; s) for a.a. x ò . S x any



(

( )

’ S (x) where S (x) # {s ò S(x); h(x ; s) #

)

h S (x)}. As

S0 (x) is supposed 0 k points, x takes the form (4.46) with some u l (x) ò S(x) and a l (x) ò [0; 1℄ su h that kl#1 a l (x) # 1. By Theorem 1.25 the multivalued mapping S0 :

± Rm is measurable, so that by Theorem 1.26 we may suppose u l measurable. Then also a l may be supposed measurable be ause x ÜÙ x is weakly measurable.29

Then supp( x )

0

to onsist from at most

*

27

Of ourse, the left-hand-side duality is between

H * and H while the right-hand-side one is between

# i H (u l ), then always h DZ  l ò L ( ) so that the right-hand side of (4.48) has a good meaning even for a l ò L ( ) \ G , as used in (4.47). # 0 for h(x ; s) # inf sòS x s " s . Yet, w*-limkÙ Æ(u k ) # 28 This fa t is obviously equivalent to h DZ for u k ò U ad implies limkÙ h Ž u k # h DZ # 0 be ause h Ž u k # 0 for any u k ò Uad ; note also that h is a G* and G. Moreover, if  l

1



( )







Carathéodory fun tion be ause

29

We may suppose that all

k

S is measurable. k points { u l ( x )} l#1

are mutually dierent otherwise we an divide

on measurable parts with this property for various

k.

Sin e all

ul

are measurable, there is

hl



ò

4.2 Optimization problems on Lebesgue spa es

Ë 271

u l (x) maximizes h(x ; -) over S(x), it holds h(x ; u l (x)) # h S (x). Simultap neously, by the des ent ondition (4.36), it also holds h ( x ; u l ( x )) ¢ a ( x ) " b  u l ( x ) . p S " 1 p Altogether, b  u l ( x ) ¢ a ( x ) " h ( x ). Thus we have got u l L p ;Rm ¢ b ( a L 1 % S 1 p p m 1 h L )   %, so that ertainly u l ò L ( ; R ), as laimed in (4.47). The point (i) As always

(

/

)

(

)

/

(

)

has been thus proved. The point (ii) then follows immediately be ause a stri tly on ave Hamiltonian

Å

attains its maximum at no more than one point.

If the Hamiltonian need not attain its maximum at a nite number of points, some optimal relaxed ontrols need not be hattering. Nevertheless, the maximum prin iple enables sometimes to establish a bit weaker result, namely that at least one optimal relaxed ontrol is hattering. Even su h weaker result might be of some usage espe ially if our task is to nd (approximately) not all optimal ontrols, but at least one optimal

ontrol, whi h is a usual standpoint, indeed. Su h kind of results is supported by the following general prin iple:

Proposition 4.28 (Chattering ontrols II).

H

Let

be

separable,

Uad

#Ö



take

S measurable and losed-valued, there exist an optimal30 p 0 ò U ad ’ YH ( ; Rm ) satisfying the maximum prin iple (4.32) with h ò H satisfying the k des ent ondition (4.36), and let, for some nite olle tion { h l } l #1 ’ H the following

the form (4.33) with

impli ation holds:

 ò U ad ; h DZ  # h S with h S (x) :# supsòS h l DZ  # h l DZ 0 ; l # 1; ::: ; k Then there exists at least one

x

( )

h(x ; s) ;

/ 7 ? 7 G

âá  is optimal :

(4.49)

 whi h is optimal and (k%1)-atomi .

0 is p-non on entrating, and therefore by Proposition 3.78, ò Yp ( ; Rm ) su h that <  0 ; h > # P P m h ( x ; s ) x (d s ) d x for any h ò H . Besides, we may suppose that, for

R p S a.a. x ò , supp( x ) is ontained in S 0 ( x ) # { s ò S ( x );  s  ¢ ( a ( x ) " h ( x ))/ b } where a and b ome from (4.36). 1% k Let us now dene the multivalued measurable mapping C : ± R by Proof.31 By Theorem 4.24,

it admits a Young-measure representation, i.e. there exists some 



C(x) # As

† h ( x ; s ) ;



h1 (x ; s) ; ::: ; h k (x ; s) ò R1%k ; s ò S0 (x)‡ :

S0 (x) is ompa t for a.a. x ò , C(x) is ompa t as well. As

Rm

L1 ( ; C0 (

)) su h that

h l (x ; u l (x)) # 1 while h l (x ; u j (x)) # 0 for j

%

x òr a1 ( S 0 ( x )), we have

#Ö l, whi h shows that a l # h l DZ

must be measurable, too.

30 31

The adje tive optimal an have an entirely formal meaning in this statement. Some ideas of this proof ome from the work by Bonnetier and Con a [136℄. f. also Balakrishnan

[49, Thm. 1.9.1℄ who did not handle measurability, however.

272

Ë

4 Relaxation in Optimization Theory

g(x) #

#

”X h ( x ; s ) x (d s ) ; X h 1 ( x ; s ) x (d s ) ; ::: ; X h k ( x ; s ) x (d s )• S0 (x) S0 (x) S0 (x) „ [ h DZ  0 ℄( x ) ;

[

h1 DZ 0 ℄(x); ::: ; [h k DZ 0 ℄(x) … ò

o(

C(x)) :

Then, by the Carathéodory theorem 1.12, this point an be obtained by a onvex ombi-

k%1 points of C(x) be ause C(x) is a subset of a k-dimensional ane h S (x)} , R with h S (x) :# supsòS x h(x ; s). In other words, there k %1 k %1 k %1 exists { u i ( x )} i #1 ’ S 0 ( x ) and { a i ( x )} i #1 ’ [0 ; 1℄ su h that  i #1 a i ( x ) # 1 and nation of at most

manifold, namely {

(

„ [ h DZ  0 ℄( x ) ;

#

H

[

)

h1 DZ 0 ℄(x); ::: ; [h k DZ 0 ℄(x) …

k %1 a (x) „ h(x ; u i (x)); i #1 i

h1 (x ; u i (x)); ::: ; h k (x ; u i (x)) … :

g and C are measurable, we an suppose the mappings a i and u i measurable a i ò L ( ) and u i ò L p ( ; Rm ) thanks to the denition of S S0 ; re all that always a ; h ò L1 ( ). Therefore also u i ò Uad . k %1 Let us dene  by <  ; h > # P  i #1 a i ( x ) h ( x ; u i ( x )) d x with h ò H , whi h is ( k %1)

atomi by the very denition. We have also  ò U ad be ause the Young measure , k %1 p m dened by x #  i #1 a i ( x ) Æ u i x , belongs to Y ( ; R ), determines ee tively the fun tional  by means of the relation (3.14), and is attainable from U ad sin e ea h u i ò U ad . S Also we have obviously h DZ  # h DZ  0 # h and h l DZ  # h l DZ  0 for any l # 1 ; :::; k . Therefore, by the hypothesis (4.49),  is optimal. Å

As both



( f. Theorem 1.27) and thus







( )

One-atomi hattering optimal ontrols are naturally of a spe ial importan e, as they are optimal for the original problem. This is another noteworthy appli ation of the relaxed problems. The essen e of su h existen e theory for the original problems

an be seen on a prototype problem

Minimize subje t to

X ' ( x ; y ( x ) ; u ( x )) d x

( ost fun tional)

A(y) # f(y; u) on ; ( x ; y ( x ) ; u ( x )) ¢ 0 :a.a. x ò ; u(x) ò S(x) :a.a. x ò ; y ò L q ( ; R ); u ò L p ( ; Rm ) ;

(state equation)

/ 7 7 7 7 7 7 7

(state/ ontrol onstraints) ? 7 7 ( ontrol onstraints)

(4.50)

7 7 7 7 7 G

R

m 2 is assumed ordered by a one D to give where A is an abstra t operator and where p m ), sense to the inequality in (4.50). Moreover, we will assume that, for any u ò L ( ; the state equation

A(y) # f(y; u)

has a unique solution

y

R

so that the state-equation

itself does not represent any onstraint on the ontrol32 and determines a ontrol-tostate mapping

32

u ÜÙ y whi h is (weak,strong)- ontinuous.

This attribute ex ludes the variational problems where

whi h (ex ept trivial ases) brings impli it restri tions on

Ay # f(x ; y(x); u(x)) takes the form ›y # u u. Similar impli it restri tions may arise in

optimal ontrol of some dierential-algebrai equations, f. Se tion 4.3.g below.

4.2 Optimization problems on Lebesgue spa es

Ë 273

a  ò L  ( ), natural growth onditions for the Carathéodory Rm Ù R, f : ,Rn ,Rm Ù Rn and : ,Rn ,Rm Ù R are

Using the notation integrands

' : ,R 

n,

'(x ; r; s) ¢ a1 (x) % brq % sp ; f x ; r; s) ¢ a p1 (x) % br

 ( 

(

q/p1

x ; r; s) ¢ a p2 (x) % br

(4.51a)

p/ p 1

% s ; % sp p2

q/p2

and

(4.51b)

/

(4.51 )

p1 ; p2 ¡ 1 and b; ò R so that '(y; u) lives in L1 ( ) while f(y; u) ò L p1 ( ; Rn ) p  ( y; u ) ò L 2 ( ; R ). Moreover, we suppose the oer ivity of (4.50) in the sense

for some and

; ¡ 0 :a.a. x ò : (r; s) ò Rn ,Rm : 1

Let us note that

lim u Lp ;Rm Ù

u



(



)

P

'(x ; r; s) £ 1 sp :

'(y u ; u) dx Ù %,

(4.52)

whi h ensures that every se-

quen e of ontrols { k } k òN minimizing for (4.50) is inevitably bounded in

L p ( ; Rm ).

Theorem 4.29 (Filippov-Roxin prin iple).33 Let there exist A" : L p1 ( ; Rn ) ٠L p1 ( ; Rn ) ontinuous and ompa t, S : ± Rm be measurable losed-valued, (4.51) 1

(4.52) hold, and the minimization problem (4.50) be feasible. Let furthermore the so alled orientor eld

Q(x ; r) #

Q dened by

†' ( x ; r; s )%

R% ; f x; r; s (

0

)

;

(

x ; r; s)% D ò R1%n%m ; s ò S(x)‡

(4.53)

be onvex and losed. Then (4.50) has a solution. Proof. First, inspired by [561℄, it will be more suitable to reformulate the onvexity and

losedness of

Q as a ondition

:a.a. x ò : r ò Rn : o ',f , with


( x ; r; R ( x ; r ))

’ Q(x ; r)

R(x ; r) # s ò S(x);

(

x ; r; s) ¢ 0 :

(4.54a) (4.54b)

q1 ; q2 ò Q(x ; r), one has s ; s ò S(x) su h that q1 £ q2 # f(x ; r; s i ), and q3i £ i 3 ( x ; r; s ) for i # 1 ; 2, and then (4.54) guarantees existen e of s ò S(x) su h that 1 1 i i i  i #1 2 2 ( ' ( x ; r; s ) ; f ( x ; r; s ) ; ( x ; r; s )) ò Q , whi h eventually results to  i #1 2 2 q i ò Q(x ; r). Conversely, the onvexity and losedness of Q implies (4.54) be ause always

o[', f , ℄(x ; r; S(x)) ’ oQ(x ; r). p m Then, we make the relaxation by hoosing the linear spa e H ’ Car ( ; R ) as Indeed, (4.54) implies the onvexity of 1

i

2

Q

from (4.53) be ause, taking

'(x ; r; s i ),

i

;

;

H # span †g0 (' Ž y0 ) % g1 - (f Ž y1 ) % g2 - ( Ž y2 ) òCarp ( ; Rm );

g0 ò C( ); g1 ò C( ; Rn ); g2 ò C( ; Rn ); y0 ; y1 ; y2 ò L q ( ; R ) ‡ :

33





(4.55)

This assertion generalizes the Filippov-Roxin ondition formulated originally for un onstrained

optimal ontrol of ordinary dierential equation [311, 704℄, f. also Cesari [193℄ or Mordukhovi h [549, 551℄.

Ë

274

4 Relaxation in Optimization Theory

Then we onsider the relaxed problem



/ 7 7 7 7

( ost fun tional)

A(y) # f Ž y DZ  on ; Ž y DZ  ¢ 0 on ; y ò L q ( ; Rn );  ò U ad ;

subje t to

where

' Ž y DZ  dx

X

Minimize

(state equation)

(4.56)

?

7 (state/ ontrol onstraints) 7 7 7 G

U ad ’ Y H ( ; Rm ) is from (4.30) with H from (4.55). p

H from (4.55), we p1 ; p2 ¡ 1 so that, by (4.51) (4.52), f ( x ; r; -) and ( x ; r; -) has a lesser growth than ' ( x ; r; -), any any solution  to p (4.56) is p -non on entrating and thus has a representation by an L -Young measure p m ò Y ( ; R ), f. Proposition 3.78. This solves the following problem: By using Proposition 3.102 modied straightforwardly for spa e

an see that

H

is separable. Furthermore, using that

X X

subje t to

A(y) # X

Rm

Rm

X

(

Rm

x (d s )d x

'(x ; y(x); s)

Minimize

x (d s )

x ; y(x); s)

R

y ò L q ( ; Let us note that, for a.a.

x (d s )

f(x ; y(x); s)

¢0

ò Yp ( ;

n );

R

;

supp  x ’ S(x) :a.a. x ò :

x ò , the probability measure

ò Yp ( ;

xò

Let us x have

limÙ



restri tion with

P m R \B 

p  ; y) ò Y H ( ; m ) solves (4.57).

 

k x; 

#

k x;  For

 £ 1,

x (d s )

p

)

(

; Rn ). Its

ds)   %. By the Lebesgue theorem, we

p

x(

# 0, where B 

is the ball in

Rm of the radius . The

x ;  ; f. the proof of Theorem 3.6. Then we put

# 

k x; 

%  Æ s0 0

x (d s ) ;

0 # 0 () # X

and

Rm \ B 

0 () ¢ PRm

ertainly P m [ ' , f , R

obviously

and therefore

Rm , L q



 i #  i (x) £

limkÙ

s

 

R

G

k k x  B  an be approximated by a k -atomi measure x ;  #  i #1  i Æ s i k 0, i#1  i # PB  x (ds), and s i # s i (x) ò B   S(x), so that w*-

#

x; 

s

for whi h P m   R

(4.57)

x must be supported on the

By the assumed oer ivity, (4.56) has a solution (

L p -Young-measure representation

? 7 7 7 7 7 7 7

:a.a. x ò ;

m );

R(x ; y(x)) dened above in (4.54b).

losed set

on

/ 7 7 7 7 7 7 7

\

s0 ò S(x):

k x (d s ). Let us note also that  i #0  i # 1 k ℄( x ; y ( x ) ; s ) x ;  (d s ) ò o[ ' , f , ℄( x ; y ( x ) ; S ( x )). B

s

 

p

Moreover,

lim

k Ù

X

Rm

v(s)

k x ;  (d s )

# X v(s) Rm

for any



# lim

k X v ( s ) x ;  (d s ) k Ù B 

x ;  (d s )



%  v(s

%  v(s ) # X v(s) 0

0

Rm

0

x (d s )

0)

%X

v(s)

Rm \ B 

v ontinuous. If v has at most p-growth, we have limÙ



v # [', f ,

℄(

P m R \B 

%  v(s

v(s)

0

p s



0)

x (d s )

x (d s )   %. Also lim Ù  0 (  ) v ( s 0 ) x ; y(x); -), by (4.54) we obtain

by the Lebesgue theorem be ause P m R Altogether, for

x (d s )

#0 # 0.

4.2 Optimization problems on Lebesgue spa es

X

Rm

',f ,


( x ; y ( x ) ; s ) x (d s )

# lim lim Ù 

Rm

ò o ', f ,

k
( x ; y ( x ) ; s ) x ;  (d s )

', f ,

X

k Ù

Ë 275


( x ; y ( x ) ; S ( x ))

’ Q(x ; y(x)) :

(4.58)

Let us put

U(x) # –s ò S(x); '(x ; y(x); s) ¢ X

'(x ; y(x); )

S(x)

f(x ; y(x); s) # X

x (d  ) ;

f(x ; y(x); )

S(x)

(

x ; y(x); s) ¢ X

S(x)

(

x (d  ) ;

x ; y(x); )

x (d  )— ;

(4.59)

U(x) is nonempty: Indeed, by (4.54), for any (q0 ; q1 ; q2 ) ò Q(x ; y(x)) s ò S(x) su h that q0 £ '(x ; y(x); s), q1 # f(x ; y(x); s), and q2 £ (x ; y(x); s).

and show that there is

Hen e, for the parti ular hoi e

(

q0 ; q1 ; q2 ) # q0 (x); q1 (x); q2 (x) # X

S(x)

',f ,


( x ; y ( x ) ; s ) x (d s ) ;

(4.60)

q0 (x) £ '(x ; y(x); s), q1 (x) # f(x ; y(x); s), and q2 (x) £ x ; y(x); s) for some s ò S(x), hen e U(x) #Ö . m dened by (4.59) is meaMoreover, the multi-valued mapping U : ± R surable. Indeed, weakly* measurable and ' , f and Carathéodory mappings imply that q from (4.60) is measurable. Furthermore, by [37, Thm. 8.2.9℄, the level sets x ÜÙ {s ò Rm ; '(x ; y(x); s) ¢ q0 (x)}, x ÜÙ {s ò Rm ; f(x ; y(x); s) # q1 (x)}, and x ÜÙ {s ò Rm ; (x ; y(x); s) ¢ q2 (x)} are measurable. By [37, Thm. 8.2.4℄, the interse tion of these level sets, whi h is just U ( x ), is also a measurable multi-valued mapping. Obviously, U ( x ) is losed for a.a. x ò . Then, by [37, Thm. 8.1.4℄, the multi-valued mapping U possesses a measurable sele tion u ( x ) ò U ( x ). In view of (4.59), f ( y; u ) # q1 # PRm [f Ž y℄(-; s) (ds) # f Ž y DZ  and ( y; u ) ¢ q 2 # P m [ Ž y ℄(- ; s ) (d s ) # Ž y DZ  ¢ 0 so that the pair (u ; y) is adR the in lusion (4.58) implies that (

-

-

missible for (4.50), and moreover

X ' ( x ; y ( x ) ; u ( x )) d x

¢ X q (x)dx # X

0



X

Rm

'(x ; y(x); s)

x (d s ) d x

# X [' Ž y DZ ℄(dx) # min(4:56) ¢ inf (4:50) :



Eventually, the oer ivity (4.52) with the assumed feasibility of (4.50) implies

1 X u(x)p dx ¢ X '(x ; y(x); u(x)) dx ¢ inf (4:50)   %:

p Therefore, u ò L ( ;

R



m ), whi h ompletes the proof that

u solves (4.50).

Employing the maximum prin iple, the Filippov-Roxin theory an be rened so that existen e an be obtained even for non onvex orientor elds. Assuming onstraints being qualied, the maximum prin iple for any solution to (4.50) reads as

Ë

276

4 Relaxation in Optimization Theory

:a.a. x ò : h 

*

h

;

 ; y ( x ; u ( x )) *

*

;

 ; y (x ; s) *

where the adjoint state

[

A



℄

*



*

# max h 

*

;



*

-

(

x ; y(x); s) " '(x ; y(x); s) ;

(4.61a)

solves the adjoint equation

*

for some multiplier

with

*

#  - f(x ; y(x); s) " 

 % [f y Ž y DZ ℄  # ' y Ž y DZ  % [ *

 ; y ( x ; S ( x ))

*





Ž y DZ ℄ 

*

(4.61b)

 £ 0. *

*

Corollary 4.30 (Filippov-Roxin prin iple rened).34 Let (4.51)(4.52)

together with the

(here unspe ied) assumptions ensuring the maximum prin iple (4.61) hold. Let also (4.54a) holds for some

R(x ; r) “ s ò S(x); h 

*

;

 ; y (x ; s) *

# max h 

*

;

 ; y ( x ; S ( x ))  : *

(4.62)

Then (4.50) has a solution. Let us note that a very spe ial ase, whi h an be however handled in a simpler way by a dire t method applied to the original problem, appears if

f x ; r; -) and

and (

(

'(x ; r; -)

is ane

x ; r; -) are onvex.

Example 4.31 (W.H. S hmidt [699℄, modied). Let us onsider m # n # 1, # (0; 1), A # "d /dx , '(x ; r; s) # "r % s , f(x ; r; s) # sin(s), S(x) # R, # 0. More2

2

3

2

A is now a 2nd-order ellipti operator, we should pres ribe boundary ondiy ò H01 (0; 1) instead of L q ( ; Rn ) in (4.50). Then the orientor eld Q ( x ; r ) is onvex, f. Figure 4.1(left) so Theorem 4.29 an be applied. Note that '(t ; r; -) is not ane so that we annot use simple weak- ontinuity arguments of a over, as

tions e.g. by onsidering

dire t method. Let us still modify this example by taking a (non onvex) ontrol onstraint

S(x) # ["3 ; "2℄ € [0; ℄. Then the orientor eld Q(x ; r) is no longer onvex,

f. Figure 4.1(right) so that Theorem 4.29 annot be applied. Yet, the adjoint problem

d  /dx # "3y ,  (0) # 0 #  (1), hen e always  £ 0 everywhere on [0 ; 1℄, so that the Hamiltonian h  ( x ; s ) #  ( x ) sin( s ) " s annot attain its maximum on ["3  ; "2  ℄ but only on [0 ;  ℄. Then the requirement (4.62) is guaranteed and

takes the form

2

*

2

2

*

*

*

*

*

2

therefore Corollary 4.30 an be used.

34

The essen e of involving an information from the maximum prin iple in Corollary 4.30 is to ex-

lude values of the ontrol whi h annot o

ur in optimal ontrols anyhow. See also [699℄ for this argumentation in on rete situations.

4.3 Optimal ontrol of nite-dimensional dynami al systems

(

repla ements '

'; f)
(x ; r; S(x))

Q(x ; r)

(

'; f)
(x ; r; S(x))

Ë 277

Q(x ; r)

f

f

non onvex orientor field

onvex orientor field

Fig. 4.1: An example of the graph [( ' ;

f)℄(x ; r; S(x)) and the onvex (left) or non onvex (right) orientor Q(t ; r) guaranteeing existen e of solutions through the Filippov-Roxin prin iple, possibly rened for the non onvex Q as in Corollary 4.30.

eld

4.3

Optimal ontrol of nite-dimensional dynami al systems

In this se tion we will treat an optimal ontrol problem for a system governed by an initial-value problem for an ordinary dierential equation (a so- alled dynami al system). We want espe ially to demonstrate the omplete analysis of the problem: a suitable formulation of the original problem, onstru tion of a orre t relaxation s heme, stability analysis, optimality onditions, approximation theory, and numeri al implementation.

4.3.a

Original problem

Throughout this se tion, a xed time interval

I :# [0; T℄

will be used in pla e of

.

As we want to fo us our attention rather to a method of relaxation than to optimal

ontrol problems themselves, we will restri t a bit the full generality and onsider our

optimal- ontrol problem in a so- alled Bolza form 35

ODE

(POC )

35

T . Minimize X ' ( t ; y ( t ) ; u ( t )) d t %  ( y ( T )) 6 6 6 0 6 6 6 dy 6 6 subje t to # f(t ; y(t); u(t)) : t ò I; 6 6 6 dt

a.a.

> 6 6 6 6 6 6 6 6 6 6 6 F

( ost fun tional) (state equation)

y(0) # y0 ; (initial ondition) ( t ; y ( t ) ; u ( t )) ¢ 0 :a.a. t ò I ; (state- ontrol onstraints) u(t) ò S(t) :a.a. t ò I; ( ontrol onstraints) y ò W 1 q (I; Rn ); u ò L p (I; Rm ); ;

This is a spe ial form of the Bolza problem on this xed time interval with a xed initial ondition.

In general, one an onsider

y0 as an additional ontrol variable and  # (y(0); y(T)), and possibly t # 0 and T .

also additional state onstraints at time

Ë

278

4 Relaxation in Optimization Theory

' : I , Rn , Rm Ù R, f : I , Rn , Rm Ù Rn , y0 ò Rn , S : I ± Rm a multivalued n Ù R, and mapping,  : R : I , Rn ,Rm Ù R are subje ted to ertain data  quali ation introdu ed later, n ; m £ 1, 1 ¢ p   %, 1   q ¢ %. Of ourse, R is expe ted to be ordered by a one D so that the ondition ( t ; r ; s ) ¢ 0 has a sense.

where



This problem ts with the framework of Se tion 4.1 if one takes the data for the problem (POC ) as

Y # W 1 q (I; Rn ) ; U # L p (I; Rm ) ; Uad # {u ò U; :a.a. t ò I : u(t) ò S(t)} ; ;

(4.63a) (4.63b)

X # L q (I; Rn ) , Rn ;

(4.63 )

 # L p (I; R ); D # { ò ; : t ò I : (t) ò D } ; dy (u ; y) # Œ " Nf (y; u) ; y(0) " y0  ; dt B(u ; y) # N (y; u) ;

(4.63d)



J(u ; y) #

T X ' ( t ; y ( t ) ; u ( t )) d t %  ( y ( T ))

(4.63e) (4.63f)

:

(4.63g)

0

u and then

For optimality onditions, we will onne ourselves to

independent of

onsider

has a nonempty interior in

 # C(I; R ). Let us note that then the one D  C(I; R ) provided D has a nonempty interior in R . 

Example 4.32 (Non-existen e of optimal ontrols: os illations).36A very simple and illustrative problem whi h orrupts existen e of solutions is:

T

Minimize

J(y; u) :# X (u(t)2 "1)2 % y(t)2 dt

( ost fun tional)

0

subje t to

dy # u; y(0) # 0; dt y ò W (I); u ò L (I): 1;4

As

J

(4.64)

( ontrolled system)

4

is non-negative, the inmum of (4.64) must be non-negative, too. A tually, this

inmum is zero. The minimizing sequen e of ontrols is, for example,

u " (t) #  Then

1 "1

t " ¡0 otherwise : if sin( / )

(4.65)

J(u " ; y " ) # O("2 ). Yet, there is no ontrol su h that J(u ; y) # 0 for ddt y # u and T

T

y(0) # 0. Indeed, then both P0 (u(t)2 "1)2 dt # 0 and P0 y(t)2 dt # 0, so that y # 0, and from

36

d dt y

T

# u we have also u # 0, whi h however ontradi ts P (u(t) "1) dt # T #Ö 0. 2

2

0

This lassi al ounterexample is essentially due to Bolza [129℄; f. also Ioe and Tikhomirov [399,

Se t. 9.1.1℄. A similar example using the ost fun tional Se t. 61℄.

P

T

0

1%y

(

2

1 % (u "1) ) dt is by Young [808,

)(

2

2

4.3 Optimal ontrol of nite-dimensional dynami al systems

Example 4.33 (Illustration of the non-existen e due to os illations).

repla ements

Ë 279

Let us illustrate

the phenomenon from Example 4.32 on a simple ele tri al ir uit in Figure 4.2.

R heat/light onve tion ( oe ient

a2 )

R

( oe ient

i

u

heat onve tion

a1 )

umax

T

Fig. 4.2:

B

A simple ele tri al ir uit to

ontrol temperature of a lamp lament;

T is a transistor, B is a battery, and R is a temperature-dependent

ib

resistor (a bulb).

y # y(t) of a lament in a lamp to be as lose yd # yd and simultaneously the heat energy

Our aim is to ontrol the temperature

as possible to the desired temperature

lost (i.e. undesired heat produ tion) on the transistor to be as small as possible. The

umax is supposed onstant. Let the ontrol variable37 be the olle toremitter voltage u # u ( t ). The (absolute) temperature y is governed by the nonlinear supply voltage

dierential equation des ribing the energy balan e in the lament

where

dy (u " u(t)) % a y(t) % a y(t) # max ; y(0) # y ; dt R(y(t)) 2

4

1

2

(4.66)

0

¡ 0 is the heat apa ity (per unit length)

of the heated lament,

a1

and

a2

are the oe ients of the heat transfer via onve tion and radiation (due to the Stefan-

R # R(r) is the temperature-dependent resistan e of y0 is the initial temperature of the lament. The sour e term on right2 hand side, namely ( u max " u ( t )) i ( t ) # ( u max " u ( t )) / R ( y ), is the Joule heat and i is the

olle tor urrent. The energy lost within a time interval I on the transistor is obviously T T P u ( t ) i ( t ) d t # P u ( t )( u max "u ( t ))/ R ( y ( t )) d t , hen e our problem is to minimize 0 0 Boltzmann law), respe tively, and the lament, and

J(u ; y) #

T u(t)(umax "u(t))/R(y(t)) X  0

%

y t " yd )  ( ( )

2

dt :

power lost on

deviation from the desired

the transistor

temperature of the lament

(4.67)

n # m # 1, '(t ; r; s) # r" yd )2 % s(umax "s)/R(r),  $ 0, $ 0, f(t ; r; s) # "1 ((umax " s)2 /R(r) " a1 r " a2 r4 ), and S ( t ) # [0 ; u max ℄. Su h problem, however, has no solution, in general. Let us show 4 2 it on a spe ial ase y d ( t ) # y 0 for some y 0 ¡ 0 su h that a 1 y 0 % a 2 y 0   u max / R ( y 0 ). ODE Obviously, it ts with the problem (POC ) if one takes the data

(

37

In fa t, the olle tor-emitter voltage is itself ontrolled by the base urrent

a tual ontrol variable.

ib whi h is therefore the

280

Ë

4 Relaxation in Optimization Theory

Then it is possible to show that the ontrol

u k (t) # 

uk ò L



I

( ) dened by

t ò [lT/k ; (l% a)T/k℄; l # 0; :::; k"1 ;

0

for

umax

elsewhere,

(4.68)

2 a # 1 " R(y0 )(a1 y0 % a2 y0 4 )u"max , drives the system arbitrarily near to y d # y 0 . More pre isely: for any " ¡ 0 one an nd k " ò N large enough so that for every k £ k " one gets  ( u k ) " y d C I ¢ " , where  ( u ) denotes the solution to (4.66). Therefore, the se ond term in (4.67) an be made arbitrarily lose to zero for y #  ( u k ) while the rst one is identi ally zero for u # u k from (4.68). In other words, we showed

with

( )

that the inmum of su h problem is zero. Yet, if this inmum were a hieved, then

y # (u) would have to be identi ally equal to yd . By (4.66), it means that umax " u)2 # R(y0 )(a1 y0 % a2 y0 4 ). However, any ontrol u satisfying this requirement makes the rst term in the ost fun tional, i.e. u ( u max " u )/ R ( y 0 ), positive. This is a

ne essarily (

ontradi tion, showing that the inmum of our problem annot be a hieved.

Example 4.34 (Nonexisten e of optimal ontrol: on entration).

Another

phenome-

non whi h an orrupt existen e of solutions an be demonstrated on the simple problem:

T

Minimize

J(y; u) :#X (2"2t% t2 )u(t) dt % (y(T)"1)2 0

subje t to

dy # u; y(0) # 0; dt y ò W (I); u ò L (I); u £ 0; 1;1

where

/

( ost fun tional) 7 7 7

7

( ontrolled system) ? 7 7

(4.69)

7 7 ( ontrol onstraint) G

1

T ¡ 1 is xed. If u ò L1 (I) would be an optimal ontrol, then u annot be identi-

ally 0 (whi h would not obviously be optimal), and we an always take some part of this ontrol and add the orresponding area in a neighbourhood of 1. This does not ae t

T

y(T) but makes P0 (2 " 2t % t2 )u(t)dt lower, ontradi ting the optimality of the

original ontrol.38 The optimal ontrol has a hara ter of a so- alled impulse ontrol, here meaning a Dira measure supported at

38

t # 1. The response on impulse ontrols is

a(t) :# t2 " 2t % 2 attains its minimum at the point t # 1 so that the optimal ontrol t # 1 provided T ¡ 1. Considering, for k ò bigger than 1/(T "1) and , the ontrol u k and the orresponding state y k given by

The oe ient

N

is for ed to on entrate around for ~

òR

u k (t) # 

k~

0

if

t ò (1; 1%1/k) ;

otherwise

.

0

y k (t) # > k~(t"1) F

~

t ò (0; 1) t ò (1; 1 % 1/k) if t ò (1 % 1/ k ; T ) ; if

if

# ~ mintòI a(t) % (~"1) % O(1/k ). Sin e mintòI a(t) # 1 and that the sequen e {( y k ; u k )}kòN will minimize J provided ~ # 1/2; then obviously limkÙ J(y k ; u k ) # 3/4 #

then we an see that

J(y k ; u k )

2

2



inf J . On the other hand, this value inf J

annot be a hieved, i.e. the optimal ontrol does not exist.

This is here be ause of the on entration ee t. More pre isely, the sequen e { uniformly integrable.

u k }kòN

’ L (I) is not 1

Ë 281

4.3 Optimal ontrol of nite-dimensional dynami al systems

typi ally dis ontinuous just at times when the ontrol is on entrated, whi h makes theory of su h ontrol systems very nontrivial.

Therefore, the need of relaxation appears very naturally even in a very simple situations. The reader an observe that the minimizing ontrol sequen es for Example 4.32

onverges weakly* to the Young measure

x

# Æ % Æ" 1

2

1

1

2

1

u

while the sequen e { k }

from (4.68) onverges weakly* in the sense of Young measures to the (relaxed) ontrol

x

# aÆ % (1" a)Æ umax . Both are, in fa t, unique optimal relaxed ontrols when speak0

ing in terms of Young measures. In real situations like in Example 4.33, su h 2-atomi

hattering ontrol an be realized in pra ti e by fast os illating ordinary ontrols

É 10" " 10" 8

quite easily be ause the swit hing-time s ale of the transistor (

6

uk

se ) is

omparatively mu h shorter than the time s ale of the heating/ ooling pro ess of the

É 10" " 10"

lamp lament (

2

1

se ). This prin iple is a tually often used in the ontrol

te hnique, exploiting spe ial swit hing transistors spe ially designed to treat the on/o regimes.

Remark 4.35 (Original versus relaxed ontrols).

The reader may ask a question why

one needs the relaxed problem if one must eventually realize approximately the relaxed ontrols by the original ones? This relation reminds the relation between the differen e and the dierential equations  the latter ones are an e ient analyti al tool to analyze a limit behaviour of the former ones. Here the aim of analysis of the relaxed problems is, beside purely theoreti al aspe ts, to establish some on rete properties of optimal relaxed ontrols, whi h may help to determine them or at least to get a theoreti al support for e ient numeri al methods. Moreover, the results valid for relaxed problems an usually be ree ted in appropriate results for the original problems, f. Corollaries 4.364.40.

Let us briey outline whi h sorts of results an be obtained for the original problems by analysing the relaxed problems. In parti ular, this an yield existen e of solutions to the original problem, the pointwise (Pontryagin's type) maximum prin iple for these solutions (if any), or some information about properties and behaviour of minimizing here



stands

-asymptoti ally admissible sequen es for the original problem (PODE OC );  for the strong topology of C ( I ; R ). Moreover, by analysing the point-

wise maximum prin iple for the relaxed problem, one an also get information about a limit behaviour of fast os illations of su h sequen es.39 To be more spe i , let us formulate a few outlined onsequen es pre isely; of

ourse, the reader is expe ted to read them again together with their proofs after going through Se tions 4.3.b, .

39

For example, if there is a unique optimal relaxed ontrol whi h is hattering, then every minimiz-

ing sequen e must inevitably exhibit a unique pattern of fast os illations whi h tends to live in a neighbourhoods of parti ular atoms, f. Figure 3.3 for the ase of a two-atomi ontrol



# i H (u )% i H (u 1

2

1

1

2

2 ).

Ë

282

4 Relaxation in Optimization Theory

The natural basi data quali ation on erning

f



tives r and

'r 

f

and

' and their partial deriva-

are the following

f t ; r; s) ¢ a q (t) % b(r) % sp q ; /

 (

p/ q

(4.70a)



f r (t ; r; s) ¢ a q (t) % b(r) % s



f r (t ; r1 ; s) " f r (t ; r2 ; s) ¢ (a q (t) % b(r1 ) % b(r2 ) % s



'(t ; r; s) ¢ a1 (t) % b(r) % s ;



' r (t ; r; s) ¢ a1 (t) % b(r) % s ;



' r (t ; r1 ; s) " ' r (t ; r2 ; s) ¢ (a1 (t) % b(r1 ) % b(r2 ) % s





;

(4.70b)



p/ q

r "r2 ;

) 1

p

(4.70d)

p





(4.70 )

(4.70e)



p

r "r2 

) 1

(4.70f)

a1 ò L1 (I), a q ò L q (I), and b ò C(R% ) in reasing. To guarantee the existen e of the ontrol-to-state mapping  : U Ù Y and the oer ivity of (PO ), we have to require additional spe ial quali ation, namely a linear-growth of f ( t ; - ; s ) and the oer ivity of ' and  with respe t to U ad , i.e.

with some

; a ò L q (I) ; ò R% : f(t ; r; s) ¢ (a (t) % sp q )(1 % r); ; a ò L (I) ; b ò R% : t ò I : r ò Rn : s ò S(t) : '(t ; r; s) £ a(t) % bsp and inf  ¡ " : /

1

1

1

(4.70g)

1

1

(4.70h)

The maximum prin iple will involve the Hamiltonian40 given by41

h y 0 ;

*

;



(

*

t ; s) #  (t) - f(t ; y(t); s) " 0 '(t ; y(t); s) : *

*

(4.71)

Corollary 4.36 (Maximum prin iple for (PODE OC )).42 Let p ò [1 ; %), q ò (1 ; %), the one  ODE D ’ R has a nonempty interior, (POC ) possesses an optimal solution (y; u) su h that 

(

y; i H (u))

40

ODE

solves the relaxed problem (R H P

OC

)

introdu ed later,43

independent of

s

Sometimes, the expression in (4.71) is alled pseudo-Hamiltonian or Pontryagin's Hamiltonian.

h (t ; r; s* ) # supsòS(t) (s* - f(t ; r; s) " '(t ; r; s)); f. h (t ; y(t); *(t)) # h Sy;1; (t) dened here by (4.75), provided *0 # 1. In fa t, we will derive the Hamiltonian (4.71) up to an integrable onstant (dependent on t ). This

Also, the Hamiltonian is sometimes dened rather as Clarke [223℄. Then obviously

41

*

does not inuen e the maximum prin iple (4.75), only it would ae t Remark 4.42.

42

The formulae (4.72), (4.74) and (4.75) represent a very lassi al version of the pointwise maxi-

mum prin iple ex ept the onstan y of the Hamiltonian in time along the optimal pair (

y; u), f. Re-

mark 4.42, whi h is irrelevant in our theory be ause the Hamiltonian resulting by our derivation is determined uniquely only up to integrable fun tions of time. Beside the original works by Boltyanski , Gamkrelidze, and Pontryagin [127, 616℄ generalizing Hestenes [382℄ and the monograph by Pontryagin, Boltyanski , Gamkrelidze and Mish henko [617℄, we refer also to Balakrishnan [49℄, Barbu [76℄, Berkowitz [110℄ and Medhin [113℄, Boltyanski and Poznyak [128℄, Cesari [196℄, Clarke [222℄, Colonius [236℄, Gabasov and Kirillova [340℄, Hartl, Sethi and Vi kson [377℄, Hestenes [384℄, Ioe and Tikhomirov [399, Se t.2.4℄, Kaskosz [420℄, Magaril-Il'yaev [509℄, Mordukhovi h [550℄, Neustadt [574℄, Zeidler [812℄, et .

43

This just meas that there is no relaxation gap, i.e.

ODE min(PODE OC ) # min(RH POC ).

It happens if the

problem is value Hadamard well-posed with respe t to suitable perturbations ( f. Remark 4.7)  in

Ë 283

4.3 Optimal ontrol of nite-dimensional dynami al systems

ò C(I , Rn ; Rk,n ) with r (t ; r) #  (t ; r)/r,  ò C(Rn ; Rn ), and (4.70) be valid. Then there are  £ 0 and  ò r a(I; R ) su h that ( ;  ) #Ö 0,  £ 0, the 

and



r



*

*

*

0

*

*

*

0

omplementarity ondition

Ž y℄ -  # 0 *

[

on

I

(4.72)

is valid, and the integral maximum prin iple

T X h y;  0 ;  *

0

(

*

t ; u(t)) dt # sup

state



ò Lq



(

I; R

*

u ò U ad

(

*

0

t ; u (t)) dt

(4.73)



R

h y; 0 ;  òCarp (I; m ) is dened by (4.71) with the adjoint n ) solving44 the ba kward terminal-value problem:

is valid, where the Hamiltonian *

T X h y;  0 ; 

*

*

d  % €f r (t ; y(t); u(t))  #  ' r (t ; y(t); u(t)) % dt  (T) # " (y(T)) : *



*

*

 * ; /

r ( t ; y ( t ))





0

*

(4.74)

? G



S is measurable losed-valued, and if S(t) is bounded in Rm uniformly with respe t to t or  0 ¡ 0,45 then also the pointwise maximum prin iple is valid:46

Moreover, if

*

:

a.a.

tòI :

h y 0 ;

*

;



(

*

t ; u(t)) # max h y 0 ;

sòS(t)

*

;



(

*

t ; s) :

(4.75)

Proof. The formulae (4.73), (4.74) and (4.75) are respe tively just (4.104), (4.105) and (4.106) below for

 # i H (u),

so that the assertion follows dire tly from Proposi-

ODE tion 4.50 for the relaxed problem (R POC ) with a suitable separable H

H ’ Carp (I; Rm )

whose existen e is guaranteed by the data quali ation (4.70a-f); f. Example 4.56. Moreover, the abstra t omplementarity (4.23a) yields the integral omplementarity

T

t ; y(t)) -  (dt) # 0 from whi h the lo al omplementarity (4.72) realizing that Ž y ¢ 0 and  £ 0 everywhere on I . P

0

*

(

*

follows when

Å

*

Corollary 4.37 (Maximum prin iple for minimizing sequen es).47 Let p ò [1; %), q ò (1 ; %), and (4.70a-d,f-h) be valid, while (4.70e) be strengthened to  ' r ( t ; r; s ) ¢ a (t) % b(r) % sp  with some  ¡ 1, let # 0 (i.e. there are no state onstraints) and 

/

1

let {( u k ;

y k )}kòN be a minimizing sequen e for (PODE OC ). Then T

:u ò Uad : lim inf X h y k k Ù

0

;1;

 k ( t ; u k ( t )) *

parti ular, if there are no state onstraints (i.e.

" h yk

;1;

 k ( t ; u ( t )) d t *

£ 0;

(4.76)

# 0) or if the problem has a linear/ onvex stru ture

( f. Example 4.57).

44

Of ourse, sin e

* is a measure, (4.74) is to be understood in the sense of distributions. Then, from d * n ) so that, in fa t,  *ò BV( I ; n ). dt  ò r a( I ;

(4.74) one an read that

R

45

The latter ondition applies, in parti ular, if

46

Let us note that, for a.a.

example, at

47

s # u(t).

# 0.

R

t ò I , the maximum on the right-hand side of (4.75) is a tually attained, for

Optimality onditions for minimizing sequen es have been also investigated by Medhin [532℄, Po-

lak and Wardi [615℄, Sumin [738℄, Hamel [375℄ et ., f. also Sumin [739℄ for paraboli optimal ontrol problems.

Ë

284

4 Relaxation in Optimization Theory

h y; 0 ;  is given again by (4.71) while the adjoint state n ) solves the ba kward terminal-value problem

where the Hamiltonian

W min(q; ) (I; 1;

R

*

*

d k  % €f r (t ; y k (t); u k (t))  k # ' r (t ; y k (t); u k (t)) ; dt *



*

 k (T) # 0 :



*

k ò *

(4.77)

Proof. The assertion follows from Proposition 4.50. Indeed, let us make a relaxation (R

ODE

H POC

) by a suitable separable

not hold, we get some {(

u k ; y k )} su h that

lim

k Ù

u ò Uad

T X h y k ;1;  0

H;

f. Example 4.56. Supposing that (4.76) does

and a subsequen e, denoted for simpli ity again by

*

k

(

t ; u k (t)) " h y k

u

By the oer ivity (4.70h), { k } is bounded in

;1;

 k ( t ; u ( t )) d t *

  0:

(4.78)

L p (I; Rm ), {y k } is bounded in W 1 q (I; Rn ), ;

W min(q; ) (I;

R

n ) so that we an suppose that and eventually also {  } is bounded in k * 1; q i H (u k ) Ù  weakly* in H , y k Ù y weakly in W (I; n ), and also *k Ù * weakly 1 ; min( q ;  ) n ). Sin e {( u ; y )} is minimizing, by Proposition 4.46 the limit (  ; y ) in W (I; k k *

solves (R



(

R

R

) and, passing to the limit in (4.77), we an also see that



*

solves (4.105)

0 # 1 and  # 0. Realizing that y k Ù y and  k Ù  also in the norm of I; Rn ), we an show that h y k 1  k Ù h y 1  in the norm of Carp (I; Rm ); f. Exam-

with

L

ODE

H POC

1;

*

*

*

;

;

*

;

;

*

*

ple 3.106. Therefore

lim

k Ù

T X 0

h yk

;1;

 k ( t ; u k ( t )) *

" h yk

;1;

 k ( t ; u ( t )) d t *

# lim i H (u k ) " i H (u); h y k k Ù

;1;

k *

#  " i H ( u ) ; h y

By (4.104), this limit annot be negative; realize that no other

;1;



*

 : *

satisfying (4.105) does

exist. This gives the sought ontradi tion with (4.78).

Å

Corollary 4.38 (Non on entration of minimizing sequen es). Let p ò [1; %), q ò (1 ; %℄, f satisfy (4.70a,b,g), ' satisfy (4.70d,e,h), and ò C(I ,Rn ; R ). If {(u k ; y k )}kòN -asymptoti ally admissible sequen e for (PODE OC ), then the ontrols do not p 1

on entrate energy, i.e. { u k  ; k ò N} is relatively weakly ompa t in L ( I ).

is a minimizing

Proof. This assertion is just the onsequen e of Propositions 4.46(iiiiv) and 3.79

H . It is imporH does exist; f. Example 4.56 p with the modi ation that H an also ontain a (separable) subspa e C ( I ) ‹ Ô (R ) with 0 m some omplete separable subring R ’ C (R ) to satisfy the assumptions of Proposiwhi h uses a relaxation by a su iently ri h but separable subspa e tant that, for given

' and f

satisfying (4.70a,b,d,e), su h

tion 3.79.

On spe ial o

asions, the relaxed problem may serve to establish existen e of solutions to the original problem. Let us just illustrate su h sort of results obtainable by two ompletely dierent te hniques: either by a suitable onstru tion of a 1-atomi solution from an arbitrary (or at least some) relaxed optimal ontrol as used in the proof

4.3 Optimal ontrol of nite-dimensional dynami al systems

Ë 285

of the Filippov-Roxin theorem 4.29 or by usage of Bauer's prin iple (Theorem 1.13) together with a hara terization of extreme points48:

Corollary 4.39 (Existen e of solutions to (PODE OC )). (4.63b) be nonempty with

S

Let

p ò [1; %), q ò (1; %℄, Uad from

measurable and losed-valued, (4.70a,b,d,e,g,h) be valid,

and at least one from the following sets of onditions on

:

a.a.

t ò I : r ò Rn :

', f , S and

be satised:

the orientor eld

Q(t ; r) # '(t ; r; s)% R%0 ; f(t ; r; s) ;

(

t ; r; s)% D  ò R1%n% ; s ò S(t) 

is losed onvex

'(t ; r; s) # ' 0 (t ; r) % ' 1 (t ; s) with ' 0 (t ; -) on ave; f(t ; r; s) # f 0 (t ; r) % f 1 (t ; s) with f 0 (t ; -) ane; S(t) is bounded (uniformly in t) and satises (3.30); # 0 ; i.e. no state onstraints :

or

(

(

)

(

)

(

)

(

)

(

)

)

(4.79)

/ 7 7 7

(4.80)

? 7 7 7 G

ODE

Then the original problem (POC ) has a solution. Proof. As to the rst option (4.79), it su es to verify the onditions of Theorem 4.29

and with A being the linear mapping y ÜÙ ddt y while, for the purpose of this proof, the initial ondition y (0) # y 0 may be involved as a state onstraint.

with

I

in pla e of

As to the se ond option (4.80), it just su es to realize that the relaxed prob-

H “ L1 (I; C0 (Rm )) onsists in minimization of a on ave ost fun tional over the onvex weakly* ompa t set U . By Bauer's extremal prin iple (Theoad ODE lem (R POC ) with H

rem 1.13), su h problem admits at least one extreme solution. Using the ane home-

N M : Y(I; S0 ) Ù U ad with M and S0 from (3.30), we an see that the points of U ad pre isely orresponds to the extreme points of Y(I; S0 ) whi h

omorphism extreme



are, by Proposition 3.9, a.e. just Dira distributions. Therefore, this extreme solution must be again 1-atomi . We shall onsider also a perturbed problem depending on perturbation parameters

"1 ; "2 ¡ 0:

ODE

(POC ; "

48

1 ; "2

)

. Minimize 6 6 6 6 6 6 6 6 6 subje t to > 6 6 6 6 6 6 6 6 6 F

T " " X ' 1 ( t ; y ( t ) ; u ( t )) d t %  1 ( y ( T )) 0

dy # f "2 (t ; y(t); u(t)) ; y(0) # y "2 ; dt " 2 ( t ; y ( t )) ¢  " 1 ( t ) for all t ò I ; u(t) ò S(t) (:a.a. t ò I ) ; q n y ò W (I; R ); u ò L p (I; Rm ): 0

1;

For usage of this te hnique even to more ompli ated situations we refer to Balder [56℄, Cellina

and Colombo [191℄, Cesari [195, Chap. 16℄, Mari onda [518℄, or Raymond [629633℄. For problems with linear ost fun tions see Neustadt [573℄ and Ole h [581℄, or also Gabasov and Kirillova [340, Se t. V.3℄.

286

Ë

4 Relaxation in Optimization Theory

The perturbed data are to approximate the original ones in the following sense:    

with some

f " (t ; r; s) " f(t ; r; s) ¢ (a0 (t) % b0 (r) % 0 sp )" ; "

(4.81a)

p

' (t ; r; s) " '(t ; r; s) ¢ (a0 (t) % b0 (r) % 0 s )" ; "

 (r) " (r) ¢ b0 (r)" ; "

y0 " y0  ¢ " ;





"

(

t ; r) "

(

(4.81b)

t ; r) ¢ " ;

(4.81 )

"

 (t) ¢ "

(4.81d)

a0 ò L1 (I), b0 : R% Ù R% ontinuous in reasing, and 0 ò R% .

Corollary 4.40 (Stability of minimizing sequen es for (PODE ò [1; %), OC )). Let p q ò (1; %℄, f and f " satisfy (4.70a,b,g), ' and ' " satisfy (4.70d,e,h), ; " ò C(I , Rn ; R ),  " ò C(I; R ),  " (t) ¡ 0 for all t ò I , and (4.81) be satised. % % Then there is E : R Ù R su h that lim

" 1 ; " 2 Ù0 "2 ¢E("1)

ODE inf  (PODE OC " 1 " 2 ) # inf  (POC ) : ;

(4.82)

;

’ R% , R% with " k # (" k ; " k ) ¢ E(" k ) with E : R% Ù R% guaranteeing

Moreover, let a positive nonin reasing sequen e { " k } k òN

1;

2;

0; 0) be given su h that " k " (4.82) and let { u } k òN with " # ( " ; " ) be a minimizing  -asymptoti ally admissible k ODE sequen e for (POC " " ). Then there is an in reasing fun tion  : N Ù N su h that any 1 2 "n sequen e { u } n òN with k n £  ( n ) is a minimizing  -asymptoti ally admissible sequen e kn

onverging to (

2;

1

;

1;

2

;

ODE

for (POC ). Proof. This assertion is a onsequen e of Corollary 4.6 and Proposition 4.47 if one realizes that for a ountable family of optimization problems a relaxation by a ommon separable

H does exist, f. Example 4.56 below. The separability of H ensures metrizH , as required in

ability of the relativized weak* topology on bounded subsets of

*

Å

Corollary 4.6.

Remark 4.41 (Lagrange and Mayer problems). Spe ial ases of the Bolza-type probODE lem (POC ) are when  # 0 ( alled the Lagrange problem) or ' # 0 ( alled the Mayer problem). The Bolza problem looks most general but, in fa t, both the Lagrange and the Mayer forms are of the same power at least if they are no distributed state onODE straints and initial state is ontrolled, too.49 In parti ular, any Lagrange problem (POC )

# 0 an be transformed into the Mayer problem by inventing an auxiliary state d dt y n% # f(t ; y ; :::; y n ; u) with the initial ondition y n % (0) # 0, and then onsidering the terminal ost fun tional y n % ( T ).

with

y n%1

and the additional dierential equation

1

1

1

1

The mentioned ontrol of the initial onditions would lead to a fully general Bolza problem involving the ost-fun tional term

49

1 (y(0); y(T)).

Transformations between these lasses of problems are thoroughly treated, e.g., in the lassi al

monograph by Cesari [196℄.

Ë 287

4.3 Optimal ontrol of nite-dimensional dynami al systems

Remark 4.42 (Constan y of the Hamiltonian along optimal traje tories). re

ondition

h y 0 ;

*

;



(

*

is

t ; u(t))

sometimes

the

maximum

is onstant in time for any optimal pair (

autonomous systems, i.e.

prin iple,

Still one monamely

that

u ; y). This a tually holds for

', f , and S independent of time and in the un onstrained smooth in the s -variable 0

$ 0 and  # 1. Assuming ' and f *

ase. In parti ular, i.e. and

ompleting

S onvex, (4.75) gives

 (t) - f s (t ; y(t); u(t)) # ' s (t ; y(t); u(t)) % N S (u(t)) : *





(4.83)

t

Then, by the following (formal) al ulations (with the -variable not expli itly written), we have

d d h (t ; u(t)) # - f ( y; u ) %  - f ( y; u ) " ' ( y; u ) t t dt y  dt dy du %   - f r (y; u) " ' r (y; u) %    - f s (y; u) "  ' s (y; u) dt dt ò NS u t #  - f t (y; u) " ' t (y; u) ; (4.84) *

*





*

;

*





*



*



0

*



(



( )) by (4.83)

d dt y

# f(y; u) and also the adjoint equation (4.74), together with # 0 for a.a. t ò I . From this we an see that h y  (t ; u(t)) is onstant in time if both f t # 0 and ' t # 0. For the state onstraint #Ö 0, the Hamiltonian

where we used

N S (u(t)) ddt u(t)

h y 0 ;

*

;

 ; *

(

*

;



*



t ; u(t)) is not onstant in time. In fa t, to obtain su h additional ondition,

one must augment the Hamiltonian (4.71) as

h y 0 ;

*

;

 ; *

(

*

t ; s) #  (t) - f(t ; y(t); s) "  (t) *

*

(

t ; y(t)) " 0 '(t ; y(t); s) : *

(4.85)

This does not ae t the maximum prin iples (4.73) and (4.75) themselves be ause

s but allows us to enhan e (4.84) to obtain ddt h y 0   (t ; u(t)) #  - f t (y; u) "  - t (y) " 0 ' t (y; u). In parti ular, for the autonomous system and the original Hamiltonian from (4.71), we obtain h y    ( t ; u ( t )) #  ( t ) ( t ; y ( t )) up to 0 does not depend on *



*

;

*



*

;

*

;

*



;

*

;

*

;

*

*

a fun tion onstant in time. Exe uting this al ulus dire tly for the relaxed problem below, we oud avoid smoothness in

s-variable and onvexity of S.

Remark 4.43 (Time-optimal ontrol).

Some problems uses the terminal time

ontrol variable and the ost fun tional just equal to

T

as a

T . When onsidering the terminal

onstraints as in Remark 4.41, we obtain a time-optimal ontrol problem. Now the data

f : [0; %) , Rn , Rm Ù Rm m and S : [0 ; %) ± R . When res aling time to a xed interval, say [0 ; 1℄, su h problems an be transformed into a Mayer-type problem with T n a position of a s alar is to be dened on not a-priori bounded time intervals, i.e.

ontrol parameter: Minimize subje t to

y n%1 (1) dy i Tf (tT; y(t); u(t)) ; y i (0) # y0i for i # 1; :::; n ; # i dt T for i # n %1 ; ( y (1)) ¢ 0 for all t ò [0 ; 1℄ ; u(t) ò S(tT) (:a.a. t ò [0 ; 1℄) ; y ò W 1 q (0; 1; Rn%1 ); u ò L p (0; 1; Rm ); T £ 0: ;

/ 7 7 7 7 7 7 7 7 ? 7 7 7 7 7 7 7 7 G

(4.86)

Ë

288

4 Relaxation in Optimization Theory

Remark 4.44 (Multi riteria problems). the multiplier

0

*

is a ve tor.

' and  ve tor-valued, and

One an onsider

Even for

'

and



s alar-valued, one an onsider a

ODE multi riteria modi ation of (POC ) to minimize (in a Pareto or Slater sense) both

P

T

0

'(t ; y(t); u(t)) dt

4.3.b

and

(y(T)), and then 0 ò R2 . *

Relaxation s heme, orre tness, well-posedness

ODE Using the previously developed theory, we will make the relaxation of (POC ) by means

of a suitable

C(I)-invariant

subspa e

H

of

Carp (I; Rm ). Throughout this se tion, we

will suppose (without any loss of generality)

H

to be a normed linear spa e with a

topology ner than the natural topology oming from

Carp (I; Rm ). We take Uad from

(4.63b) and, likewise we did in (4.30), we put

U ad # b lH with

Uad

B

*

;

B i H ( U ad )

’ YHp (I; Rm ) L p (I; Rm ).

denoting, of ourse, the norm bornology on

is de omposable in the sense (4.41) so that

U ad

(4.87) Let us note that

is always onvex; f. also Re-

mark 3.13. Furthermore, we will assume that the two-argument Nemytski mapping

f

N f : W 1; q (I; Rn ) , L p (I; Rm ) Ù L p (I; Rn ) admits a ontinuous extension N : p W 1; q (I; Rn ) , Y H (I; Rn ) Ù L q (I; Rn ) and extend the original initial value problem

to

f

dy/dt # N (y; )

with

y(0) # y0 .

Moreover, we will onne ourselves to the

ase when the extended Nemytski mapping is ane with respe t to the (relaxed)

ontrol, namely when it takes the form

W 1 q (I; Rn ) ’ L ;



(

f

N (y; ) # f Ž y DZ ;

see Lemma 3.101. Sin e

I; Rn ), this semi-ane extension will ertainly exist if f ò CAR H

;

p; q

(

I , Rn , Rm ; Rn ) :

(4.88)

Then the extended initial-value problem takes the form

dy # f Ž y DZ  ; y(0) # y : dt

(4.89)

0

In other words, we extend

L q (I; Rn ) , Rn



from (4.63e) to

 : YH (I; Rm ) , W 1 q (I; Rn ) Ù p

;



dened by

 ( ; y) #

œ

dy " f Ž y DZ  ; y(0) " y dt

0



:

(4.90)

Analogously, supposing

' òCAR H

;

for some

p1

p;1

(

¡ 1,

I , Rn , Rm ; R)

and

òCAR H

;

p; p1

(

I , Rn , Rm ; R )

we an extend also the Nemytski mapping

in the ost fun tional, so that the ost fun tional

J

N'

(4.91) appearing

from (4.63g) extends to

4.3 Optimal ontrol of nite-dimensional dynami al systems

Ë 289

J : YH (I; Rm ) , W 1 q (I; Rn ) Ù R dened by p

;



T X ['

J ( ; y) #

0

Ž y DZ ℄(dt) % (y(T)) ;

(4.92)

'Ž y DZ  is to be understood possibly in the sense of measures on I . And also the p  mapping B from (4.63f) is to be extended analogously, being valued in L 1 ( I ; R ). Thus

where

we ome to the following relaxed problem:

(R

ODE

H POC

T X ['

. Minimize 6 6 6 6 6 6 6 )

0

Ž y DZ ℄(dt) % (y(T))

dy # f Ž y DZ  ; y(0) # y ; dt Ž y DZ  ¢ 0 a.e. on I; p  ò U ad ’ YH (I; Rm ) ; y ò W

subje t to > 6 6 6 6 6 6 6

F

0

1;

q (I;

Rn

)

:

Lemma 4.45 (Corre tness of the extended state problem). Let p ò [1; %) and q ò p m (1 ; %℄, let H be a C ( I )-invariant separable subspa e of Car ( I ; R ), f satisfy (4.88) and additionally50 also the growth ondition (4.70g). Then: The extended state equation (4.89) possesses for any

(i)

y #  () ò W 1 q (I; Rn ). p m 1 q n The mapping  : YH ( I ; R ) Ù W ( I ; R ) thus p m %

ontinuous if restri ted on YH % ( I ; R ) with any % ò R .

p

;

tion (ii)

 ò YH (I; Rm ) a unique solu-



;



dened is (weak*,weak)-

;

 # i H (u) with u ò L p (I; Rm ), then y #  () solves the original initial-value probODE p m 1 q n lem in (POC ). In other words,  Ž i H #  where  : L ( I ; R ) Ù W ( I ; R ) denotes

(iii) If



;



the original ontrol-to-state mapping.

 ò YH (I; Rm ), there is a sequen e {u k }kòN bounded in L p (I; Rm ) su h that i H (u k ) Ù  weakly* in H . To prove the existen e of the solution to (4.89), we shall just pass to the limit with the solutions y k that orresponds to u k , Proof. By the very denition of

p

*

whi h means

dy k # f(t ; y k ; u k ) ; y  (0) # y : (4.93) dt p n By the lassi al theory, (4.70g) ensures for any u k ò L ( I ; R ) the existen e of just q n one solution y k ò W ( I ; R ); see Proposition 1.36. Besides, f ( t ; r; u k ( t )) satises the 0

1;

growth ondition

f t ; r; u k (t)) ¢ (a1 (t) % 1 u k (t)p

/

 (

a k # a1 % 1 u k p p m bounded in L ( I ; R ).

with is

50

/

q

1 % r) ¢ a k (t)(1 % r)

)(

(4.94)

q bounded in Therefore,

L q (I) independently of k be ause {u k }kòN t  y k ( t ) #  y 0 % P f (  ; y k (  ) ; u k (  )) d   ¢  y 0  % 0

Let us note that (3.192b) with Remark 3.104 turns out here to (4.70a), whi h would not guarantee the

existen e of the solution of our initial-value state problem, however. For this reason we must impose the stronger growth ondition (4.70g).

Ë

290

4 Relaxation in Optimization Theory

t

a k ()(1 % y k ()) d, whi h shows via the Gronwall inequality that {y k }kòN is L (I; Rn ). Then, from (4.93) with (4.94), we an also see that {y  } ò is 1 q n bounded even in W ( I ; R ). Hen e, taking possibly a subsequen e (denoted, for sim-

P

0

bounded in



;

pli ity, by the same index), we an suppose that

yk Ù y

weakly (or, for

q # %, weakly*) in W 1 q (I; Rn ) : ;

(4.95)

Let us now pass to the limit in (4.93). The left-hand side obviously onverges to

dy/dt thanks to (4.95). The right-hand side f(t ; y k ; u k ) an be written in the form f Ž y k DZ  k for  k # i H (u k ). Realizing that (4.95) implies y k Ù y strongly in L (I; Rn ), we q n

an use Lemma 3.101 to obtain f Ž y k DZ  k Ù f Ž y DZ  weakly in L ( I ; R ). This shows q n that y ò W ( I ; R ) from (4.95) satises d y/d t # f Ž y DZ  . q ( I ; R n ), hen e Also, y (0) # y be ause y k (0) # y and y k Ù y weakly in W n also strongly in C ( I ; R ), and in parti ular y k (0) Ù y (0). Altogether, the existen e of 

1;

0

1;

0

a solution to (4.89) has been demonstrated. Now, we are to prove the uniqueness of this solution. Supposing

y1 ; y2

are two

solutions to (4.89), we have

d( y " y dt 1

Then, supposing

!! !! y 1 ( t )

# (f Ž y " f Ž y ) DZ  : 1

" y (t)!!!! # 2

h p (t ; s) # sp .

!! t !!X [( f !! ! 0 t X a2

!!

Ž y " f Ž y ) DZ ℄() d!!!! 1

2

% b (y 2

0

Realizing that

y1

1 )

and

taking into a

ount the initial onditions for all

tòI

!

% b (y 2

y2

2 )

% h p DZ y "y d 2

1

2

are apriori bounded in

L

I; Rn ) and y1 (t) # y2 (t)



y1 (0) # y0 # y2 (0), one gets

(

as a onsequen e of the Gronwall inequality generalized by a ontinuous

extension for the ase naturally

2

h p ò H , one gets by (3.192 )51

¢ with

2)

y #  ().

h p DZ  òr a(I). The point (i) has thus been demonstrated, putting

 restri ted on YH % (I; Rm ) with % ò R% arbitrary. p m Let us take a sequen e {  k } k òN ’ Y H % ( I ; R ) su h that  k Ù  weakly* in H , and denote y k #  (  k ). In other words,

Now we will show the ontinuity of

p

;



*

;



dy k # f Ž y k DZ  k ; y k (0) # y : (4.96) dt As previously, supposing h p q ò H , we an obtain the estimate d y k /d t  ¢ (a % h p q DZ  k )(1 % y k ). Again, we an dedu e that y k L I Rn ¢ C( a % 0

/

1

51

1



/

f t ; r1 ; s)

Let us note that (3.192 ) turns out here to  (

2 sp )r1 " r2  with some a2 ò L1 (I), b2 : %. and 2 ò

R

R% Ù R%

( ;

" f(t ; r ; s) ¢ (a (t) % b (r 2

2

1

)

2

1 )

% b (r 2

2 )

%

arbitrary ontinuous in reasing ( f. Remark 3.104)

4.3 Optimal ontrol of nite-dimensional dynami al systems

Ë 291

1 h p q DZ  k L1 I ) with a suitable C : R% Ù R% , and then also dy k /dt L q I;Rn ¢ a1 %

1 h p q DZ  k L q I C( a1 % 1 h p q DZ  k L1 I ). As the sequen e {h p q DZ  k }kòN is bounded in p L q (I) if  k ranges YH % (I; Rm ), we an dedu e as previously that the sequen e {y k }kòN 1 q n is bounded in W ( I ; R ), and then we an take a weakly onvergent (possibly sub-) sequen e. Let us denote its limit by y . Now our only task is to show that y #  (  ), but /

( )

/

( )

(

/

)

/

( )

;

;



the limit passage in (4.96) is entirely the same as performed previously for the spe ial

 k # i H (u k ). By the already proved uniqueness of the solution to (4.89), even the whole sequen e { y k } k òN onverges to y . This ompletes the proof of the point (ii).

ase

The last fa t to prove, namely the point (iii), follows immediately by Lemma 4.11.

Proposition 4.46 (Corre tness of the relaxation s heme). Let H be a C(I)-invariant Carp (I; Rm ), p ò [1; %), q ò (1; %℄, (4.70g), (4.88) and (4.91) be valid, ò C(I , Rn ; R ), (PODE OC ) admits a bounded  -asymptoti ally admissible se-

separable subspa e of

quen e,52 and (i)

(R

H

' and  be oer ive in the sense (4.70h). Then: p-non on entrating.

ODE POC ) has a solution, and every solution is ODE

(ii) Every solution to (R P

H

OC

)

-asymptoti ally admis-

an be attained by a minimizing

ODE sible sequen e for (POC ).

(iii) Conversely, a limit of every minimizing

-asymptoti ally

admissible weakly* on-

verging sequen e for (POC ) (when embedded via i H ) solves (R H P

ODE

ODE

OC

).

Proof. First, let us noti e that Lemma 3.101 and the ompa tness of the embedding

W 1 q (I; Rn ) ’ L

I; Rn ) guarantee that J ( ; y) : YH (I; Rm ) , W 1 q (I; Rn ) Ù R, dened by (4.92), is the weakly* ontinuous extension of the original ost fun tional J from ;



p

(

;



(4.63g). To verify the oer ivity ondition (4.21), let us just estimate, by (4.70h),

(u) # J(u ; (u)) £

T X a(t) dt 0

% b u pLp I Rm % inf (Rm ) ; ( ;

)

(u) Ù % for u L p I;Rm Ù % and u ò Uad . p m Then the level sets of  are ontained in some Y H % (I; R ) ’ H

from whi h we get

(

)



;

p priately large. We an use the weak* ompa tness of U  YH ; % ( I ; ad

with

% ò R appro-

 and R with R(u) # B(u ; (u));  () # J ( ;  ()) and R () # B ( ;  ()). Then one

weak* ontinuity of all involved mappings, i.e. both see Lemma 4.45 and realize that

R

*

m ) together with the

















gets the points (i)(iii) by using Proposition 4.1. As to the non on entration laimed in (i), let us suppose that the optimal relaxed

 is not p-non on entrating. Then it diers from its p-non on entrating modŽ i ation  whi h does exist thanks to the separability of H , see Proposition 3.81. By Ž Lemma 4.23,  ò U ad and drives the ontrolled system to the same state y as the ontrol  be ause f , having the p/ q -growth, has a growth lesser than p sin e q ¡ 1. Yet,

ontrol

52

Re all that throughout the whole se tion

 refers to the strong topology of C(I;

R

).

Ë

292

4 Relaxation in Optimization Theory

by Lemma 4.22(ii) and (4.70h),

Ž 

a hieves lower ost than the ontrol



whi h thus

Å

annot be optimal, a ontradi tion.

Further natural question on erns stability of the relaxed problem to the perODE turbed problem (POC ; "

1 ; "2

), denoted naturally as

min(RH PODE OC " 1 " 2 ). ;

;

Proposition 4.47 (Stability of relaxed problem). Let H ’ Car (I; Rm ) be C(I)-invariant, pq p n m n " n m ( I,R ,R ; R), p ò [1; %), q ò (1; %℄, f " ; f òCARH ( I,R ,R ; R ), ' ; ' òCAR H " ; ò C ( I , R n ; R  ),  " ò C ( I ; R  ),  " ( t ) ¡ 0 for all t ò I (in parti ular, the one D ’ R  p

;

;

;

;1



f " uniformly with respe t " ¡ 0, the oer ivity ondition (4.70h) be fullled both for ' and for ' " , and (4.81)

must have a nonempty interior), (4.70g) be valid both for to

hold. Then the relaxed perturbed problem (R H P there is

E:R ÙR %

%

f

and

ODE OC ;

" 1 ; " 2 ) always possesses a solution and

su h that

lim

" 1 ; " 2 Ù0 "2 ¢E("1)

min(RH PODE"1 "2 ) # min(RH PODE ) ; OC ;

;

(4.97)

OC

Limsup Argmin(RH PODE"1 "2 ) ’ Argmin(RH PODE ) : OC ;

" 1 ; " 2 Ù0 "2 ¢E("1)

;

(4.98)

OC

ODE

Sket h of the proof. The fa t that (R H POC ; " ; " ) has a solution follows simply from Propo1 2 sition 4.46. Then the stability (4.97) and (4.98) follow readily from Proposition 4.5 modied for the ase

 " repla ed by  "1

;

"2 (u)

T

# P ' "1 (t ; y "2 ; u) dt with y " from (4.99) be0

low, so that our task is only to verify the assumption (4.7) modied for a ve tor-valued perturbation parameter

" # ("1 ; "2 ).

" ¡ 0 and y0" are bounded thanks to (4.81d), we an see by the Gronwall inequality that the olle tion { y " } " ¡0 is bounded n 1 q n in L ( I ; R ), where y " ò W ( I ; R ) denotes the unique solution to the initial-value First, as (4.70g) holds uniformly with respe t to



;

problem

dy " # f " (t ; y " ; u) ; dt

By (4.81a) and (3.192 ) (used for

f"

y " (0) # y0" :

(4.99)

and modied in the spirit of Remark 3.104), we

an further estimate the dieren e between (4.99) and the unperturbed equation as follows

!! ! !! y " ( t )" y ( t )!!!

!! t

# !!!!X f " ( ; y " ; u) " f( ; y; u) d % y " " y !

¢

0

0

t " X  f (  ; y " ; u ) " f (  ; y " ; u )d  0

0

t

!! !! !! !

% X f( ; y " ; u) " f( ; y; u)d % y " " y 0

0

0

t

¢ X (a () % b (y " ()) % u()p )"d 0

0

%

0

t X a2 () 0

from whi h we obtain

0

% b (y " ()) % b (y()) % u()p y " ()"y() d % " ; 2

2

2

lim"ÿ y " " y C I Rn # 0 by the Gronwall inequality. 0

( ;

)

Ë 293

4.3 Optimal ontrol of nite-dimensional dynami al systems

By (4.81 ) we then get



"

Ž y" "

Ž y C I R ¢ " Ž y " " Ž y " C I R % Ž y " " Ž y C I R ¢ " % o % ( y " " y L ( ;

( ;

)

( ;

y

)



I

Rn ) ) ;

( ;

is the modulus of ontinuity53 of on I , { r  ¢ % } where % is so large that ¢ % and y " C I Rn ¢ %. This veries (4.7a) for R " (u) # " Ž y " with y " #  " (u)

o%

where

)

Rn )

C(I;

( ;

)

solving (4.99). By (4.81b) and (3.192 ) used for



 "1

;

" 2 ( u ) "  ( u )

¢

' " , one an further estimate

T ! " ! X !!! ' 1 ( t ; y " 2 ; u ) " ' ( t ; y " 2 ; u )!!! d t 0 T %X !!!!'(t ; y "2 ; u) " '(t ; y; u)!!!! dt 0

¢ a % b (y "2 ) % up L1 I " % a % b (y "2 ) % b (y) % up L1 I y "2 " y C I Rn : 0

0

2

Thus (4.7b) modied for

0

( )

2

2

1

2

( ;

( )

)

" # ("1 ; "2 ) has been veried.

Å

As for (4.7 ), it follows simply from (4.70h) ombined with (4.81b).

Remark 4.48 (Delayed ontrols).54 The ontrol onstraint of the type (4.63b) need not be always satisfa tory. E.g., one an onsider a problem with one additional delayed

ontrol (

t0 ¡ 0 is a xed time delay):

Minimize

T X '(t ; y(t); u(t); u(t 0

subje t to

"t

0 ))

dt

dy # f(t ; y(t); u(t); u(t " t )) for t ò I; y(0) # y ; dt u(t) ò S (:a.a. t ò ("t ; T)); y ò W q (I; Rn ); u ò L p ("t I ; Rm0 ): 0

0

0

0

1;

/ 7 7 7 7 7 7 7 ? 7 7 7 7 7 7 7

(4.100)

G

m # 2m0 , S(t) # S0 , S0 , and u $ (u1 ; u2 ) ò L p (I; Rm0 )2 , namely

ODE It an obviously be transformed into the form (POC ) with

one additional onstraint on the new ontrol

u2 (t) # u1 (t " t0 )

:a.a. t ò (t ; T) : 0

ODE The relaxed problem takes again the form (R POC ) but the expli it form of

H

(4.87) is now quite deli ate matter. For the spe ial ase

S0 ’

U ad

from

Rm0 ompa t and H #

ò C(I , Rn ; R ), is uniformly ontinuous on ea h ompa t I , {r ¢ %}. In parti ular, there is o % : R Ù R% su h that lim"ÿ o % (") # 0 and  (t ; r ) " (t ; r ) ¢ o % (max(t " t ; r " r )) whenever max(r ; r ) ¢ %. 53

2

54

Let us note that, sin e

%

1

2

0

1

1

1

2

2

1

2

Delayed ontrols has been treated by Rosenblueth [651653℄ and Vinter [654℄, and by Warga and

Zhu [794, 815℄.

294

Ë

4 Relaxation in Optimization Theory

L1 (I; C(S0 , S0 )), the onvex set of admissible relaxed ontrols U ad ’ Y(I; S0 , S0 ) was expli itly des ribed in Rosenblueth's works [651653℄ as

U ad # – ò Y(I; S0 , S0 ); :h ò L1 (t I ; C(S0 )) : T X X h(t ; s2 ) t (ds1 ds2 ) dt t0 S0 ,S0

T

#X

X h(t ; s1 ) t"t0 (ds1 ds2 ) dt — : t0 S0 ,S0

Remark 4.49 (Relaxation via a onvex ompa ti ation by J.E. Rubio).

A

ompletely

different approa h is an attempt to ompa tify the pairs of ontrol-state whi h

dy dt

# f(t ; y(t); u(t)) on I with y(0) # y0 . Let us H ’ Car(I , (Rn ,S0 )) with S0 ’ Rm ompa t and the embedding i H : ( y; u ) ÜÙ H . For ò C 1 (I , Rn ), when putting

satises the state equation, i.e. here present it briey by onsidering *

f

(

(

t ; r; s) #

)

%



t ; y(t)) dt #

(

r ( t ; r ) f ( t ; r; s )



t (t ; r) ;

we note that

T X 0

f

T

(

t ; y(t); u(t)) dt # X

0

d dt

(

T; y(T)) "

0; y ) :

(

0

$ 0) and with  $ 0 and

ODE Considering (POC ) without the state onstraints, (i.e.

S(t) # S0 , based on [778℄ one an think about its metamorphosis into Minimize subje t to

i H ( y; u ) ; '

# (T; y(T)) " (0; y0 ) : ò C u(t) ò S0 (:a.a. t ò I ) ; 1 q n m ( y; u ) ò W (I; R ) , L (I; R ): f

i H ( y; u ) ;

;

(1)

(

I ,R

/ 7 7 7 n) ; 7



? 7 7 7 7

(4.101)

G

and then the relaxed problem takes the form: Minimize

 ; '

subje t to

 ;

T; y(T)) "  òr a% (I , Rn , S0 ) : f

#

(

0; y

(

0)

: òC

(1)

(

I , Rn ) ;

/ 7 ? 7 G

(4.102)

In a series of works started by [707, 708℄ and summarized in the monograph [709℄, it is shown that the measures admissible in (4.102) are weakly* attainable by sequen es of the pairs of the original state- ontrols.55 The set of these measures is obviously

r a(I , Rn , S ) and thus it forms a onvex - ompa ti ation of q ( I ; R n ) , L ( I ; R m ); d y # f ( y; u ) ; y (0) # y }. Optimality onthe set {( y; u ) ò W dt

onvex subset of

0

1;



0

ditions based on the geometry of this onvex ompa ti ation was then formulated in [777, Thm 2.2℄. An approximation by dis retising the set of measures

55

r a% (I , Rn , S

y(t) # y T xed, and then it is proved that any  for (4.102) an be attained by an admissible sequen e for (PODE OC ) but with this terminal-

A tually, [709℄ onsiders the terminal-state onstraint

admissible

0)

state onstraint satised only asymptoti ally in the spirit used already in Proposition 4.1.

4.3 Optimal ontrol of nite-dimensional dynami al systems

Ë 295

(i.e. an inner approximation) or by taking only a nite number of onstraints by taking nite number of test fun tions

's (i.e. an outer approximation) would lead to

a semi-innite mathemati al programming (SIP) be ause either the number of linear

onstraints or the number of variables still remain innite; here it is semi-innite linear programming. The outer approximation may under-relax the problem but in the limit when number of onstraints in reases onverges to the orre t relaxation. Combining both approximation then leads to a linear mathemati al programme (LP) that

an be e iently implemented on omputers even for a relatively very large number of variables and onstraints, as the approximate problems presumably have. The Rubio's onstru tion has later been re-invented under the name linear-matrix-inequality (LMI) relaxations and the measures o

urring in (4.102) alled o

upation measures, f. e.g. [29, 219, 220, 478, 479, 511℄, together with a ombination of the method of moments for numeri al approximation when the nonlinearities are polynomial as in Se t. 3.3.d.

4.3.

Optimality onditions

The further aim of ours is to exploit the results from Se tions 4.1 and 4.2 to ompose ODE the optimality onditions for (R POC ). Of ourse, we must strengthen (4.88) and (4.91)

H

to

f òCAR H di (I , Rn , Rm ; Rn ) ;

p; q

and

;

' òCAR H di (I , Rn , Rm ; R): ;

p;1

;

(4.103)

Proposition 4.50 (Maximum prin iple). Let H be a C(I)-invariant subspa e of Carp (I; Rm ), p ò [1; %), q ò (1; %), the one D ’ R has a nonempty interior, n k , n ) with r ò C(I , R ; R r ( t ; r ) #  ( t ; r )/ r , (4.70g), (4.70h) and (4.103) be valid, ODE and let (  ; y ) ò Argmin(R H P ). Then there are  £ 0 and  ò r a(I; R ) su h that (  ;  ) #Ö 0,  £ 0, the omplementarity ondition (4.72) is valid, and the integral 





*

*

*

*

0

OC

*

*

0

maximum prin iple

T X [ h y;  0 ;  *

0

*

DZ ℄(dt) # sup

*

u ò U ad

is valid, where the Hamiltonian

h y 0 ;

*

;



*

T X h y;  0 ;  0

(

*

t ; u(t)) dt

(4.104)

ò H is given by (4.71)56 with  ò L q (I; Rn ) solving 

*

the ba kward terminal-value problem:57

d % f r Ž y DZ  #  ' r Ž y DZ  % dt *



*

*



0

56

In fa t, we derive uniquely

# 0.

h y;0 ; *

*



only up to



r

Ž y  ;  (T) #   r (y(T)): *

*

*



0

‹ 1òH

for

(4.105)

ò L1 (I) arbitrary. We hoose simply

* Of ourse, (4.105) is to be understood in the sense of distributions. In fa t,  always belongs to 1/ " ; p; 1 L (I; n ). Moreover, if # 0 and if ' ò CARH;di (I , n , m ; ) for some " ¡ 0, then even * ò W 1;min(q;1/(1"")) (I; n ) be ause 'r Ž y DZ  ò L1/(1"") (I; n ) thanks to the growth ondition (3.197b). In

57

R

R

R

R R R

this latter ase, the solution to (4.105) an be understood in the usual Carathéodory sense.

Ë

296

4 Relaxation in Optimization Theory

S is measurable losed-valued, and if S(t) is bounded in Rm uniformly with respe t to t or  0 ¡ 0, then h y   DZ  is absolutely ontinuous and the following point0 1 wise maximum prin iple is valid in the sense of L ( I ): Moreover, if

*

;

h y 0

*

;



;

*

*

*

;

DZ 
(t) # sup h y 0 ;

sòS(t)

*

;



(

*

:

t ; s))

a.a.

tòI :

(4.106)

Proof. We will use Theorem 4.15 to the problem (RPOC ) with the data from (4.63) and with

F

#

FH

from (4.29). Su h problem transformed via



*

: FH Ù H *

*

is equiv-

ODE alent to (R POC ). The smoothness assumptions of Theorem 4.15 are guaranteed via H

 was D has a nonempty interior in C(I; R ),

Lemma 3.103 while the ontinuity58 of the extended ontrol-to-state mapping proved in Lemma 4.45. Also note that the one



int( D ) #Ö .

as required in Theorem 4.15 be ause



For larity, we divide the derivation of the above laimed optimality onditions into separate steps.

 (-; y), B (-; y), and J (-; y).) In view of Lemma 3.103 the parq n  tial dierentials ›   (  ; y ) ò L( H ; L ( I ; R ) , R), ›  B (  ; y ) ò L( H ; C ( I ; R )), and ›  J (  ; y ) ò L( H ; R) are given respe tively by the formulae: Step 1. (Dierentials of





*

*

*

with

òH

*



[› 

 ( ; y)℄(  ) # "f Ž y DZ  ; 0 ;

[› 

B ( ; y)℄(  ) # 0 ;

[› 

J ( ; y)℄(  ) #











T X ['





0

Ž y DZ  ℄(dt) #  ; ' Ž y 



. Moreover, all the dierentials are (weak*,weak)- ontinuous59 as required

by Theorem 4.15.

Step 2. (Dierentials

of

the partial dierentials

L( W

1;

q (I;

R

n ); C(I;

R

 )),

 ( ; -), B ( ; -), and J ( ; -).) In view of Lemma 3.103 1 q n q n n › y  (  ; y ) ò L( W ( I ; R ) ; L ( I ; R ) , R ), › y B (  ; y ) ò 1 q n and › y J (  ; y ) ò L( W ( I ; R ) ; R) are given respe tively by



;



;



the formulae:

dy " (f r Ž y DZ ) - y ; y (0) ; dt [› y B (  ; y )℄( y ) # „ r Ž y … - y ;

[› y

 ( ; y)℄( y ) #



œ















J  ; y)℄( y ) #

[› y (

58





T  X [' r 0

Ž y DZ ℄(t) - y (t) dt %  (y(T)) - y (T) 





R

Y # W 1; q ( I ; n )  is (strong,strong)- ontinuous, as re-

In fa t, the linear and the nonlinear parts should be treated separately: we endow

by the norm of

L (I;

Rn

) so that the ontrol-to-state mapping

quired by Theorem 4.15 be ause of Lemma 1.59. Let us note that the respe tive dierential of the nonlinear part remains ontinuous with respe t to this weaker norm, while the dierential of the linear part, being onstant, an be treated in the original strong topology of

59

As for [›

 ( ; y)℄, this requires g

- (

f

W 1; q ( I ;

Rn

).

Ž y) ò H for any g ò L q (I; Rn ) Ê L q (I; Rn ) 

*

, whi h is just

J  ; y)℄, it follows simply from ' Ž y ò H .

ensured by (3.196) whi h is ee tive due to (3.46). As to [› (

4.3 Optimal ontrol of nite-dimensional dynami al systems

y ò W 1 q (I; Rn ). ;

with



 ( ; y)

Let us note that › y



Ë 297

a tually possesses a bounded in-

verse,60 as required in Theorem 4.15.

Step 3. (The adjoint problem.) The abstra t adjoint equation (4.23b) bears the form

´

; [›y  ( ; y)℄( y )µ # 0 ´›y J ( ; y); y µ % *

*









´

; [›y B ( ; y)℄( y )µ

*

(4.107)





y ò W 1 q (I; Rn ) and some 0 £ 0,  $ (1 ; 2 ) ò L q (I; Rn ) , Rn , and  £ 0 su h that (  0 ;  ) #Ö 0, <  ; B ( y;  )> # 0; note that the last identity just results to the *

;

for all

*

*



*

*

*



*

*

*

omplementarity ondition (4.72). Using the formulae from Step 2, the identity (4.107) takes the form

T * X 1 0

dy "  - f r Ž y DZ 
- y dt %  - y (0) dt T  ' r Ž y DZ 
(t) - y (t) dt %   (y(T)) - y (T) %X ( 

*

1

T X

#

*



-

*



*

0

0

2







0



Ž y) - y -  (dt) :



*

r



0

(4.108)



Using the by-part integration, we an easily see that (4.108) will be valid for every

y ò W 1 q (I; Rn ) provided  # (1 ; 2 ) ò L q (I; Rn ) , Rn ;

*

*



*



butions):

satises (in the sense of distri-

d  %  - €f r Ž y DZ  # " ' r Ž y DZ  "  dt  (T) # "  (y(T)) and  (0) #  : *

*

1

1



*



*

0

*

*

1

0



*

*

1

2

- (



r



instead of

"

(4.109a) (4.109b)

From the last equality, we an eliminate the formal multiplier *

Ž y) ; 2 , and *

write simply

*

. Thus we ome just to (4.105). Let us note that, thanks to (3.197b), 1

the right-hand side of the linear ordinary dierential equation in question, namely



0 ' r Ž y DZ  % r Ž y)
 , belongs to r a(I; Rn ) whi h is ontained61 in W 1 q (I; Rn ) . As › y  has a bounded inverse as shown in Step 2, our terminal-value problem possesses q n always a (unique) solution  ò L ( I ; R ), as required. *





*

;

*





*

Step 4. (The Hamiltonian.) The abstra t Hamiltonian (4.23d) an be now written in the form

f y 0 ;

*

;



*

# h y 0  ;

*

;

*

with

h y 0

*

;

;



*

ò H determined with help of the formulae from

Step 1 by the identity

 ; h y;  0 ;  *



*

#  ; [›  ( ; y)℄(  ) "  ; [› B ( ; y)℄(  ) "  *

*





#  ; " f Ž y DZ  "  ;  ' Ž y #  ; "  *

*

1



whi h is to hold for any

h y 0 ;

*

;

òH 

*



0



*

1

- (

›  J (  ; y ) ;

0

 

f Ž y) " 0 ' Ž y ; *

. This gives the expression (4.71) for the Hamiltonian

 if we write shortly, as in Step 3, *

p; q

*







*

in pla e of

R R R

"

R

*

1

. As

f y 0 ;

*

;

 is determined *

f ò CARH;di (I , n , m ; n ) ensures f r Ž y DZ  ò L q (I; n,n ), and therefore the initial-value problem d y/dt " (f r Ž y DZ ) - y # f and y(0) # y0 denes the bounded linear operator (f; y0 ) ÜÙ y : L q (I; n ) , n Ù W 1;q (I; n ) being just [›y  ( ; y)℄"1 . 1; q n ) ’ C ( I ; n ) is of the type (D) and therefore the 61 Here we use the fa t that the embedding W (I; 60

Indeed,

R

R

;

R

R

R

adjoint operator realizes the ontinuous embedding; f. Se tion 1.3 .

Ë

298

4 Relaxation in Optimization Theory

ò H essentially does not hange by T adding arbitrary integrand of the form ‹1 ò H be ause ( ‹1) # onst.# P ( t ) d t . h y 0

uniquely up to onstants, our Hamiltonian

;

*

;



*

0

Step 5. (Lo alization of the maximum prin iple.) Eventually, the maximum prin iple (4.23 ) an be transformed into the form (4.106) by means of Theorem 4.21(i). Let us verify the des ent ondition (4.36). It is satised trivially when all

S(t) are

t

bounded independently of . Also, (4.70g) and (4.70h) allow us to estimate

h y 0

*

;

;



(

*

t ; s) ¢ "0 a(t) " 0 bsp %  (t) a1 (t) % 1 sp *

*

*

/

q

% y C I Rn  ;

1

( ;

)

0 ¡ 0 be ause then one an estimate  (t) 1 sp q (1 % 1 p y L I ;Rn ) ¢ 2  0 b  s  % C with C large enough depending on  0 , 1 , on y L I ;Rn , and on  L I ;Rn ; re all that q ¡ 1. Having (4.36) at our disposal, we an readily use *

whi h gives (4.36) provided 

*

*

(

*



)

*



(

/

(

)

Å

Theorem 4.21(i) to get (4.106).

Remark 4.51 (Setting the state equation alternatively). dene  : U , Y Ù X as Y # L q (I; Rn ); X # W 1 q (I; Rn ) ; ;

(u ; y);

T

y # X y 

0

)

*

and, for all

Instead of (4.63a, ,e), one an

y ò W 1 q (I; Rn ) : ;



dy % f(y; u) - y dt " y(T) - y (T) % y dt

y (0) :







This integral identity indeed overs both the state equation

0

-



d dt y

(4.110)

# f(y; u) on I

and

y(0) # y0 and, under the same data quali ation as before, the y ò L q (I; Rn ) to the state equation (u ; y) # 0 does exists, even belongs to W 1 q (I; Rn ) as before, and is unique for a given u.62 Then, instead of the al ulations

the initial ondition solution ;

(4.108)(4.109), the abstra t adjoint equation (4.107) results to

d %  - f r Ž y DZ 
- y dt "  (T) - y (T) dt T # X  ' r Ž y DZ 
(t) - y (t) dt %   (y(T)) - y (T) %

T X

*

y-

*



*







0

*



*

0

0



0





T X ( 0



r

Ž y) - y -  (dt) : *



(4.111)

y ò W 1 q (I; Rn ). From this, we an read that  ò X # W 1 q (I; Rn ) satises the n terminal-value problem (4.105) in the sense of R -valued measures on I . for any

;

*

*

;



Corollary 4.52 (Chattering ontrols I). be fullled,

H

be separable,

Let all the assumptions made in Proposition 4.50

S be measurable and losed-valued, and, for a.a. t ò I , any

# f(u ; y) a.e. on I an be seen by taking an arbitrary y ò C (I; Rn ) with y(0) # 0 # y(T) and making the by-part integration in time. This reveals also that y ò W q (I; Rn ) and, after by-part integration in time, (4.110) results to ( y (0)" y ) -  y(0) # 0. Taking now y arbitrarily gives also the initial ondition y (0) # y . Having two solutions to  ( u ; y ) # 0 and  ( u ; y ) # 0, we an use that 62

The equation

d dt y

1

1;

0

they satises also as before.

d dt y 1

0

# f(u ; y

1 ) and

d dt y 2

# f(u ; y

1

2

2 ) and show uniqueness by Gronwall's inequality

4.3 Optimal ontrol of nite-dimensional dynami al systems

Ë 299

0 ¡ 0 and any r; r ò Rn , the fun tion "0 '(t ; r; -) % r - f(t ; r; -) attains its maximum on S(t) at no more than k points. Then every optimal ontrol for (RPODE ), whi h admits the

orresponding multiplier  0 positive, is k -atomi . *

*

*

*

OC

*

 must satisfy the maximum prin ih # h y 0  whi h satisfy

Proof. By Proposition 4.50, every optimal ontrol

ple (4.106). Then it su es to apply Proposition 4.27(i) with

;

*

;

*

0 ¡ 0, f. Step 5 of the proof of Proposition 4.50.

Å

*

the des ent ondition (4.36) if

Corollary 4.53 (Chattering ontrols II).63

Let all the assumptions made in Proposi-

tion 4.50 be fullled, H be separable,

S be measurable and losed-valued, and there is at least one optimal solution for whi h  0 ¡ 0. Then there exists at least one ( n %1)-atomi *

hattering optimal ontrol.

0 , whi h does exist due to Proposi0 and

Proof. Let us take some optimal relaxed ontrol tion 4.46, for whi h

0 ¡ 0. Furthermore, let  *

*

be the adjoint state related with

h y 0  be the Hamiltonian from (4.71). Then it su es to apply Proposition 4.28 with h # h y 0  ò H and h l # [f Ž y℄l for l # 1; :::; n. Note that h satises the des ent ondi;

*

;

*

;

*

;

*

tion (4.36), f. Step 5 of the pre eding proof. Let us verify the ondition (4.49), onsider-

 ò U ad su h that h DZ  # h S with h S (t) :# supsòS t h(t ; s) and h l DZ  # h l DZ  0 . First, let us noti e that the last ondition ensures that  (  ) #  (  0 ); in other words, both  and  0 drive the ontrolled system to the same state y . As we supposed  0 ¡ 0 we an write ing some other relaxed ontrol

( )





*

' Ž yDZ #

# whi h implies

1 0

*

1

0

*

„

*

„

*

- (

f Ž y DZ ) " h y 0

- (

f Ž y DZ 0 ) " h y 0

;

*

;

;



*

*

;



J ( ; y) # J (0 ; y) # min(RH PODE OC ).



DZ  % … *

DZ 0 % … # (' Ž y) DZ 0 ;

Therefore



is optimal for (R

ODE H POC ),

verifying thus the hypothesis (4.49). Then our assertion follows from Proposition 4.28 with

k # n.

Å

We would like to point out that the estimates annot be improved in the sense that one an onstru t examples that do not admit any hattering relaxed ontrol with less atoms than stated in Corollaries 4.52-4.53; f. also the example in Subse tion 4.3.e. In some ases where the ontrolled system is only slightly nonlinear in terms of the states, the relaxed problem an be proved onvex. Then the rst-order optimality

63

Su h kind of results (but a more pessimisti (

n%2)-atomi estimate)

was outlined also by Cesari

'(t ; r; s) $ '(t ; r)), then the existen e of an (n%1)-atomi ontrol was established too; f. [196, Se t. 1.14A℄. Supm ompa t-valued, in [196℄ su h results were derived dire tly from Proposition 4.50 posing S : I ± by means of the te hnique of the proof of Proposition 4.28. The ( n %2)-atomi ontrols have been also [196, Se t. 1.14B℄. If the ost fun tional would not depend expli itly on the ontrol (i.e.

R

used by Berkowitz [110, Chap. IV℄ and Carlson [170℄.

300

Ë

4 Relaxation in Optimization Theory

ondition (i.e. the maximum prin iple) is not only ne essary but also su ient for optimality. Let us illustrate it on the additively oupled ase, whi h allows to opy the arguments behind the abstra t Proposition 1.62 in the on rete situation.

Proposition 4.54 (Su ien y of the maximum prin iple).64 For the additive ansatz '(t ; r; s) # g(t ; r) % h(t ; s); f(t ; r; s) # G(t ; r) % H(t ; s); let us assume that

# 0;

and

(4.112)

g : I , Rn Ù R, G : I , Rn Ù Rn , h : I , Rm Ù R, H : I , Rm Ù Rn

are Carathéodory fun tions satisfying the growth onditions

;a ò L q (I) ;b ò R : ;a ò L (I) :



1

with some



G(t ; r) ¢ a(t) % br;

g(t ; r) ¢ a(t);





H(t ; s) ¢ a(t);

(4.113a)

h(t ; s) ¢ a(t)

(4.113b)

q ò (1; %), and a smoothness onditions

;a ò L (I) ;b : R Ù R ontinuous : g (t ; r) ¢ a(t) % b(r);  g ( t ; r ) " g ( t ; r ) ¢ ( a ( t ) % b ( r ) % b ( r )) r " r  ; % " ;a ò L (I) ;b : R Ù R ontinuous : G (t ; r) ¢ a(t) % b(r);  G ( t ; r ) " G ( t ; r ) ¢ ( a ( t ) % b ( r ) % b ( r )) r " r  : 1







1

2

1

1

2

1

2







1

2

1

2

1

2

(4.113 ) (4.113d) (4.113e)

S : I ± Rm is supposed bounded, measurable, and in the m m m form S ( t ) # M ( t ; S 0 ) for some S 0 ’ R ompa t and M : I , R Ù R a Carathéodory " 1 mapping su h that both M ( t ; -) and M ( t ; -) are Lips hitz ontinuous uniformly with respe t to t ò I . Let us assume G ( t ; -) twi e ontinuously dierentiable, and let g ( t ; -) be The multivalued mapping

uniformly onvex in the sense

:r; r ò Rn : max(r;  r ) ¢ R âá 



g(t ; r ) " g(t ; r) " g (t ; r)( r " r) £ (t) r "r2 



with

(i)

(4.114a)



R[y(u)℄(t) and with the modulus  £ 0 satisfying

(t) £ with



b(t)

2

eB t sup G (t ; r) ( )



r ¢R

with

 

b(t) :#

T X a (  ) d  and t

B(t) :#

T X A() d t

(4.114b)

a(t) :# sup r ¢R g (t ; r) and A(t) :# sup r ¢R G (t ; r).65 Then:  is onvex on U ad , and 



 

 

(ii) the maximum prin iple (4.105)(4.105) with some

0 ¡ 0 and  £ 0 *

 ient ondition for the relaxed ontrol to be optimal.

64

For a generalization for unbounded ontrols see [675℄.

65

Note that (4.113 .d) ensures

a ; A ò L1 (I).

*

*

is also a suf-

4.3 Optimal ontrol of nite-dimensional dynami al systems

Ë 301

Proof. The maximum prin iple involves the adjoint equation

d # " (t)G (t ; y(t)) " g (t ; y(t)) ;  (T) # 0 : dt *

*





*

(4.115)

The assumption (2.2) ensure that the terminal-value problem (2.3) possesses pre isely

 ò W 1 1 (I; Rn ). ; ò U ad and y; y ò W 1 q (I; Rn ) solve the initial-value problem in (RP) with

one solution



;

;

Let and

*





, respe tively. Then using the by-part integration and the adjoint equation (4.115),

we an al ulate:

( 

)

" ( ) " [›(

)℄(

" (

T X X

( 

)

T

#X

"

)

0



"

Rm

)



*

t H(t ; s) % h(t ; s)[

( )

y (t)) " g(t ; y(t)) " X

”g(t ;

 (t)H(t ; s)[

t

"

t ℄(d s ) d t

t

"

t ℄(d s )• d t

*

Rm



0





d( y (t) " y(t))  dt dt T d # X g(t ; y (t)) " g(t ; y(t)) %  (t)(G(t ; y (t)) " G(t ; y(t))) % ( y (t) " y(t))dt dt T

# X g(t ; y (t)) " g(t ; y(t)) %  (t) œ G(t ; y (t)) " G(t ; y(t)) " *







0

*

*







0

T

# X  g(t ; y (t)) " g(t ; y(t)) " g (t ; y(t))( y (t) " y(t)) 





#:
g ( t )

0

%  (t)  ( G ( t ; y ( t )) " G ( t ; y ( t )) " G ( t ; y ( t ))( y ( t ) " y ( t ))) dt *





(4.116)



#:
G ( t )


g (t)

Estimating the se ond-order orre ting terms


G (t),

and

the in remental

formula (4.116) enables us to investigate onvexity of the extended ost fun tional From the adjoint equation (4.115) we an estimate

d * dt  

.

¢ A(t) (t) % a(t) so that by *

the Gronwall inequality one gets

 (t) ¢ *



T " P T A()d d• ePtT A()d ”X a (  )e t t

:  (t) ¢ " G(t ; y(t)) "

To simplify the notation, we an also (a bit more pessimisti ally) estimate 

*

b(t)eB t : By the Taylor expansion, we an estimate G(t ; y (t)) G (t ; y(t))( y (t)" y(t)) ¢ sup r ¢R 12 G (t ; r)  y (t)" y(t)2 . Then (4.114) ensures ( )







 





1   ( t )  G ( t ; y ( t ))  y ( t ) " y ( t ) 2 1 sup G (t ; r) y (t) " y(t) £ 0 : r ¢R 2


g (t) %  (t)
G (t) £ (t) y (t) " y(t)2 " *



£ Œ(t) "

b(t)

2

eB t

( )

*

2





2





 

so that the se ond right-hand term in (4.116) is non-negative. From (4.116) we obtain

: ; ò U ad : 

whi h just says that



ondition then follows.

(

is onvex on



)

" (

U ad .

)

" [›(

)℄( 

"

)

£ 0;

The su ien y of the 1st-order optimality

302

Ë

4 Relaxation in Optimization Theory

Example 4.55 (Conventional relaxed ontrols).66Let us apply our theory to the problem ODE m (POC ) with S ( t ) losed and bounded uniformly in time, i.e. S ( t ) ’ S for a ball S in R . 0

Then the growth we keep

pò

0

p of the ontrols is irrelevant; nevertheless, for notational simpli ity,

R% formally in our problem. The most general s heme will be reated 0

by taking the nest possible relaxation from the onsidered lass, reated obviously by

H # Carp (I; Rm ). We may and will endow this H

by the universal (semi)norm ( f.

(3.141)):

h H #

Then the restri tion operator

inf



:(t ; s)òI ,Rm: p  h ( t ; s )¢ a ( t )% b  s 

a L1 I % b :

(4.117)

( )

h ÜÙ hI ,S0 : Carp (I; Rm ) Ù L1 (I; C(S0 ))

is linear and

ontinuous.67 Besides, this restri tion mapping is also surje tive so that the adjoint

L1 (I; C(S0 )) Ê Lw (I; r a(S0 )) Ù H # Carp (I; Rm ) is ontinuous and inje tive, and embeds the set of Young measures Y( I ; S 0 ) ’ L w ( I ; r a( S 0 )) ( f. also Examp m m ple 3.44) into Y ( I ; R ). Thus U H ad is anely homeomorphi with { ò Y(I; R ); :a.a. t ò I : supp( t ) ’ S(t)} provided S satises some additional quali ation, e.g. (3.30). *

operator

 *

*

*



*

ODE Then (R POC ) an be equivalently written in terms of the lassi al relaxed ontrol

H

(= Young measures) as follows

T X X

Minimize

0

Rm

'(t ; y(t); s)

t (d s ) d t %  ( y ( T ))

/ 7 7 7 7 7 7 7 7 7

dy # X f(t ; y(t); s) t (ds) (:a.a. t ò I); y(0) # y ; dt Rm ? 7 7 ( t ; y ( t )) ¢ 0 (: t ò I ) ; 7 7 7 7 7 supp( t ) ’ S ( t ) (:a.a. t ò I ) ; 7 7 m q n y ò W (I; R ); ò Y(I; R ): G

subje t to

0

(4.118)

1;

Therefore, under the respe tive data quali ation, we are authorized to apply Propositions 4.46 and 4.50 to this on rete problem; note that the oer ivity ondition (4.70h) is fullled automati ally be ause

S(t)

are here bounded uniformly for

t ò I.

In par-

ti ular, by Proposition 4.46 we have guaranteed the existen e of an optimal relaxed

ontrol

and, by Proposition 4.50, we have at our disposal the pointwise maximum

prin iple (4.106), whi h an be now written in the form

h y; 0 ;  X S(t) *

66

(

*

t ; s)

t (d s )

# max h y 0  (t ; s) sòS(t)

;

*

;

(4.119)

*

Su h kind of relaxation was frequently used in the literature; let us mention for example Balder

[5052℄, Barron and Jensen [79℄, Berkowitz [111℄, Carlson [170℄, Gamkrelidze [345℄, Ghouila-Houri [352℄, Goh and Teo [358, 758℄, M Shane [528℄, Medhin [531℄, Papageorgiou and Papalini [592℄, Rishel [644℄, S hwarzkopf [724℄, Warga [786791℄, Williamson and Polak [798℄, et .

67

there are

a " ò L (I) and b " ò 1

"

"

" " 1 " " h I ,S0 " "L (I;C(S0 )) ¢ max(1 ; T maxsòS0  s  ) h H be ause for any " ¡ 0 % su h that a 1 % b # h % " and  h ( t ; s ) ¢ a ( t ) % b  s p , and then

Indeed, we have the estimate

R

p

" L (I) " T P a " (t) 0

# PT maxsòS0 h(t ; s) dt ¢ ¢ max(1; T maxsòS0 sp )( h H % "). " " " " 1 " " h I ,S0 " "L (I;C(S0 ))

0

H

"

"

% b " maxsòS0 sp dt ¢ a " L1 I % Tb " maxsòS0 sp ( )

Ë 303

4.3 Optimal ontrol of nite-dimensional dynami al systems

for a.a.

tòI

h y 0

with the Hamiltonian

*

;

;

 dened by (4.71) with *



*

solving the adjoint-

equation problem68

d % X f r (t ; y(t); s)  (t) t (ds) dt S t #  X ' r (t ; y(t); s) t (ds) % *



*

( )

*

0





S(t)

Example 4.56 (A universal approa h).

(

t ; y(t))  ;

y(T) # 0  (y(T)) :

*

*



Let us investigate a general situation when

S(t) need not be bounded uniformly with respe t to t ò I . Supposing (4.70h), we have L p (I; Rm ) but not L (I; Rm ), and therefore we are for ed to 

got the oer ivity only in

employ the general theory from Se tion 3.4. To extend as mu h problems as possible, we shall ertainly take the nest onvex ompa ti ation from the investigated lass. This means we put here

H # Carp (I; Rm ) ;

(4.120)

endowed with the universal (semi)norm (4.117). Then, in fa t, only the natural growth

onditions, i.e. (4.70a-f), are imposed on the Carathéodory integrands

f and ' be ause

the assumptions (3.192a), (3.196), and (3.197a) are void, as shown in Remark 3.107. However,

H

from (4.120) is not separable, whi h eventually prevents any usage of a great

part of our results. This is the reason why smaller subspa es ations, are more advantageous. In fa t,

H , reating oarser relax-

H should only ontain all possible integrands

that an appear in the investigated problem(s). Having in mind only a single problem with the data

f

and

' satisfying (4.70a-f), one an put

H # span † g - (f Ž y) % g - (f r Ž y) - y % g0 - (' Ž y) % g 0 - (' r Ž y) - y ; 









y; y ò C(I; R ); g0 ; g 0 ò C(I); g; g ò L n







q



(

I; R



n

)‡

:

(4.121)

C(I)-invariant linear subspa e of Carp (I; Rm ) and also (4.103) is satised. Moreover, H from (4.121) is separable69 if endowed with the norm Let us note that su h

H

is a

(4.117). If we are not interested in optimality onditions, we an avoid the data quali ation (4.70 ) and (4.70f) and take a smaller

H , namely

H # span †g - (f Ž y) % g0 - ('Ž y); y ò C(I; Rn ); g0 ò C(I); g ò L q (I; Rn ) ‡ ; 

whi h is a separable

C(I)-invariant

linear subspa e of

(4.122)

Carp (I; Rm ). Then (4.88) and

ODE (4.91) are guaranteed. Moreover, dealing with a olle tion of the problems (POC ; " ; " ), 1 2

we an take a linear hull of all subspa es onstru ted in (4.121) or (4.122). If taking the

olle tion ountable, we do not lose the separability of the resulted subspa e

68

H.

The solution of the adjoint problem is to be understood in the distributional sense sin e

*

is in

general a measure.

C- and L q -spa es involved in (4.121) and from the separate (strong,strong)- ontinuity of the mappings ( g; y ) ÜÙ g - ( f Ž y ), ( g; y;  y) ÜÙ g - (f r Ž y) - y, (g; y) ÜÙ g - (' Ž y), and ( g; y;  y) ÜÙ g - ('r Ž y) - y; for the ontinuity with respe t to the y-variable we refer to Remark 3.107 while the ontinuity with respe t to g - and  y-variables is an easy exer ise; f. also Proposition 3.102. 69



This follows from the separability of the

304

Ë

4 Relaxation in Optimization Theory

Example 4.57 (Linear/ onvex problems).

A

great deal of problems appearing in

appli ations have got a linear/ onvex stru ture (

t ; r; -) D - onvex, 

and

S

'(t ; r; -)

onvex,

f(t ; r; -)

ane, and

onvex-valued, measurable losed-valued. If our growth

assumptions as well as the oer ivity assumption (4.70h) with

p ¡ 1

are fullled,

then su h problems do not require any relaxation at all. In fa t, it su es to endow

L p (I; Rm ) by the weak topology (re all that always p   %). In other words, we an p m make the onvex  - ompa ti ation of the original spa e of ontrols L ( I ; R ) by means of the subspa e

H # L p (I) ‹ (Rm ) ; 

whi h auses

*

(4.123)

YH (I; Rm ) Ê L p (I; Rm ) and thus U ad Ê Uad ; f. also Examples 3.50 and p

3.73. It is well known70 that the ost fun tional in the original problem is weakly lower ODE semi ontinuous, so that the original problem (POC ) essentially oin ides with the reODE laxed problem (R H POC ) if one admits a non-ane extension of the ost fun tional, i.e.

the term (

' Ž y) DZ  is repla ed by N 'Žy (). In parti ular, by Proposition 4.2(i) there is

no relaxation gap. On the other hand, if the data satisfy the mild assumptions (4.70a)(4.70g) with

p # q ¡ 1, then one an make also a ner relaxation by taking H , i u

ontrol also for the nely relaxed problem whenever u is the optimal ontrol for the the natural hoi e

e.g., as in (4.121). We an apply here Corollary 4.36 be ause H ( ) is an optimal relaxed

oarsely relaxed (i.e. here original) problem.71 This yields the lassi al maximum prin-

h y 0  òCarp (I; Rm ) resulting from the ner relaxation s-variable, h y 0  : Uad Ù R is weakly upper semi ontinu-

iple (4.75). As the Hamiltonian is here on ave in the

;

*

;

*

*

;

*

;

ous,72 and therefore we an use Proposition 4.10 to transfer this maximum prin iple on the oarser relaxation, whi h gives again the maximum prin iple (4.75). If additionally

'(t ; r; -) is dierentiable, then h y 0 ;

*

;



t ; -) is smooth and on ave, so that we an h ò N S t (u(t)) or, more expli itly

(

*



rewrite the ondition (4.75) into the form ( y;  * ;  *) s 0

( )

f s (t ; y(t); u(t))  (t) " 0 ' s (t ; y(t); u(t)) ò N S

:a.a. t ò I :



*

*



t

( )

(

u(t)) :

(4.124)

Alternatively, one an get (4.72), (4.74), and (4.124) by a dire t appli ation of Corollary 1.60 to the original problem using the original geometry indu ed on

L p (I; Rm ).

Uad

from

As a result, in the linear/ onvex ase, the nely relaxed problems admit a two-fold understanding: either as auxiliary problems imposing a suitable geometry just for a

70

See, e.g., Buttazzo [161℄ where sequential weak lower semi ontinuity is demonstrated. However,

the oer ivity of the ost fun tion together with metrizability of the weak topology of

L p (I;

Rm

) on

bounded subsets implies the weak lower semi ontinuity, as well.

71

If the Hamiltonian is stri tly on ave, then even every solution to the nely relaxed problems has

this form; f. Proposition 4.27(ii).

72

This is obvious if

*0

# 0 be ause the Hamiltonian is then ane. For  ¡ 0 we refer again, e.g., to *

0

Buttazzo [161℄, using also the des ent ondition (4.36) together with metrizability of the weak topology of

L p (I;

Rm

) on bounded subsets.

4.3 Optimal ontrol of nite-dimensional dynami al systems

Ë 305

derivation of the pointwise maximum prin iple for the oarsely relaxed problems, or as usual ontinuous relaxation s hemes but with a spe ial property that some (or all) optimal relaxed ontrols are 1-atomi ; f. also the proof of Corollary 4.39.

Remark 4.58 (Renement of Filippov-Roxin's theory).

The onvexity ondition (4.79)

in the form (4.54a) an be, if used arefully in the proof in Theorem 4.29, ombined with the maximum prin iple (4.75) to weaken the onvexity ondition as

:r ò Rn : o ',f , for a.a.


( t ; r; R ( t ; r ))

’ Q(t ; r)

with

R(t ; r) # s ò M(t);

(

t ; r; s) ¢ 0 ;

t ò I with M(t) being an arbitrary estimate of the set of maximizers in (4.75), i.e. M(t) “ s ò S(t); Hy 0 ;

*

;



(

*

t ; s) # Hy 0 ;

*

;



(

*

t ; u(t))  :

This may sometimes enable us to get rened existen e results even if the onventional orientor eld

4.3.d

Q is non onvex; f. [341, 549, 561℄. Cf. also [699℄ for integral equations.

Approximation theory

ODE Further task we want to pursue is a numeri al approximation of (R POC ). Rather than

H

presenting a general theory, we want to demonstrate appli ations of the results developed previously in Se tion 3.5 to build one on rete (semi)dis retisation. The reader

an anti ipate that we hoose onvex inner approximation of the set of relaxed admissible ontrols

U ad . We will make only dis retisation in the t-variable but not s-variable

so that we get in general only the s heme of Type II; see the lassi ation from Se tion 3.5. For s hemes of Type I see Remarks 4.62, 4.66 and 4.67 below. Let

d ¡ 0

be a time step. We will suppose

T/d

integer and use an equi-distant

I . For d1 £ d2 ¡ 0, we also suppose d1 /d2 integer so d2 is a renement of the partition with d1 . Then we dene the p p m m proje tor Pd : Car ( I ; R ) Ù Car ( I ; R ) by partition of the time interval that the partition with

[

Pd h℄(t ; s) #

1 d

X I kd

h( ; s) d

if

t ò I kd :# [(k"1)d ; kd); k # 1; :::; T/d;

(4.125)

f. also Se t. 3.5.b. On this rather abstra t level, we will assume that there are some linear subspa e

V ’ C p (Rm ) and a linear subspa e G su h that

G ‹ V ’ H ’ l(G ‹ V); G0 ’ G ’ L where  l refers to the natural topology73 of

G0 #

73

℄ Gd d ¡0

with

G d # Žg ò L





I ; H is G-invariant;

( )

(4.126)

Carp (I; Rm ) and

I ; :k # 1; :::; T/d ; gI d ò C(I kd )DZ ;

( )

k

Quite equally it would su e to onsider any ner topology, e.g. the topology indu ed by the norm

(4.117) or any ner form, if exists.

306

Ë

4 Relaxation in Optimization Theory

f. (3.164). Also, we will assume that

H

as well as its norm is ompatible with

Pd

from

(4.125) in the sense that74

Pd : H Ù H

:h ò H :

is a ontinuous proje tor

;

(4.127a)

lim h " Pd h H # 0 :

(4.127b)

d Ù0

H d # Pd H ’ H . By Propositions 3.83(i) and 3.86(i), Pd Y H (I; Rm ) is p m a onvex, weakly*  - ompa t subset of Y ( I ; R ). Supposing that S ( t ) forming the H

ontrol onstraints in U ad from (4.63b) is onstant,75 i.e.

:a.a. t ò I : we an easily see76 that even

S(t) # S0 ;

(4.128)

Pd U ad ’ U ad . Also, it holds *

d1 £ d2 ¡ 0, so that the onvex in reases for d Ù 0; f. Proposition 3.83(iii). Pd2

p

*

Then we denote

whenever

here obviously

approximations

Pd1 Ž Pd2 #

Pd U ad ’ Y H (I; Rn ) p

*

To ensure the onvergen e of the approximate problems, we onsider problems without state onstraints, f. Remark 1.52. For simpli ity, we suppose that the state equation as well as the ost fun tional do not require any additional approximation to be handled ee tively.77 Thus we ome to the following (semi)dis retised relaxed problem:

d ODE (R POC ) H

T X ['

. 6 Minimize 6 6 6 6

0

> 6 subje t to 6 6 6 6 F

Ž y DZ ℄(dt)%(y(T))

dy # f Ž y DZ  ; y(0) # y ; dt p  ò Pd U ad ’ YH (I; Rm ) ; y ò W 0

*

1;

q (I;

Rn

)

:

Proposition 4.59 (Convergen e of numeri al approximations). Let all the assumptions of Proposition 4.46 with # 0 together with (4.126)(4.128) be valid. Then (R

d ODE H P ) has a solution OC

lim min(RdH PODE ) # min(RH PODE ) ;

d Ù0

OC

(4.129a)

OC

Limsup Argmin(RdH PODE ) ’ Argmin(RH PODE ) : d Ù0

74

In fa t, it su es to suppose

OC

OC

Pd H

(4.129b)

’ H be ause (4.125) ensures Pd h H ¢ h H so that Pd L H H ¢ 1 (

;

)

provided - H is the norm (4.117), whi h an be always supposed.

75

A generalization for pie ewise onstant

partition of

76

I is straightforward.

ÜÙ S(t) whi h is pie ewise onstant on some equi-distant

Let us note that, supposing (4.128) and taking a sequen e {

dened by (3.165) also belongs to

77

t

Uad .

u k }kòN

’ Uad , the sequen e {u k }   ò, ( ;

)

In fa t, this an be true only in very simple ases. In general, we need always a numeri al solver

for the system of ordinary dierential equation as well as a numeri al-quadrature formula to evaluate the ost fun tional. We negle t this need to make the presentation learer.

4.3 Optimal ontrol of nite-dimensional dynami al systems

Ë 307

d ODE

Proof. The existen e of a solution to (R H POC ) follows by the same arguments as for the

ODE

ase (R H POC ), see the proof of Proposition 4.46. Then the laimed onvergen e follows

 d # . Also note that (1.101 ) is satised by Proposition 3.83(iv) be ause U ad is a B - oer ive onvex - omn pa ti ation of U ad sin e H ontains a oer ive integrand, e.g. ' Ž y with y ò C ( I ; R ), dire tly by the arguments of Remark 1.51 even simplied as

f. (4.70h) and Proposition 3.67(ii).

V

If

from (4.126) is nite-dimensional, the set of admissible relaxed ontrols

d ODE for (R H POC ) is, in fa t, a onvex subset of a nite-dimensional linear variety78

Pd U ad *

and as su h, it an be implemented dire tly on omputers; then we fa e the approximation of Type I (in a

ord with the lassi ation in Se tion 3.5). In the opposite

ase, we have obtained the approximation of Type II and a further theoreti al eort

d ODE is needed to implement the semi-dis retised problem (R POC ) on omputers. Namely, H d ODE we have to pose and analyze the optimality onditions for (R POC ). Of ourse, thanks H to the onvexity of

Pd U ad , *

we are able to perform it in an entirely parallel way

ODE how it was done for (R POC ). Now, the maximum prin iple will involve the dis rete

Hamiltonian

h dy  ò H d ;

*

H

dened by

h dy  # Pd  ;

*

*

- (

f Ž y) " (' Ž y)  :

(4.130)

Proposition 4.60 (Maximum prin iple for approximate problems).79

Let all the assum-

ptions of Proposition 4.50 together with (4.126)(4.128) be valid, and (  d ;

d ODE ). Then the pointwise maximum prin iple

y d ) be a so-

lution to (R H P

OC

h dyd

;



*

DZ  d
(t) # sup h dyd sòS0

;



(

*

t ; s)

L1 (I), where the Hamiltonian h dy;  ò H is given by (4.130) and n ) is a solution80 to the adjoint ba kward terminal-value problem:

is valid in the sense of



ò Lq

*



(

I; R

(4.131)

*

d % f r Ž y d DZ  d  (t) # ' r Ž y d DZ  d ; dt *



*

 (T) # 0:



*

(4.132)

Proof. By the same arguments as in the proof of Proposition 4.50, realizing additionally that (R

d ODE H POC ) is un onstrained (so that one an re kon

0 # 1 and  # 0), we get the *

*

adjoint terminal-value problem (4.132) together with the inequality

: ò Pd U ad :

 d

*

Pd (G ‹ V) # {g : I

"  ; h yd  £ 0 ;

*

(4.133)

Ù R pie e-wise onstant on I} ‹ V is nite-dimensional.

78

Let us note that the spa e

79

Su h kind of maximum prin iple has been also stated by Chryssoverghi and Ba opoulos in [212℄

and, for ellipti optimal ontrol problems, by Chryssoverghi and Kokkinis in [214℄ or, for a paraboli optimal ontrol problem, also in [213, Thm. 3.2℄.

80

Likewise (4.105), in general (4.132) is to be understood in the sense of distributions.

Ë

308 with

4 Relaxation in Optimization Theory

hy  #  ;

 d

*

*

- (

f Ž y) " (' Ž y). Sin e Pd  d #  d and Pd  # , we an obviously write *

*

"  ; h y  # Pd  d " Pd  ; h y  #  d "  ; Pd h y  #  d "  ; h dy  ; ;

*

*

*

;

*

;

*

;

*

whi h allows us to rewrite (4.133) as

: ò Pd U ad :

 d

*

;

(4.134)

*

d d * d , we have <  ; h y;  > # <  ; P h y;  > # < P  ; h y;  > for any  ò ; ; d d n ). As a result, the inequality (4.134) holds even for every  ò U , whi h gives ad d d the maximum prin iple <  d ; h y ;  > # max  ò U ad <  ; h y d ;  >. This maximum prin iple d Sin e

p Y H (I;

Pd h dy  # h dy 

"  ; h dyd  £ 0 :

R

*

*

*

*

*

*

*

an be equally written in the form

T X 0

h dyd

;



*

DZ  d
(dt) # sup

u ò U ad

T d X h yd ;  0

(

*

t ; u(t)) dt :

Then one an just use Theorem 4.21(i) to lo alize this integral maximum prin iple to get eventually (4.131); note that, thanks to (4.70g) and (4.70h),

h dy  ;

*

satises the de-

Å

s ent ondition (4.36).

Corollary 4.61 (Chattering solutions to (RdH PODE OC )).

Let

the

assumptions

of

H be separable. Then: d ODE There exists a solution (  ; y ) to (R H P ) with  being ( n %1)-atomi .

Proposi-

tion 4.60 be valid and (i)

OC

 ; y) solves (RdH PODE ),  d m Ù fun tion h y  ( t ; -) : R  is k-atomi .

*

(ii) If (

OC

;

*

is the orresponding adjoint state and, for a.a.

 ò Pd h dy;  and

n%1) suitable onditions involving the integrands h # d ODE for 1 ¢ l ¢ n , solves (R POC ); we use here the hain of identities H satises (

Ž y # Pd  ; ' Ž y #  ; Pd (' Ž y) #  ;  *

*

- (

U ad , whi h h l # [f Ž y℄l

*

Proof. Likewise in the proof of Corollary 4.53, we an show that any

 ; '

t ò I , the

R a hieves its maximum at no more than k points, then

*

f Ž y) " h dy  ;

*

#  ; f Ž y DZ  " 1; h dy  DZ  #  ; f Ž y DZ  d " 1; h dy  DZ  d *

*

;

#  d ; 

*

- (

f Ž y) "

*

h dy;  *

;

*

#  d ; Pd (' Ž y) #  d ; ' Ž y :

Then the point (i) follows from Proposition 4.28 modied for our pie ewise homogeneous ase (details are omitted). Eventually, the point (ii) again follows from Proposi-

Å

tion 4.27(i).

By means of Corollary 4.61(i) we an eventually implement the semi-dis retised

d ODE relaxed problem (R POC ) on omputers: we an onsider only su h pie ewise homoH geneous relaxed ontrols whi h are (

n%1)-atomi ,

whi h form a nite-dimensional

4.3 Optimal ontrol of nite-dimensional dynami al systems

Ë 309

(non onvex) manifold.81 Then Corollary 4.61(i) ensures that this manifold ontains at

d ODE least one solution to (R H POC ).

Remark 4.62 (Dis retisations of Type I). Requiring the regularity of the data ' Ž y pH ¢  1 ( I ; C ( S ))  L ( I ; C  2 ( S )) with  ;  ò (0 ; 1℄, we C and f Ž y p H n ¢ C for H # W

S

an perform a full dis retisation by means of the proje tor P # P P ( f. Se ts. 3.5.b d d1 d2 1

;1

2

and 3.5. with

0



1

0;

1

2

repla ed by I ). For implementation see also Remark 4.66 below.

Remark 4.63 (Dis retisations of Type III).

One an think also about a dire t dis retisa-

ODE tion of the original problem (POC ) by making the original set of admissible ontrols

Uad from (4.63b) smaller, e.g. Ud #

† u ò U ad ;

uI d k

onstant

; k # 1; :::; T/d ‡ :

Then the relaxed problem serves only for an asymptoti al analysis. Alternatively, we ODE

an get su h approximation as the restri tion of the relaxed problem (R H POC ) to the

i

(generally) non onvex nite-dimensional variety H (

U d ), and thus su h s heme an

be viewed as a dis retisation of Type III a

ording to the lassi ation from Se tion 3.5. Su h s heme is used quite often, alled a ontrol parametrization method.82 Error estimates

s-variable of the test intef ; f. [661℄. Also the dimensionality of the resulting problem

require additionally a ertain smoothness in the

grands and thus also of

is, under the omparable rate of error of minima, higher than the (semi)dis retisation of Type II presented here. The smoothness requirements as well as the dimensionality are similar as if we would have made additional aggregation of the pie e-wise homogenized Young measures ( f. Remark 4.62) to get a dis retisation of Type I.83

Remark 4.64 (Adaptive support-estimation strategy).

In omparison with Type II or

III, the dis retisations of Type I lead to problems whi h are onvex if the relaxed problem is onvex but have a very large number of variables, in parti ular if

onsidering again

Pd #

m Á 1. Yet,

PdI1 PdS2 from Se ts. 3.5.b and 3.5. , most of the oe ients in

the Young-measure representation (3.168) of the optimal relaxed ontrols are zero, be ause typi ally the optimal ontrols have a hattering hara ter and these Young measures have supports only at few points where the dis rete Hamiltonian (4.130) is maximized. In onvex relaxed problems, the maximum prin iple (4.131) is a su ient

ondition and, if one knows (at least approximately) the orre t Hamiltonian, i.e. the

orre t adjoint state



*

from (4.132), one an onsider only those points in

S where this

Hamiltonian is (approximately) maximized. This may de rease dramati ally the number of degrees of freedom. Of ourse, one does not know a-priori the adjoint state

81

The dimension is

n

- (

n%1) - m

- (

. Yet,

T/d) be ause on T/d time subintervals we need to pres ribed n%1) ve tors from m and n mutually independent

the support of the underlying Young measure as (

R

weights appearing in the onvex ombination of Dira measures supported at these ve tors.

82

See, e.g., Goh and Teo [358, 758℄.

83

For a theoreti al omparison of su h s hemes, the reader is referred to [671℄.

310

Ë

4 Relaxation in Optimization Theory

d k # (d1 k ; d2 k ) with d1 k ¡ d1 k%1 , d2 k ¡ d2 k%1 , k # 1; :::, and, knowing the solution on d k -dis retisation and the orresponding adjoint state  , to a tivate only su h supporting points on the k d next rened d k %1 -dis retisation where the fun tion h d ( x ; -) is maximized with some y  one an exploit a series of su

essively rening dis retisations ;

;

;

;

;

;

*

; d

*

toleran e. This multi-level strategy dealing with the adaptively tuned dis retisation of Type I an thus be organised due to the ow diagram:

INITIALIZATION

BEGIN

CHOOSE INITIAL DISCRETIZATION

ACTIVATION

ACTIVATE THE GRID POINTS WITH GREAT VALUE OF THE HAMILTONIAN

OPTIMIZATION ROUTINE SOLVE THE DISCRETE PROBLEM CONSIDERING THE ACTIVE GRID POINTS

NO

IS THE MAX. PRINCIPLE SATISFIED AT ALL GRID POINTS ?

YES NO

CORRECTION TAKE GREATER TOLERANCE

FINAL DISCRETIZATION LEVEL

YES

END

REFINEMENT REFINE THE DISCRETIZATION AND TAKE THE ORIGINAL TOLERANCE

This was proposed in [179℄ and shown to have an ability to be more e ient than the plain dis retisation from Remark 4.62 for additively oupled problems like (4.112) whi h leads to a linear-quadrati programming (LQP). In more general ases, some iterative solvers must be used for the dis rete problems. In this multi-level approximation strategy for the relaxed problems, a usage of an iterative linear-programming-based algorithm was devised in [89℄. Moreover, adaptive meshing of

as in [173℄ an advan-

tageously be ombined with this adaptive support-estimate algorithm, f. [84℄.

4.3.e

Illustrative omputational simulations: os illations

ODE Let us illustrate the pre eding results on a on rete problem (POC ) with 3 1 y0 # ( 16 ; 10 ), T # 1, # 0, S(t) # f # ([f℄1 ; [f2 ℄) in the form

'(t ; r; s) # A3i#1(s " u i (t))2 % 2i#1 (r i " [yd (t)℄i )2 with

u1 (t) # 31 t % 23 ; yd (t) # „(t "

1 4

)(

n # 2, m # 1,

R (i.e. no ontrol/state onstraints) and ' and

u2 (t) # 2t( 23 " t) % 13 ;

t"

3 4

)

; (t "

1 2

1 " t)(t "

)(

u3 (t) # "t 1 5

)…

;

(4.135a)

4.3 Optimal ontrol of nite-dimensional dynami al systems

f

t ; r; s) # r2 % s " 32 t4 % 95 t3 "

45

f

t ; r; s) # "r1 % (s " 1) % 43 t6 " 62 t5 " 45

t4 %

[ ℄1 ( [ ℄2 (

11

t2 %

32 15

t"

23 15

;

Ë 311 (4.135b)

2

23 15

373 270

t3 "

457 90

t2 %

t"

281 90

47 48

:

(4.135 )

p # 6 and q # 3 so that a relaxation H ’ Car6 (0; 1; R2 ) is possible. The data (4.135) are

This problem satises the assumptions (4.70) with by hoosing a suitable subspa e

inf

#0

ODE (POC ) and we know the exa t solution to the relaxed problem ODE (R POC ). By Corollary 4.53 there is at least one 3-atomi solution ( ) whi h is given H

hosen so that

; y

here, in terms of Young measures, by

# ( " t )Æ u1 t % ( " t " t )Æ u2 t % ( y(t) # „(t " )(t " ); (t " )(1 " t)(t " )… t

where

2

1

5

6

3

( )

1

1

1

2

10

3

2

( )

1

3

1

1

4

4

2

5

is the Young-measure representation of

%

1 10

.

1 10

t % 21 t2 )Æ u3

t

( )

(4.136a) (4.136b)

This solution is even unique,84

whi h shows that, in parti ular, the estimate of number of atoms in Corollary 4.53

annot be improved. An illustrative omputational experiment al ulations for (R

d ODE H POC ) presented here

0; 1) with the time-step d # 2"

4

has used an equi-distant partition of the time interval (

.

By Corollary 4.61, one an rely on the existen e of at least one 3-atomi solution to

d ODE the semi-dis rete problem (R POC ) whi h an a tually be implemented on omputers. H

An initial (intentionally rather badly) guessed 3-atomi ontrol sponding response85

y # () is shown86 in Figure 4.3.

 as well as the orre-

Three sele ted iterations are shown in Figure 4.3, namely the zero (initial) one, 20th, and the nal 430th one) obtained by a sequential-quadrati -programming (SQP) optimization routine.87 For omparison, the (unique) optimal ontrol to the nonODE dis rete relaxed problem (R POC ) and the orresponding response is displayed by dotH

ted lines.

84

min(RdH PODE OC ) # 0, we have y # y d determined ’ {u i (t); i # 1; 2; 3} lo alized uniquely. Then the onvex

As both terms in the ost fun tional must vanish if

uniquely and also the support

supp(

t)

ombination of the Dira s in (4.136a) is determined from (4.89) also uniquely be ause the ve tors

dyd /dt " f(t ; yd (t); u i (t)) with i # 1; 2; 3 form here a (3 , 2)-matrix of a full rank. 85

Of ourse, this response has been omputed only numeri ally by an expli it Euler method but with

a small time step 1/3200 so that it an be onsidered numeri ally as exa t. For sti systems or for higher a

ura y, more sophisti ated methods (as e.g. Runge-Kutta) would have to be employed.

86 3

87

 is displayed. a l i H (u l ) is displayed only by u l (t), while the weights a l (t) are not indi ated.

Only the support of the Young measure orresponding to

l#1

This means,



#

The SQP routine NLPQLD by S hittkowski [716℄ has been exploited. The time-dis retisation of the

ontrol has used

d # 2"4 .

312

Ë

4 Relaxation in Optimization Theory

PSfrag repla ements

Starting point S(

1

; y) for the optimization routine

supp( ν )

0.6

y2

T =1

0 PSfrag repla ements

0

PSfrag repla ements

S-1

A urrent point S(

1

T =1

y1

-0.2

; y) after 20 iterations

supp( ν )

0.6

y

T=1

0

1

PSfrag repla ements

S-1

The solution S(

1

T=1

0

PSfrag repla ements



y

2

-0.2

; y) after 430 iterations

supp( ν )

0.6

T=1

0

y1

PSfrag repla ements

S-1 Fig. 4.3:

T=1

0

y2

-0.2 The starting point, and intermediate iteration, and the nal iteration (

 ; y) for the optimiza-

RdH PODE OC ); only supports of the 3-atomi Young measure but not the probability

tion routine solving (

distributions are displayed. Cal ulation and visualization: ourtesy of

Mar ela Mátlová-Vítková

(for-

merly Cze h A ademy of S ien es)

As the (unique) relaxed optimal ontrol for (R

ODE

H POC

) has, in fa t, an

representation and also the approximate relaxed ontrols ported.

d

L



-Young-measure

remain boundedly sup-

By a detailed analysis as in [661℄, one an see that the dis retisation error

min(RdH PODE OC )

" min(RH PODE OC ) is of the order at least O ( d ), whi h agrees with the experi-

mental results, as shown on Figure 4.4.

4.3 Optimal ontrol of nite-dimensional dynami al systems

Discretization error

G

u

Fig. 4.4:

a

−5

ra

te

e

d

The dis retisation error

ODE min(RdH PODE OC ) " min(RH POC )

n

10

Ë 313

sl

in dependen e on

o

p

e

O(

d.

d)

repla ements Figure 4.3 (nal iteration)

−6

10

−3

−4

2

−5

2

−6

2

Remark 4.65 (Warga's algorithm).

2

Time step

d

The implementation on basis of Corollary 4.61(i) is

d ODE H POC ). Warga [792℄ p n) proposed a steepest des ent algorithm whi h uses the onvex geometry of Y ( I ; H

not the only numeri al approa h to the semi-dis rete problem (R

R

but, after ea h iteration, the resulted relaxed ontrol (whi h may possibly not be implementable) is repla ed by a hattering ontrol exhibiting the same ee ts, i.e. driving the ontrolled system to the same state under the same ost, whi h an be already implemented on omputers; by the Carathéodory theorem 1.12 it an be shown that there is at least one (

n%2)-atomi ontrol with this property.88 Thus in our ase, Warga's al-

gorithm would handle 4-atomi relaxed ontrols.

Remark 4.66 (Dis retisation of Type I). stru ture in terms of

r,

Sin e the data (4.135) have linear/quadrati

the dis retisation by the proje tor

Pd # PdI1 PdS2

outlined in

Remark 4.62 results to a linear/quadrati onvex mathemati al-programming problem. Therefore, a global minimizer an be found by a nite solver,89 whi h is ertainly a great advantage resulted from the onvex stru ture kept in the fully dis retised problem. In more sophisti ated appli ations, taking su h advantage may be ome a ne essity; f. Example 7.3.13. Presented sample al ulations use again the time-

d1 # 2"4 and additionally the proje tor PdS2 ( f. (3.167)) whi h makes dis retisation of S # ["1; 1℄ by 61 equi-distant points so that the mesh-size parameter is d 2 # 1/30; note that, for simpli ity, S has been restri ted now only on the interval ["1 ; 1℄ without hanging the set of solutions though the original S # R ould be also dis retised by, however, a non-equidistant mesh. The dis retisation proje tor

PdI1

with the time step

resulted solution is shown on Figure 4.5. One an see that the optimal solution of the

88

For appli ation of Warga's algorithm see Chryssoverghi and Ba opoulos [212℄ or, in ase of a

paraboli optimal ontrol problem, also Chryssoverghi [210℄.

89

The solution shown on Figure 4.5 has been al ulated by the a tive-set-strategy linear-quadrati

programming routine QLD by S hittkowski [716℄.

314

Ë

4 Relaxation in Optimization Theory

dis rete problem need not be now three-atomi though, of ourse, in the limit it inevitably approa hes the (unique) optimal three-atomi solution.

1

supp( ν )

0.3

y

1

T=1

0

T =1

0

y2

PSfrag repla ements

S-1

-0.3

Fig. 4.5:

Cal ulated optimal solution (

and visualization: ourtesy of

 d ; y d ) to (RdH PODE OC ), d

Mar ela Mátlová-Vítková

Remark 4.67 (Coarser relaxations).

$ (d ; d 1

2)

# (1/16; 1/30). Cal ulation

(formerly Cze h A ademy of S ien es).

If one uses a su iently oarse relaxation, it may

happen that one gets immediately the nite-dimensional onvex dis retisation (i.e. of Type I) when only applying the spatial dis retisation by

Pd

from (4.125). E.g., appli a-

ODE tion of (4.121) to (POC ) with the data (4.135) leads to the hoi e

H # C(I) - {h0 } % L3

/2

I ‹ {h1 ; h2 }

( )

with

h0 (t ; s) # Ai#1 (s" u i (t)) ; h1 (t ; s) # s ; h2 (t ; s) # (s"1)2 : 3

2

(4.137)

Q : C(I) , L3 2 (I)2 Ù H : (g0 ; g1 ; g2 ) ÜÙ 2l#0 g l - h l , the adjoint mapping Q : H Ù r a(I) , L3 (I)2 makes the equivalen e Y Hp (I; Rm ) Ê w*-b lr a I , L 3 I 2 m B i ( U ad ) where the embedding is dened by i ( u ) # Q i H ( u ); it is easy to see that i ( u ) # ( h 0 Ž u ; h 1 Ž u ; h 2 Ž u ) be ause /

Considering

*

*

*

( )

i ( u ) ;

(

( )

g 0 ; g 1 ; g 2 ) # Q i H ( u ) ; ( g 0 ; g 1 ; g 2 ) # i H ( u ) ; Q ( g 0 ; g 1 ; g 2 ) *

#

2 T H X g l ( t ) h l ( t ; u ( t )) d t l #0 0

#

( h 0 Ž u ; h 1 Ž u ; h 2 Ž u ) ;

(

g0 ; g1 ; g2 ) :

Also, if

 ò Y H (I; Rm ) is p-non on entrating and has thus a Young-measure represen-

tation

, then it is an easy exer ise to verify the formula

p

Q  # (h0 DZ ; h1 DZ ; h2 DZ *

i.e., the parti ular omponents of

orresponding

hl.

Q  *

)

;

are just the momenta of

(4.138) with respe t to the

We already met this ee t when aggregating Young measures by

h l # 1 ‹ v l ; f. the formula (3.168). p Pd Y H (I; Rm ) is nite-dimensional be ause the linear spa e Pd H has a nite dimension, namely 3 T / d . However, the implementation of the resulted dis retisation is not so easy in general. For example, if S ( t ) $ S 0 is bounded, then

means of spe ial nite-element fun tions It is now lear that

*

4.3 Optimal ontrol of nite-dimensional dynami al systems

from the formula (4.137) one an dedu e90 that

Q

*

L

with the subset in



 ò L 

I

3

( )



Pd U ad *

Ë 315

is anely homeomorphi via

of the form

I 3 ; :a.a. t ò I :  (t) ò o[Pd h℄(t ; S0 ) ;

( )



Pd h℄(t ; s) # ([Pd h0 ℄(t ; s); [Pd h1 ℄(t ; s); [Pd h2 ℄(t ; s)). It should be however emphaS0 # [a ; b℄ ò R and u i pie ewise onstant on the partition T d of I , it is pra ti ally impossible to des ribe expli itly the onvex hull of 3 2 2 3 the urve [ P h ℄( t ; S 0 ) # {(A i #1 ( s " u i ( t )) ; s ; ( s " 1) ); a ¢ s ¢ b } in R . The di ulty of d this task depends essentially on the parti ular nonlinearities h l involved in a problem. A spe ial situation o

urs if m # 1 and the nonlinearities are polynoms  then one

where [

sized that, even in the spe ial ase

an use the stru ture from Se t. 3.3.d.91

H still smaller than (4.137) would ause similar troubles: for example the H # L3 2 (I) ‹ {h1 ; h2 } would require the extension of the ost fun tional only by lower semi ontinuity, while still a smaller H would additionally require a multivalued Taking

/

hoi e

extension of the ontrol-to-state mapping.92

4.3.f

Illustrative omputational simulations: os illations and on entrations

We already saw that the on entration ee ts an be ombined with os illation ones. The situation from Figure 3.9 on p. 152 an be illustrated on the following Bolza-type optimal- ontrol problem [459℄, enhan ing the Example 4.34:93

T

Minimize

J(y; u) :# X (2"2t% t2 )u(t) % y22 (t) dt % (y1 (T)"1)2 0

subje t to

dy # u ; y (0) # 0 ; dt dy max(0; u) min(0; u) # % ; y (0) # 0 ; dt  1" y $ (y ; y ) ò W (I; R ) ; u ò L (I) ; 1

1

2

2

1

with some

2

1;1

2

1

/ 7 7 7 7 7 7 7 7 ? 7 7 7 7 7 7 7 7 G

(4.139)

 ò (0; 1). Obviously, y2 L2 I tends to be as small as possible. Let us note 2"2t%t2 ¡ 0 has the minimum at t # 1, whi h for es the minimizing ( )

that the polynom

sequen e to on entrate around this time instant. An example of a minimizing sequen e is the ontrol

90

One must use also the well-known properties of integrals of multivalued mappings:

l PA S(t) dt for any A ’ I measurable provided S : I ±

Rm

P A

o S(t) dt #

is measurable, losed-valued, and inte-

grably bounded; see Aubin and Frankowska [37, Thm. 8.6.4℄ for details.

91

The relaxation of optimal- ontrol problem by algebrai moments has been used in [535, 606℄.

92

Cf. the approa h by Buttazzo [160, 161℄. For a omparison with the ner relaxation using the Young

measures see Mas olo and Miglia

io [519℄, or also [664℄.

93

For a dierent relaxation of the example (4.139) using res aling time was devised by Kamps hulte

[417℄, loosing onvexity of the relaxed problem, however.

Ë

316

4 Relaxation in Optimization Theory

t ò (1; 1 % "), t ò (1 % " ; 1 % "), otherwise ;

~/ " . 6 u " (t) # > "~/"

6 F

y2"

is small, namely

y2 L2

I

( )

(4.140)

if

0

for whi h the orresponding state while

if

y " # (y1" ; y2" ) has the omponent y1" as in (4.140) # O("). For ~ # 1/2, we get the inmum of the

problem (4.139) is 3/4.

R # " ; % of R and the analyti al solution to su h relaxed problem is  ò r a 0; 1 , R , f.

The relaxed problem an take the two-point ompa ti ation

S #

[



([

℄

℄ )

the notation (3.51), given by

Æ 1 ‹ Æ" dt ‹ Æ0 ~

 (dtds) # 

{

}



% (1")Æ % Æ  ; 

0

if

t#1

otherwise

(4.141a)

;

and the orresponding states given (a.e.) as

y1 (t) # 

0 1"

if if

0¢t 1 ; 1 t¢1

y2 (t) # 0 :

(4.141b)

The numeri al approximation of Type I (i.e. onvex / nite-dimensional) an be made by the proje tors from Remark 3.86 and Example 3.5. . The results from omputational implementation of su h dis retisation are presented in Figure 4.6.

Fig. 4.6: The (support

of the) DiPerna-Majda measure and the orresponding response approximating

R#

d # 0:0125 of the time interval [0; 1:5℄ and the "; %℄ by 20 points. Cal ulations and visualization: ourtesy of Martin

the exa t solution (4.141) with the dis retisation dis retisation of

Kruºík

1

[

(Cze h A ademy S ien es).

Con entrations may o

ur not only at isolated points but they an be smeared out along the whole interval and, simultaneously, they an os illate like on Figure 3.10 on p. 157. This an be demonstrated on the following problem [459, 693℄:

u(t)2 1%u(t)4 0 %(y1 (t)" t)2 % y2 (t)2 dt 1

Minimize

subje t to

J(y; u) :# X 2 u(t) %

dy # u ; y (0) # 0 ; dt max(0; u) min(0; u) dy # % ; y (0) # 0 ; dt  1" y $ (y ; y ) ò W (I; R ) ; u ò L (I) ; 1

1

2

2

1

2

1;1

2

1

/ 7 7 7 7 7 7 7 7 7 7 7 ? 7 7 7 7 7 7 7 7 7 7 7 G

(4.142)

repla ements

Ë 317

4.3 Optimal ontrol of nite-dimensional dynami al systems

0      1 and ¡ 0. Let us note that this fun tional is oer ive on L (I). Here again the minimum does not exist and we an see that the inmum is (1 " 2 /3), uk viz [693, Prop. 4.1℄. The example of a minimizing sequen e for # 0 is on Figure 4.7. 1

with

2

u

k

y 1; k

/k2

k

0

2

k

:::

3

k

:::

1

t

3/k

T

2/k

1")/k

(

y 2; k

1/k

2

0

"k Fig. 4.7:

One element of a minimizing sequen e {(

u k ; y k )}kòN

t

T for the problem (4.142) with



# 0 on-

verging to (4.143).

Taking again R # [" ; %℄, the analyti al solution to the relaxed problem is  òr a([0; 1℄ , R) is given by

 (dtds) # 

dt ‹ Æ" % (1")Æ % Æ dt ‹ Æ 



0



if if

0

0¢ t ¢ 1" ; 1"   t ¢ 1 ;

(4.143a)

and the orresponding states given as

t

y1 (t) # 

0¢ t ¢ 1" ; 1" ¢ t ¢ 1

if

1"

if

y2 (t) # 0 :

(4.143b)

f. [693, Prop. 4.1℄. Although the data varies ontinuously in time, the optima relaxed

ontrol jumps at

t # 1"

if

  1, whi h has

a similar hara ter as Tartar's broken

extremal, f. [575℄. The numeri al approximation of Type I an be made as in the previous example.

 # 0:5

# 0:3

On Figure 4.8, we an see numeri al results for and , and the adaptive repla ements support-estimation strategy from Remark 4.64 has been used in [693℄.



% γS

supp(

d1

0 1−β

0.5

d ) 

1.0

t

d2

1.0

d2

d2

d2

# 1/32 y 2 t 1.0

y1

0.5

# 1/32

0

0.5

DiPerna-Majda measure and the orresponding response approximating

d # 0:05 of the time interval [0; 1℄ and two dis reti# 1/10 (dash line) and 1/34 (full line) for R # ["; %℄. Cal ulations and visualization:

the exa t solution (4.143) with the dis retisation sations

# 1/10

# 1/10

" Fig. 4.8: The (support of the)

d2

ourtesy of

Martin Kruºík

1

(Cze h A ademy S ien es).

Ë

318 4.3.g

4 Relaxation in Optimization Theory

Optimal ontrol of dierential-algebrai systems

A nontrivial appli ation of the relaxation method is to optimal ontrol of systems governed by dierential-algebrai equations (DAE). We onne ourselves to so- alled

ausal semi-expli it systems,94 also alled Hessenberg-form DAEs. We will deal with the following optimal ontrol problem in the Bolza form:

DAE

(POC )

T X ' ( t ; y ( t ) ; w ( t ) ; u ( t )) d t

. Minimize 6 6 6 6 6 6 6 6 6 subje t to 6 6 6 6 6

0

% (y(T))

( ost fun tional)

dy # f(t ; y(t); w(t); u( t)) ; (state equation - dierential part) dt 0 # g(t ; y(t); w(t); u(t )) ; (state equation - algebrai part) y(0) # y ; (initial ondition) ( t ; y ( t ) ; w ( t ) ; u ( t )) ¢ 0 ; (state- ontrol onstraints) u(t) ò S(t) (:a.a. t ò I ) ; ( ontrol onstraints) y ò W q1 (I; Rn1 ); w ò L q2 (I; Rn2 ); u ò L p (I; Rm );

> 6 6 6 6 6 6 6 6 6 6 6 6 6 6 F

0

1;

' : I , Rn1 , Rn2 , Rm Ù R, f : I , Rn1 , Rn2 , Rm Ù Rn1 , g : I , Rn1 , Rn2 , Rm Ù Rn2 ,  : Rn1 Ù R, y0 ò Rn1 , S : I ± Rm a multivalued mapping, and : I , Rn1 , Rn2 , Rm Ù R are subje ted to ertain data quali ation introdu ed  later, n 1 ; n 2 ; m ;  £ 1, 1 ¢ p   %, 1   q 1 ¢ % and 1   q 2 ¢ %. Of ourse, R is expe ted to be ordered by a one D so that the ondition ( t ; r; v; s ) ¢ 0 has a sense. The pe uliarity of that the w -variable is not subje ted to a time derivative and may follow the speed or amplitude of the ontrol variable u . Thus this fast part of the state ( y; w ) may exhibit the fast-os illation and the on entration ee ts like the ontrol

where



variable and the relaxation must be done arefully, ounting an impli it onstraint

w and u by respe ting the algebrai part g(t ; y(t); -; -) # 0 depending also on the slow part of the state, i.e. on y .95 This suggests to make a relaxation for ( w; u ) jointly. Yet, respe ting the mentioned impli it onstraint g ( t ; y ( t ) ; - ; -) # 0, this would lead to a onvex ompa ti ation depending on the state y be ause onvex ompa ti ations in w and u separately would need a bi-ane extension of f ( t ; y ( t ) ; - ; -) and g ( t ; y ( t ) ; - ; -) and its joint ontinuity, whi h is rather overambitious, as we saw in relating values of

Se t. 3.6. , f. also Remark 4.72 below. Therefore, relying that no impli it restri tion is imposed on the ontrol

u (related

with the assumed ausality) and exploiting the underlying ODE, we translate the

94

y; w℄(t) does not depend on the derivatives t but on u(t) only. For some results in more general ases

The adje tive  ausal here means that the solution [

du/dt ; : : : ; d k"1 u/dt k"1 (

)

(

)

at a urrent time

see [469℄.

# 0 ould be, in prin iple, treated as a state onstraint in

95 The algebrai part g ( t ; y ( t ) ; w ( t ) ; u ( t )) DAE ( OC ). Yet, su h approa h would forget the

P

impose any restri tion on the ontrol

spe ial hara ter of the algebrai part whi h does not

u and would yield a dierent optimality onditions than (4.150)

(4.152) or (4.161) below, involving an additional multiplier and would bring te hni al troubles with failure of ontinuity into

L -type spa e usually required.

4.3 Optimal ontrol of nite-dimensional dynami al systems

Ë 319

DAE results from the previous Se tions 4.3.a- to (POC ). Assuming rather for notational sim-

pli ity that all equations in the ontrolled system have the same index, we formulate the results for the index at most 3, exploiting the onditions (1.47), (1.48), or (1.52) here modied for the ontrol problem. In parti ular, (1.47) reads as

; w òCAR

;

p; q2

(

I , Rn1 , Rm ; Rn2 ) : g(t ; r; v; s) # 0 ã v # w(t ; r; s):

(4.144)

DAE ODE The problem (POC ) is equivalent to (POC ) in the sense that (

u ; y) ò Argmin(PODE OC )

when we use

ã

(

u ; y; w) ò Argmin (PDAE OC )

' # 'ODE , f # fODE , and

#

w # w(y; u)

with

ODE ODE in (POC ) and the exponent

q # q1 with

'ODE (t ; r; s) :# '(t ; r; w(t ; r; s); s) fODE (t ; r; s) :# f(t ; r; w(t ; r; s); s); ODE (

with

', f ,

t ; r; s) :#

(

(4.145)

(4.146a) and

t ; r; w (t ; r; s); s)

(4.146b) (4.146 )

DAE from (POC ); again, we will often omit the expli it dependen e on

and

t as well as write w(y; u) instead of Nw (y; u). Let us further dene the manifold M(t ; x) ’ Rn2 , Rm where the admissible pairs (w; u) respe ting the algebrai state equation take values as

M(t ; r) :#  (v; s) ò Rn2 , S(t); g(t ; r; v; s) # 0  :

(4.147)

One an apply the Filippov-Roxin existen e theory as in Corollary 4.39, leading to:

Proposition 4.68 (Existen e of solutions to (PDAE OC ): index-1 ase). Let (4.144) hold and ODE (POC ) with ' # ' ODE and f # f ODE from (4.146) and q # q and U ad from (4.63b) be 1

nonempty with

S measurable and losed-valued, and (4.70a,b,d,e,g,h)96 hold. Moreover,

let

:

a.a.

t ò I : r ò Rn1 :

QM (t ; r) :#

the orientor eld

†  ' ( t ; r; v; s )%

R

QM (t ; r) ’ R,Rn1 ,R

% ; f(t ; r; v; s) ; 0

(

t ; r; v; s)% D  ò R1%n1 % ; 

(

be losed onvex, where is

dened by

v; s) ò M(t ; r) ‡

(4.148)

M from (4.147). Then the problem (PDAE OC ) has a solution.

Proof. 97 Let us note that the data quali ation allows us to use Proposition 1.37 so that, for a xed ontrol

96

f and g. E.g. (4.70g) an be granted by asf t ; r; v; s) ¢ (a1 % vq2 /q1 % sp/q1 )(1 % r) with some a1 ò L q1 (I).

Using (4.70) imposes ertain quali ation on the data

suming 

97

u ò L p (I; R ), the initial-value problem for the DAE in question has

w t ; r; s ¢ C 1 % (

)

(

r

 ) and  (

Cf. [702, Proof of Prop. 1℄ for details.

Ë

320

4 Relaxation in Optimization Theory

a unique solution. We will employ the transformation (4.146) and aim to use CorolODE lary 4.39 for the transformed problem (POC ). Then (4.79) results, after the substitution

n1 # n and (4.146), and w # w(y; u), to

Q(t ; r) # † '(t ; r; w(t ; r; s); s)% R%0 ; f(t ; r; w (t ; r; s); s) ; (

t ; r; w (t ; r; s); s)% D  ò R1%n1 % ; s ò S(t) ‡ 

# † '(t ; r; v; s)% R% ; f(t ; r; v; s) ;

(

0

t ; r; v; s)% D  ò R1%n1 % ; 

v # w(t ; r; s); s ò S(t) ‡ # QM (t ; r) QM from (4.148). Then, the assumed onvexity of QM (t ; r) results to the onvexity Q, so that, by Corollary 4.39, the (now auxiliary) problem (PODE OC ) has a solution ( u ; y ). DAE Using the impli ation á in (4.145) and putting w # w( y; u ), we get a solution to (POC ).

with of

By ombining the transformation (4.146) with Corollary 4.36, one an formulate DAE also the maximum prin iple for (POC ).98 We will assume

g v (t ; r; v; s) 

is a regular (

!!  "1 ! !![ g v ( t ; r; v; s )℄ !!!

n2 ,n2 )-matrix and

¢ (r);

wr (t ; r; s) " wr (t ; r ; s)!!!! ¢ ~ r " r 

!! !!





1



with some

(4.149a) (4.149b)



ò C(Rn1 ), ~1 ò R, and with w from (4.144).

Proposition 4.69 (Maximum prin iple for (PDAE be ontinuous OC ): index-1 ase).99 Let n n ,  ODE 1 1 with ), and let (4.144) hold and (POC ) with ' # ' ODE v s $ 0 and r ò C ( I ,R ; R and f # f ODE from (4.146) and q # q satisfy the assumptions (4.70) hold. Moreover, let DAE ( u ; y; w ) solve (POC ). Then there are  £ 0 and  òr a(I; R ) with ( ;  ) #Ö 0,  £ 0, (y) -  # 0 on I su h that the following maximum prin iple holds in the sense of L (I): 



0

( ; )

1

*

*

*

0

*

*

*

0

1

*

hy  ;

0 ; *

(

*

t ; w(t); u(t)) #

where the manifold

max h v s òM( t ; y ( t )) y;  0 ;  *

( ; )

(

*

t ; v; s)

for a.a.

tòI ;

(4.150)

M is from (4.147) and the Hamiltonian h y;  : I , Rn2 , Rm Ù *

R is

dened by

h y  (t ; v; s) :#  (t) - f(t ; y(t); v; s) " 0 '(t ; y(t); v; s) *

;

with

 ò BV(I; Rn1 ), *

*

*

solving, together with

 ò L1 (I; Rn2 ), *

(4.151)

the adjoint terminal-value

problem for the linear dierential-algebrai system

98

There is a quite ommon belief in literature [349, 492, 555, 556, 722℄ that one an apply the standard

maximum prin iple to DAEs as usual. This is however not always true, as shown on ounterexamples [263, 702℄.

99

The maximum prin iple involving the manifold

M is from (4.147) has been used e.g. in [351, 702℄.

A dierent manifold and a dierent Hamiltonian has been devised in [350, Theorem 7.1.6℄.

Ë 321

4.3 Optimal ontrol of nite-dimensional dynami al systems

d % f (y; w; u)  % g r (y; w; u)  #  ' r (y; w; u) % r (y)  dt r with  (T) #   r (y(T)) ;   f w (y; w; u)  % g w (y; w; u)  #  ' w (y; w; u) ; *



*



*

*





*

0

*

*



(4.152a)

0



*



*

*



(4.152b)

0

Rn1 -valued measures on I . Moreover, in the un$ 0), one has  # 1,  # 0, and  ò W I; Rn1 .

a tually (4.152a) holds in the sense of

*

onstrained ase (i.e. if

1;1

*

0

(

)

ODE

Proof. 100 We use Corollary 4.36 for the transformed problem (POC ) so that automati ally (

y; i

u

ODE H ( )) solves the relaxed problem (R H POC ), as exploited there. The Hamiltonian

(4.71) gives now

hode y  (t ; s) #  (t) - f(t ; y(t); w (t ; y(t); s); s) " '(t ; y(t); w (t ; y(t); s); s) *

;

(4.153)

*

while the adjoint equation (4.74) gives the adjoint terminal-value problem

d # ['ODE ℄r " [fODE ℄r  # ' r (y; w(y; u); u) % wr (y; u) ' w (y; w(y; u); u) dt " f r (y; w(y; u); u)  " wr (y; u) f w (y; w(y; u); u)  % r (y)  *

*





*



for the adjoint state





*





*

(4.154)

 ò W 1 1 (I; Rn1 ) with  (T) #  (y(T)). Using also the substitution *

;

*



w :# w(t ; y(t); s) turns the Hamiltonian (4.153) into the form h y  (t ; v; s) #  (t) - f(t ; y(t); v; s) " '(t ; y(t); v; s) *

;

*

for

(

v; s) ò M(t ; y(t)) :

(4.155)

Moreover, the maximum prin iple (4.106) then turns into (4.150), the initial-value problem (4.14) together with

w :#

w(t ; y; u) and (1.47) gives just the DAE in (PDAE OC ),

and eventually (4.154) results to

d # ' r (y; w; u) % wr (y; u) ' v (y; w; u) dt " f r (y; w; u)  " wr (y; u) f v (y; w; u)  % *







with

 (T) #  (y(T)). *



*



 *

r (y)

*



(4.156)





Due to (4.149a), there is

*

solving (4.152b), namely

R



#

*

[(

g w ) ℄"1 ((' v ) " 



 *  "1 is in L (I; n2 ,n2 ) and, by (4.8a,e), (' ) " ( f v )  ). Let us note that, by (4.8g), [ g w ℄ w   * 1 n * n 2 ), as laimed. By (1.47), we have 2 ) so that ertainly  ò L 1 ( I ; (f )  ò L (I; 

R

w

R

g(y; w(y; u); u) # 0 so that g r (y; w; u) % g v (y; w; u)w r (y; u) # 0. Using it for (4.152b) 

multiplied by





wr , one gets 

' v (y; w; u) wr (y; u) "  f v (y; w; u) w r (y; u) 



* 



#  g v (y; w; u) wr (y; u) # " g r (y; w; u) : *





Substituting it into (4.156) gives (4.152a). This shows that ( as laimed.

100

Cf. [702, Proof of Prop. 2℄ for details.

*

 ; *

*



) solves the DAE (4.152),

Ë

322

4 Relaxation in Optimization Theory

The ondition (4.144) often annot be fullled be ause the DAEs in question have a higher index. We will demonstrate the needed modi ations rst for the index-2 ase,

g of C 1 (

assuming

)

- lass, and

g v (t ; r; v; s) # 0 

and

g s (t ; r; v; s) # 0 ; 

(4.157)

g # g(t ; r; v; s) depends r.101 Then, using the al ulations (1.49), like (1.50), we modify (4.144),

f. (1.48) for the former relation. In fa t, (4.157) means that only on

t

and

assuming

; w òCAR

;

p; q2

(

I , Rn1 , Rm ; Rn2 ) : v # w(t ; r; s)

ã G(t ; r; v; s) # 0 with G # g t % g r f : 



(4.158)

ODE We again use the transformation (4.146) towards (POC ), and dene the manifold

M(t ; r) :# (v; s) ò Rn2 , S(t); G(t ; r; v; s) # 0 :

Proposition 4.70 (Optimal ontrol of index-2 DAEs).

Let

(4.157)

(4.159)

and

(4.158)

hold.

Then: (i)

QM (t ; x), dened by (4.148) now with M from (4.159), is onvex x and a.a. t, then (PDAE OC ) has a solution. n If also the assumptions of Proposition 4.69 holds and let, for some : R 1 Ù R

If the orientor eld for all

(ii)

ontinuous,

Gv 

be a regular ( n 2

,n

2 )-matrix

! !

! with !!

"1

Gv


(

!

t ; r; v; s)!!!! ¢ (r)

(4.160)

G from (4.159). Then, for any (u ; y; w) solving (PDAE OC ), there are  0 £ 0 and  ò  r a(I; R ) with (0 ;  ) #Ö 0,  £ 0, (y) -  # 0 on I and the maximum prin iple 1 1 n (4.150) is satised with M from (4.160), h y  from (4.151), and  ò W (I; R 1 ) 1 n 2 solving, together with  ò L ( I ; R ), the terminal-value problem for the following *

with

*

*

*

*

*

*

*

;

;

*

*

adjoint DAE:

d % f (y; w; u)  % G r (y; w; u)  #  ' r (y; w; u) % r (y)  dt r with  ( T ) #   r ( x ( T )) ;   f v (y; w; u)  % G v (y; w; u)  #  ' v (y; w; u) ; *



*



*

*





*

0

*

*



0



*



*

*



0

with

(4.161a) (4.161b)

G dened in (4.159).

Proof. It just opies the arguments for Propositions 4.68 and 4.69.

101

The ase

gv

#Ö 0 but singular would lead to various indi es in parti ular equations, whi h would

require a suitable ombination of the presented results.

4.3 Optimal ontrol of nite-dimensional dynami al systems

The derivation of

w and M be omes ompli ated quite rapidly for in reasing

index. Let us show it only for the index Assuming

f

of

C1 (

)

Ë 323

- lass and

g

C2 (

of

)

3 whi h also appears in nontrivial appli ations.

- lass, we now have to suppose, in addition to

(1.48), also

gr fv $ 0 



gr fs $ 0 : 

and



(4.162)

The former ondition, already used as (1.52), means that the DAEs do not be ome a differential equation for the variable

v and again, the latter ondition implies the ausalG in (4.160) as

ity of the DAEs. Then, for the index-3 DAEs as in (1.54), we modify

G # g tt % g rr f 2 % g r f r f % 2g tr f % g r f t : 













(4.163)

Remark 4.71 (Singular perturbations). Repla ing the algebrai part of the ontrolled d system g ( y; w; u ) # 0 by the dierential equation " dt w # g(y; w; u) and adding an DAE ODE initial ondition for w , the problem (POC ) turns into the form (POC ). Assuming " ¡ 0

w being y. The asymptoti s for " Ù 0 is a natural (and very nontriv-

small, this models dynami al systems with two time s ales, the evolution of qualitatively faster than of

ial) question. Su h problems are also alled singularly perturbed and for their limit analysis in the ontext of optimal ontrol see e.g. Z. Artstein [28, 3032℄.

Remark 4.72 (Dire t relaxation of (PDAE OC )). in the fast omponent

w

The mentioned on entrations/os illations

suggests to relax both

u

w

and

simultaneously. This would

lead to the relaxed problem

Minimize

T X 0

subje t to

' Ž y DZ 
(dt) % (y(T))

dy # f Ž y DZ  ; y(0) # y ; dt 0 # g Ž y DZ ; Ž yDZ ¢ 0; q p y ò W q1 (I; Rn1 );  ò Y H2 (I; M(y)) ; 0

1;

;

/ 7 7 7 7 7 7 7

(4.164)

? 7 7 7 7 7 7 7 G

I; M(y)) denotes the onvex - ompa ti ation of {(w; u) ò L q2 (I; Rn2 ) , L p (I; Rm ); (w(t); u(t)) ò M(t ; y(t)) :a.a. t ò I} using H a separable linear subspa e of q p n m the anisotropi  spa e Car 2 ( I ; R 2 ,R ) of Carathéodory integrand with dierent growth restri tion in variables w and s , ontaining the linear hull of ' Ž y , f Ž y , g Ž y , and Ž y. The pe uliarity now is that the set U ad (y) of admissible relaxed ontrols and fast

where

q ;p

Y H2

(

;

states (whi h annot be learly distinguished from ea h other) depends on the slow

y. For index-1 DAEs, the algebrai part is dire tly ontained in the manifold M(y) q p g Ž y DZ  # 0 is fullled automati ally for any  ò Y H2 (I; M(y)). Exploiting w the Nemytski mapping indu ed by w and its ontinuous extension N ( y;  ) of the w Nemytski mapping N ( y;  ) indu ed by w from Lemma 3.100, we may think about

state

and thus

;



Ë

324

4 Relaxation in Optimization Theory

an alternative relaxation

T X

Minimize

0

' Ž y DZ  DZ 
(dt) % (y(T))

dy # f Ž y DZ  DZ  ; y(0) # y ; dt w  # N (y; ); Ž yDZ DZ ¢ 0; q2 p q n n2 m 1 1 yòW (I; R );  ò Y H2 (I; R );  ò Y H (I; R ) ;

subje t to

0



1;

/ 7 7 7 7 7 7 7

(4.165)

? 7 7 7 7 7 7 7 G

H2 ’ Carq2 (I; Rm ) and H ’ Carp (I; Rm ). Here we use semi-bi-ane extension f Ž y DZ  DZ  and ' Ž y DZ  DZ  from Remark 3.113.

with suitable

Example 4.73 (Me hani al des riptor systems).102

A on rete

example

of index-3

DAEs o

urs in so- alled me hani al multi-body des riptor systems using redundant

oordinates being subje ted to some holonomi kinemati onstraints. For a general formulation see e.g. [735℄. Prominent appli ations are industrial robots and their optimal ontrol is typi ally related with traje tory planning. the following autonomous

ase:

M(q)

dq dq d q (0) # q ; % K Œq;  # J(q) w % B(q; u) ; C(q) # 0 ; q(0) # q ; d t dt dt 2

0

2

1

(4.166)

q : I Ù Rn is a time-varying position (traje tory) of the robot, M : Rn Ù Rn,n n n n a regular mass matrix depending on q , the for e K : R , R Ù R involves Coriolis, n 

entrifugal and possibly also fri tion ee ts, C : R Ù R des ribes kinemati onn k , n denoting the Ja obian matrix and straints assumed smooth with J :# C : R Ù R H :# C : Rn Ù Rk,n,n (used later) its Hessian, w : I Ù R is the orresponding

where





Lagrange multiplier expressing the rea tion for es to these onstraints, being in position of the fast variable. The ontrol

u:I ÙS’

B :R ,R ÙR performed by the hoi e: n 1 # 2 n , n 2 # k , n

transmission fun tion

m

Rm a t as applied for es through a

n . Transformation to the DAE in

dq ; g ( t ; r; v; s ) # C ( r ) ; f ( t ; r; v; s ) $ [ f ; f ℄( r; v; s ) with dt f (r; v; s) # r and f (r; v; s) # M " (r )J(r ) v % B(r ; s) " K(r) : y # Œ q;

1

1

2

1

1

2

2

1

1

1

DAE

(POC ) an be

(4.167a) (4.167b)

The onditions in (4.162) are fullled due to the following orthogonality:

g r f v # „C r1 ; C r2 … - „[f1 ℄v ; [f2 ℄v … # (J; 0) - (0; M "1 J  ) # 0 ; 











g r f s # „C r1 ; C r2 … - „[f1 ℄s ; [f2 ℄s … # (J; 0) - „0; B s … # 0 : 

Therefore, if

102













g dierentiated twi e with respe t to time, a

ording to (1.54) we get

For optimal ontrol of me hani al des riptor systems f. e.g. [350, 555, 556, 722℄.

Ë 325

4.4 Ellipti optimal ontrol problems

G(t ; r; v; s) # €g rr f 2 % g r f r f % 2g tr f % g r f t % g tt  (t ; r; v; s) 







# H(r )r % JM " (r 1

of ourse, the term with

2 2

1







 v % B(r ; s) " K(r) 1

J(r1 )

1)

Hr22 $ r H(r1 )r2 ò Rk 2

means [

# 0;

(4.168) 2

Hr22 ℄ # ni#1 nj#1 [ r2i r2 j C  ℄r2 i r2 j ;

;

;

;

 # 1; :::; k. Now the variable v appears in this expression and, if JM "1 J  is regu-

lar, we an express

w # w(y; u) # [JM "1 J  ℄"1(y1 ) „ J(y1 )M "1 (y1 )K(y)" B(y1 ; u) " H(y1 )y22 … : Hen e the manifold

M from (6.4), now time-independent, an be expli itly obtained

in the form:103

M(r) :# (v; s) ò Rk,m : s ò S ; JM "1 (r1 )(K(r)" J(r1 ) v"B(r1 ; s)) # H(r1 )r22 : This proves that the DAEs (4.166) are indeed of the index 3. The ompatibility onditions (1.51) and (1.55) now read simply as position

q0

C(q0 ) # 0 and J(q0 )q1 # 0, i.e. the

initial

of the robot fullls the kinemati onstraints while its initial velo ity

q1

lies in the tangent spa e.

4.4

Ellipti optimal ontrol problems

In this se tion we want to demonstrate appli ations of the presented theory to relaxation of optimal ontrol problems where the state equation is governed by a nonlinear 2nd-order ellipti partial dierential equation in the divergen e form; su h nonlinearity is referred as quasilinear. We want only to illustrate basi te hniques, so that we

onne ourselves to a derivation of a orre t relaxed problem and orresponding optimality onditions; the stability analysis, approximation theory, as well as various

onsequen es for the original problem are more or less parallel to the Se tion 4.3 and are thus left as exer ises.

4.4.a

The original problem and its relaxation

We will onsider both distributed and boundary ontrol but, for simpli ity, we will not impose any state onstraints in most of the exposition; f. Remark 4.84. The boundary

ontrol will a t through a nonlinear onditions of the Robin type, though the reader

103

Here

we

assume

nondegenera y

supy1 òRn det(JM "1 J  (y1 )) ¡ 0.

of

the

holonomi

onstraints

in

the

sense

that

Ë

326

4 Relaxation in Optimization Theory

an ertainly imagine a modi ation for the ase of Diri hlet boundary onditions, as well.104 For

being a domain in Rn , n £ 2, with a Lips hitz boundary

, we will deal with

the following optimal- ontrol problem

ELL

(POC )

Minimize X ' ( y; u d ) d x % X  ( y; u b ) d S (fun tional) . 6 6

6 6 6 6 subj. to div a(y; ›x y) # (y; ›x y ; ud ) on ; (state equation) 6 6 6

"n - a(y; ›x y) # b(y; ub ) on ; (boundary ondition) > 6 6 u (x) ò Sd (x) (: x ò

); (distributed- ontrol onstraints) 6 d 6 6 6 6 ub (x) ò Sb (x) (: x ò ); (boundary- ontrol onstraints) 6 6 1; q m ); u ò L p1 ( ; n1 ); u ò L p2 ( ; n2 ) ; y ò W (

; d b F

a.a. a.a.

R

R

R

x-dependen e of a, b, , and n . Here, p ; p ò 1; %), q ò (1; %), and the Carathéodory mappings a : , Rm , Rm,n Ù Rm,n ,

: , Rm , Rm,n , Rn1 ٠Rm , b : , Rm , Rn2 ٠Rm , ' : , Rm , Rn1 ٠R, and  : , Rm , Rn2 ٠R, and the set-valued onstraint mappings Sd : ± Rn1 and S b : ± Rn2 will be subje ted to ertain data quali ation spe ied later. The

ase n # 1 is ex luded be ause it would not t with the following relaxation s heme sin e the ( n "1)-dimensional Lebesgue measure on is not non-atomi , whi h would

where, for notational simpli ity, we omit

1

2

[

ex lude usage of the theory from Chapter 3. ELL The problem (POC ) ts with the framework of Se tion 4.1 if one takes the data for

the problem (POC ) as follows:

Y # W 1 q ( ; Rm ) ; U # L p1 ( ; Rn1 ) , L p2 ( ; Rn2 ) ; ;

Uad # Ud , Ub

(4.169a)

with

Ud # ud ò L p1 ( ; Rn1 ); :a.a. x ò : ud (x) ò Sd (x) ;

X # W

1;

q

Ub # ub ò L p2 ( ; Rn2 ); :a.a. x ò : ub (x) ò Sb (x) ;

(

(u ; y) ò W

; R 1;

q

(u ; y);

J(u ; y) #

(

m

*

)

; R

(4.169b)

;

(4.169 )

m

*

)

dened for all

y ò Y by 

y # X a(y ; ›x y): ›x y % (y; ›x y ; ud )- y dx % X b(y; ub )- y dS ; 





X ' ( y; u d ) d x





% X (y; ub ) dS :

Let us re all that the state equation

(u ; y) # 0

(4.169d) (4.169e)

with



from (4.169d) is a so- alled

ELL weak formulation of the state equation with the boundary ondition from (POC ).

104

Many authors dealt with ellipti optimal ontrol problems, though mostly in lesser generality

(typi ally with bounded ontrols and/or linear highest-order term). E.g., we refer to Alibert and Raymond [17, 18℄, Bonnans and Casas [131, 132℄, Bonnans and Tiba [135℄, Buttazzo [160℄, Buttazzo and Dal Maso [163℄, Casas [180, 181℄, Casas and Fernández [183185℄, Lions [494℄, Lou [497, 498℄, Ma kenroth [508℄, Raitums [625, 626℄, et .

4.4 Ellipti optimal ontrol problems

Ë 327

ELL Of ourse, the problem (POC ) need not have any solution in general, and thus a

relaxation is desired. To exploit the theory of ellipti equations from Se tion 1.4.b, we will impose the following quali ation on erning the data

a, b, and . As for a, we

again assume (1.63a) and (1.65a). We will use Lemma 3.101 about semi-ane ontinuous extension for the following Nemytski mappings

N ' : L q ( ; Rm ) , L p1 ( ; Rn1 ) Ù L1 ( ) ;

(4.170a)

N  : L q ( ; Rm ) , L p2 ( ; Rn2 ) Ù L1 (

(4.170b)

*

Ž

Ž

)

Ž

;

N b : L q ( ; Rm ) , L p2 ( ; Rn2 ) Ù L(q ")

; Rm ) ; N : L q ( ; Rm ) , L q ( ; Rm,n ) , L p1 ( ; Rn1 ) Ù L q 

(4.170 )

(

*

(

*

R

") ( ; 

m

(4.170d)

)

 ¡ 0; let us remind the notation (1.42) and (1.40) now for q and qŽ, respe

tively. Let us note that that the mapping N in (4.170d) works with the state argument ( y; ›x y ) from an anisotropi  spa e with dierent exponents q and q . A

ording to *

with some

*

(3.192), we an formulate the basi growth/ oer ivity and ontinuity properties

sp1 ¢ '(x ; r; s) ¢ 

'(x ; r; s) " '(x ; r ; s) ¢ 

sp2 ¢ (x ; r; s) ¢

1(



(x ; r; s) " (x ; r ; s) ¢



b(x ; r; s) ¢



b(x ; r; s) " b(x ; r ; s) ¢



(

q Ž"  )



(

x ; r;  ; s) ¢

 q

*

x) % Cr

q



qŽ



(

*

*

(4.171b)





qŽ



/



%

/(

q Ž"  )

Ž

(4.171e)

Ž



*

(

(

% Cq

x) % Cr

q

*

/(

q

*

(4.171d)



/

qŽ





 q q

 r " r  ;

;



x) % Crq "1 % C r q "1 % Csp2









Ž

(

(4.171 )

Ž Ž Ž x) % Crq "1 % C r q "1 % Cs1 q

q "1"  ") (x) % Cr

*

(4.171a)

"1 % C r q "1 % Cs1/q r" r  ; 

Ž

*

(

*

x) % Crq % Csp2 ;

x ; r;  ; s) " (x ; r ;  ; s) ¢

 (

(

q

x) % Crq "1" % Csp2



 (

x) % Crq % Csp1 ; *

1(

")



% Csp1

 r " r  ;

(4.171f)

") ;

(4.171g)



/(

q

*



"1 % C r q "1 % Csp1 /q r" r  *

*



x) % Cq"1 % C  q"1 % Csp1



/



q



  "   ; 

(4.171h)

 ¡ 0 arbitrarily small, C ò R, and with p ò L p ( ) or p ò L p ( ) with spe i exponents p . Moreover, in addition to the quali ation (1.63a) and (1.65a) of a , like

with some

(1.65b) and (1.66), we assume the uniform oer ivity in the sense

;" ¡ 0 :ud ò L p1 ( ; Rn1 ); ub ò L p2 ( ; Rn2 ); y ò W X a ( y; ›x y ) : ›x y

1;

q

(

; Rm ) :

% (y; ›x y; ud )- y dx % X b(y; ub )- y dS £ """"" y"""""W 1 q Rm " 1" ;

(

and the stri t monotoni ity (1.66), i.e. here

:ud ò L p1 ( ; Rn1 ) :ub ò L p2 ( ; Rn2 ) :y; y ò W 

1;

q

(

; R m ) ; y #Ö y : 

;

)

(4.172a)

Ë

328

4 Relaxation in Optimization Theory

X a ( y; ›x y )

" a( y ; ›x y ) : ›x (y" y ) %  (y; ›x y; ud ) " ( y ; ›x y ; ud )-(y" y ) dx 











% X b(y; ub ) " b( y ; ub )-(y" y ) dS ¡ 0; 

(4.172b)



so that the ontrolled system has a uniquely determined response for ea h ontrol pair. Let us rst formulate the Filippov-Roxin-type existen e theory for the original ELL problem (POC ):

Proposition 4.74 (Existen e for (PELL OC )).

Let (1.63a), (1.65a), (4.171), and (4.172) hold. Let

also the orientor elds

R% ; x; r;  ; s  ò R %m ; s ò Sd x  : x ò ; Qb x ; r :#   x ; r; s % R% ; b x ; r; s ò R %m ; s ò Sb x  : xò ; m m , n . Then the optimal- ontrol problem are onvex for any r ò R and  ò R Qd (x ; r; ) :# (

)

' ( x ; r; s )% (

(

)

(

0

(

0

1

)

(

1

))

(

)

(

)

(

a.a.

a.a.

(4.173a)

)

(4.173b)

)

ELL

(POC )

pos-

sesses a solution. The proof of the above assertion straightforwardly modies the arguments in the ELL proof of Theorem 4.29. For this, we need to onstru t a suitable relaxation of (POC ). To

this goal, we will pro eed by the routine way, parallel with the pre eding se tion. As we need to relax here both distributed and the boundary ontrols, we take here two separable linear subspa es

Hd ’ Carp1 ( ; Rn1 ) supposing

Hd to be C( )-invariant and Hb

U ad d # b lH ;

’ YHpd1 ( ; Rn1 )

i (U ) d ;Bd Hd d *

Hb ’ Carp2 ( ; Rn2 ) ;

and

to be

and

C(

(4.174)

)-invariant, and then put

U ad b # b lH ;

i (U ) b ;Bb Hb b *

’ YHpb2 ( ; Rn2 ); (4.175)

where

Bd

;R

L p2 (

and

Bb

L p1 ( ; Rn1 ) and U ad d , U ad b is onvex in Hd , Hb .

refer respe tively to the norm bornologies on

n 2 ). Thanks to the spe ial form (4.169b),

*

;

*

;

Again, we will onne ourselves to the ontinuous ane extensions with respe t to the ontrols, whi h will be guaranteed if the subspa es

Hd and Hb will be su iently

large. Like in Example 4.56, one an here take

Hd # spang-( Ž yŽ›x y) % g ('Ž y ); y; y ò W 1 q ( ; Rm ); g; g ò C( ; Rm ); ;









Hb # spang-(bŽy) % g (Ž y ); y; y ò L 







qŽ

(

; Rm ); g; g ò C( ; Rm ) : 

These spa es are separable if normed by the norms in

Car

p2

(

;R

Carp1 ( ; Rn1 )

(4.176a) (4.176b) and in

n 2 ), respe tively, f. Proposition 3.102. Of ourse, we an take any larger sep-

Carp1 ( ; Rn1 ) and Carp2 ( ; Rn2 ) for Hd and Hb , respe tively. q ( ; R m ) Ù R by Then we dene the extended ost fun tional J : H d , H b , W

arable linear subspa es of

*

*

1;



J (d ; b ; y) #

X



[

' Ž y DZ d ℄(dx) % X [ Ž y DZ b ℄(dS)

(4.177)

4.4 Ellipti optimal ontrol problems

with on



' Ž y DZ d and



mapping

and

 Ž y DZ b

Ë 329

being understood, if needed, in the sense of measures

, respe tively. Furthermore, one an dene the extended state-equation

 : Hd , Hb , W 1 q ( ; Rm ) Ù W 1 q ( ; Rm ) *

*

;

;

*



by

y µ # X a(y; ›x y) : ›x y % ( Ž yŽ›x y DZ d )- y dx % X (b Ž y DZ b )- y dS :

´  (d ; b ; y);











We an see that

(4.178)

 (d ; b ; y) # 0 is just equivalent to saying that y ò W 1 q ( ; Rm ) is ;



the weak solution of the boundary-value problem

div a(y; ›x y) # Ž yŽ›x y DZ d "n - a(y ; ›x y) # b Ž y DZ b

on on

; :

§

(4.179)

This leads us to the following relaxed optimal ontrol problem Minimize subj. to

X



[

' Ž y DZ d ℄(dx) %

X

[

 Ž y DZ b ℄(dS) ;

/ 7 7 7 7 7

div a(y; ›x y) # Ž yŽ›x y DZ d on ; "n - a(y; ›x y) # b Ž y DZ b on ; d ò U d ; b ò U b ; y ò W q ( ; Rm ) 1;

with

U d ’ YHd1 ( ; Rn1 ) p

and

U b ’ YHb2 ( ; Rn2 ). p

(4.180)

? 7 7 7 7 7 G

Then our additional oer ivity and

monotoni ity assumptions (4.172) guarantees the orre tness of the resulted relaxation s heme.

Lemma 4.75 (Corre tness of the extended state problem). Let p ; p ò [1; %), q ò (1 ; %) (1.63a), (1.65a), (4.171e-h), and (4.172) be satised, let H d and H b be from 1

(4.176). Then: (i)

The extended state problem (4.179) possesses for any

2

 # (d ; b ) ò YHd1 ( ; Rn1 ) , p

YHb2 ( ; Rn2 ) a unique weak solution y #  () ò W 1 q ( ; Rm ). p

;



W

1;

R

q ( ;

R

R

mapping  : YHpd1 ( ; n1 ) , YHpb2 ( ; n2 ) Ù m ) thus dened is (weak*,weak)- ontinuous if restri ted on the losure

(ii) The relaxed- ontrol-to-state



of bounded sets.

d # i Hd (ud ) with ud ò L p1 ( ; Rn1 ) and b # i Hb (ub ) with ub ò L p2 ( ; Rn2 ), ELL then y #  (  ) solves the original boundary-value problem in (POC ). In other words, p n p n 1 q 1 1 2 2  Ž (i Hd , i Hb ) #  where  : L ( ; R ) , L ( ; R ) Ù W ( ; Rm ) denotes the

(iii) If



;



original ontrol-to-state mapping.

d ò YHd1 ( ; Rn1 ) and b ò YHb2 ( ; Rn2 ), there are sep n p n quen es { u d k } k òN and { u b k } k òN bounded in L 1 ( ; R 1 ) and L 2 ( ; R 2 ) su h that i Hd (ud k ) Ù d weakly* in Hd and i Hb (ub k ) Ù b weakly* in Hb , respe tively. To p

Proof. By the very denition of ;

p

;

*

*

;

;

prove the existen e of the solution to (4.179), we shall just pass to the limit with the solutions

yk

that orresponds to (

ud k ; ub ;

;

k ), whi h means ea h

y k ò W 1 q ( ; Rm ) is ;

the weak solution to the boundary-value problem

div a(y k ;›x y k ) # (y k ; ›x y k ; ud k ) "n - a(y k ;›x y k ) # b(y k ; ub k ) ;

on

;

;

on

:

¯

(4.181)

Ë

330

4 Relaxation in Optimization Theory

This means that, for any

X a ( y k ; ›x y k ) : ›x y



y ò W 1 q ( ; Rm ), it holds ;



% (y k ; ›x y k ; ud k ) - y dx % ;



X

b(y k ; ub

;

k)

y dS # 0 :

-

(4.182)



ud k ; ub k ), our assumptions ensure the existen e of y k ò W 1 q ( ); see Proposition 1.41. The sequen e {y k }kòN is 1 q bounded in W ( ), whi h an be shown by putting y # y k into (4.182) and using the uniform oer ivity (4.172a). We used also that the sequen es { u d k L p1 ;Rn1 } k òN and { u b k L p2 ;Rn2 } k òN are bounded. Then, taking possibly a subsequen e (denoted, for By the lassi al theory, for any (

;

;

;

just one weak solution ;



(

;

(

;

)

)

simpli ity, by the same index), we an suppose that

yk Ù y

weakly in

W 1 q ( ; Rm ) : ;

(4.183)

 ¡ 0, we an rely on that the lower-order part is ompa t in W 1 q ( ; Rm ) . Then, using the uniform monotoni ity (1.65a) of the highest-order part "div a(y; ›x y), we an improve the weak onvergen e (4.183) into the strong onverUsing (4.170 ,d,) with ;

*

gen e. Then it is easy to pass to the limit in the integral identity (4.182) to obtain

X a ( y; ›x y ) :

for any

›x

y % ( Ž y Ž ›x y DZ d ) - y dx % X (b Ž y DZ b ) - y dS # 0 



(4.184)



y ò W 1 q ( ; Rm ). Thus we an see that y is the weak solution to the relaxed state ;



problem (4.169) and, by (4.172b), it is determined uniquely and (4.183) holds even for the whole sequen e. The statement (i) has been thus proved. One an easily verify that the whole pro edure works also for

p

dk ò YHd1

;

% ( ;

Rn1

)

; Rn2 ) with % ò R% arbitrary but xed. Hen e the point (ii) holds true, as well. Also the point (iii) is obvious. Å p2 k and  b ò Y Hb ; % (

Proposition 4.76 (Corre tness of the relaxation s heme). Let p ; p ò [1; %), q ò (1 ; %), U ad be nonempty,105 let (4.174), be valid, and (4.171) Then: 1

(i)

2

The relaxed problem (4.180) has a solution.

d and b , i.e. every solution to (4.180), are p1 - and p2 -non on entrating, respe tively, and an be attained by a minimizing adELL 1 missible sequen es for the original problem (POC ) whi h have relatively L -weakly

(ii) Every every optimal relaxed ontrols

ompa t energy. (iii) Conversely, a limit of every minimizing admissible weakly* onverging sequen e for ELL

(POC )

(embedded via i H d

, i Hb into Hd , Hb ) solves (4.180). *

*

Proof. First, let us noti e that the extended ost fun tional

J



in (4.180) takes the form

J (d ; b ; y) # # X b(y) - v dS

: v ò H ( ; Rm ) 2

d ò U ad ;  ò Y H2 ( ; Rm , R ) :

/ 7 7 7 7

(4.188)

? 7 7 7 7 G

ELL The minimum of (4.188) exists and is ertainly below the inmum of (POC ). For the

 ò Y H2 ( ; Rm , R ) satisfying <  ; h v > # P b ( y ) - v d S , there exists a sequen e {( y k ; u k )} k òN attaining  weakly* in H and with y k solving the the boundary-value problem
a ( y k ) # ( y k ; u k ) on . equality (i.e. no relaxation gap), it is important that, for any

*



This does not seem known, however; f. [710℄. Another appli ation might be in optimal ontrol of the Navier-Stokes equation with the onve tive term (

y - ›)y written as %div(y‹y). If % ¡ 0 is onstant, this allows 1 2 n y -
v % %(y‹ y):›x y " (y; u)- v dx # 0 on Wdiv ( ; R ); 0

for the very weak formulation P

;

f. p. 48 for the notation. Yet, again, the zero relaxation gap is not obvious.

;

4.4 Ellipti optimal ontrol problems

4.4.b

Ë 333

Optimality onditions in semilinear ase

The further task on erns the optimality onditions. The problem will be onsiderably simplied106 if we onne ourselves to to a semilinear system when onsidering

a lin-

ear, i.e. [

a(x ; )℄ij #

n H a ijkl ( x )  kl k ; l #1

with

aòL



(

; R m,n (

2

)

)

:

(4.189a)

a(x) instead of a(x ; ). Assuming again the uniform monotoni ity a(x) will be assumed positive denite uniformly in x. We will thus naturally

onsider q # 2 and strengthen data quali ation (4.186) as:

Then we will write (1.65a),

Žp

1 2 m n1 m n2 ' òCAR Hd di ( , R , R ; R);  òCAR Hb di ( ,R ,R ; R) ;

p

*

2 ;

;1

2 ;

;

")

p 1 ;(2

òCAR Hd di 2;

*

;

Žp

2 b òCARHb di 2 ;

;1

;



Ž"

;(2

(

,R ,R



)

;

(

m

m,n

,R ; R n1

m

)

;

(4.189 )

, Rm , Rn2 ; Rm ) :

Without any loss of generality, the spa es

(4.189b)

(4.189d)

Hd

Hb

and

an be supposed separable

and suitably normed; e.g. we an always take respe tively the universal norms from

Carp1 ( ; Rn1 ) and Carp2 ( ; Rn2 ), f. Example 3.76. These data quali ation already enables us to formulate the optimality onditions in terms of the pointwise maximum prin iples both for the distributed and the boundary ontrols.

Proposition 4.80 (Maximum prin iples). Let p ; p ò [1; %), (4.174), (4.172) with q # 2, (4.189) hold, and let (d ; b ; y) be an optimal relaxed ontrol, i.e. a solution to (4.180). Ž Besides, let 2 £ 3 and 2 ¢ 3 hold.107 Then the following integral maximum prin iples 1

2

*

hold:

y; 

y; 

hd DZ d ℄(dx) # sup X hd

ud òUd

X X

[

[

y; 

*

hb DZ b ℄(dS) # sup *

ub òUb

where the distributed Hamiltonian

y; 

hd

X

y; 

hb

*

(

*

(

x ; ud (x)) dx ;

x ; ub (x)) dS ;

*

and the boundary Hamiltonian

(4.190a)

(4.190b)

y; 

hb

*

are

given respe tively by the formulae

y; 

hd

*

(

x ; s) #  (x)- (x ; y(x); ›x y ; s) " '(x ; y(x); s) ; *

106

The general quasilinear ase has been treated by Casas and Fernández [183℄.

107

This ondition omes from the requirement 

(4.191a)

2 ¢ q used in Lemma 3.103, whi h sounds here ¢ q "  and 2qŽ ¢ qŽ " . , We an see that this requirement an be fullled only for three n with n ¢ 3) or, if the boundary ontrol would not be (or less) dimensional problems (i.e. ’ R

onsidered, for n ¢ 5. as

2q

*

*



Ë

334

4 Relaxation in Optimization Theory

y; 

hb

*

(

x ; s) #  (x)- b(x ; y(x); s) " (x ; y(x); s) ; *

(4.191b)

 ò W 1 2 ( ; Rm ) solving in the weak sense the adjoint boundary-value problem *

with

;

"divŒa ›x  %[  Ž y Ž ›x y DZ d ℄   %[ r Ž yŽ›x y DZ d ℄  # ' r Ž y DZ d n - Œa ›x  %[  Ž y Ž ›x y DZ d ℄   %[b r Ž y DZ b ℄  #  r Ž y DZ b 

*

*



*



*



on

;

/ 7 7 7 7 7

:

? 7 7 7 7 7 G

*



*



on

(4.192)

' and  is satised and if Sd and Sb are measur and

Moreover, if the oer ivity (4.171a, ) of

able losed-valued, then also the following pointwise maximum prin iples on hold:

y; 

y; 

hd DZ d
(x) # sup hd *

sòSd (x)

y; 

y; 

hb DZ b
(x) # sup hb *

sòSb (x)

*

x ; s)

in the sense of

L1 ( );

(4.193a)

x ; s)

in the sense of

L1 ( ):

(4.193b)

(

*

(

Proof. We will apply Theorem 4.15 together with Theorem 4.21. Note that, by Lemma 4.75,

ontinuous if

 is (weak*,weak)- ontinuous, and therefore also (strong,strong)Y # W 1 2 ( ; Rm ) is endowed with the relativized norm of L2 ( ; Rm ),

*

;

the respe tive dierentials of the nonlinear terms remaining ontinuous with respe t to this weaker norm, as well.108 For larity we divide the proof into separate steps.

Step 1. (Smoothness of

 (-; -; y)

and



J (-; -; y).)

By the assumption (4.189d) and by us-

ing Lemma 3.103, the state-equation mapping

 (-; -; y) : Hd , Hb Ù W 1 2 ( ; Rm ), *

*

;



dened by (4.178), is linear and weakly* ontinuous, and has the Fré het derivative ›

 (  d ;  b ; y ) ò L( H d , H b ; W 1 2 ( ; R m ) *

*

;

*



 b ); y µ #

´€›   (  d ;  b ; y ) (  d ;







) given by the formula

X [

Ž yŽ›x y DZ  d ℄ y dx % X [b Ž y DZ  b ℄ y dS 







 d ò Hd ,  b ò Hb , and y ò W 1 2 ( ; Rm ). Moreover, by Lemma 3.103 this dierential is jointly ontinuous with respe t to the strong topologies on H d and H b and the 1 2 m weak topology of W ( ; R ); re all that we have the ompa tness of the embedding 1 2 m q "  m 1 2 m q Ž"  ( ; R m ), reW ( ; R ) ’ L ( ; R ) and of the tra e operator W ( ; R )Ù L spe tively. By (4.189d), always Ž y Ž›x y ò H d and b Ž y ò H b so that ›   (  d ;  b ; y ) is for any

*



*

;





*

*

;

*

;

;



(weak*,weak)- ontinuous, as required in Theorem 4.15.

J d ; b ; y) ò L(Hd , Hb ; R) is given *

Analogously, by (4.189b) and Lemma 3.103, ›  (

*

by the formula

€›  J (  d ;  b ; y ) (  d ;



108

b ) # 

X '

Ž y DZ  d dx % X  Ž y DZ  b dS 



(4.194)

In fa t, we again treat the linear and the nonlinear parts separately, using the original norm of

W 1;2 ( ; as well.

Rm

) for the linear part. Note that the proof of Lemma 1.59 works in this generalized situation,

Ë 335

4.4 Ellipti optimal ontrol problems

 d ò Hd ,  b ò Hb . By (4.189b), always ' Ž y ò Hd and  Ž y ò Hb so d ; b ; y) ò that ›  J (  d ;  b ; y ) is weakly* ontinuous, whi h implies here that ›  J ( Hd , Hb ’ Hd , Hb Ê L(Hd , Hb ; R) and we an obviously write *

for any

*







**

**

*

*

J d ; b ; y) # (' Ž y;  Ž y) ò Hd , Hb :

› (

Again, by Lemma 3.103 this dierential is jointly ontinuous.

 (d ; b ; -) and J ( d ; b ; -).) By (4.189b, ) and Lemma 3.103 and by the assumed linearity of a , we an see that these mappings are dierentiable and 1 2 m 1 2 m 1 2 m › y  (  d ;  b ; y ) ò L( W ( ; R ); W ( ; R ) ) is given, for any y ; y ò W ( ; R ), Step 2. (Smoothness of



;

;

*

;





by the formula

yµ # X

´[› y  (  d ;  b ; y )℄( y ) ;







a(x) % (  Ž yŽ›x y DZ d )›x y  :

”Œ›x y :







›x

y 

% ( r Ž yŽ›x y DZ d ) y - y • dx % X (b r Ž y DZ b ) y - y dS ; 









J d ; b ; y) ò L(W 1 2 ( ; Rm ); R) Ê W 1 2 ( ; Rm ) ;

while › y (

;

*



yµ #

´› y J (  d ;  b ; y ) ;



X (' r



Ž y DZ d ) - y dx %

X



(



is determined by

 r Ž y DZ b ) - y dS : 

(4.195)



Again, by Lemma 3.103 and our data quali ation (4.189b, ), these dierentials are jointly ontinuous with respe t to the strong topologies on topology of

W

1;2

(

; R

m ).

Let us note that › y

 (d ; b ; y)

Hd

*

and

Hb

*

and the weak

possesses a bounded inverse as required in The-

orem 4.15. This follows by the lassi al results about solutions of linear ellipti boundary-value problems with the oe ients

b r Ž y DZ b ò 

Ž Lq



"qŽ ) ( 

/(2

r Ž y-›x y DZ d ò L q 

*

"q

/(2

*

)

(

)

and

); the integrability omes from the growth onditions

(4.189 ,d), while the solvability and boundedness of the inverse is due to (4.172).

Step 3. (The adjoint problem.) The adjoint equation (4.23b) with

0 # 1 and  # 0 an *

*

be written here as

´[› y  (  d ;  b ; y )℄( y ) ;  µ *





#

´› y J (  d ;  b ; y ) ;

yµ ; 

whi h results here to the integral identity

X Œ›x 

*

: a%

*

- (

 ŽyŽ›x y DZ d ) : ›x y % Œ 



*

- (

% X Œ

*

r Ž yŽ›x y DZ d ) " ' r Ž y DZ d  - y dx 





- (

b r Ž y DZ b ) "  r Ž y DZ b  - y dS # 0 : 





However, this is nothing else than the weakly-formulated boundary-value problem (4.192). By the lassi al results about linear ellipti equations, we an see that this adjoint boundary-value problem has always a unique solution

 ò W 1 2 ( ; Rm ); note that the *

;

Ë

336

4 Relaxation in Optimization Theory

right-hand side in (4.192) determines a tually a ontinuous fun tional on

qŽ

W 1 2 ( ; Rm ) ;

; Rm ) and  r Ž y DZ b ò L ( ; Rm ) thanks to the Ž growth ondition (3.197b) with ( q; p ;  ) # ( q "  ; p 1 ; 1) and ( q; p ;  ) # ( q "  ; p 2 ; 1), ' r Ž y DZ d ò L q 

be ause always



*



(

*

respe tively.

Step 4. (The Hamiltonians.) The abstra t Hamiltonian (4.23d) an be now written in the

f

form y;  *

# hdy  ;

y;  ¼  d ; hd ½ *



*

with

y; 

hd ò Hd determined by the identity

# ´ ; [›d  (d ; b ; y)℄(  d )µ " *

[› 





#X



„ [

y; 

hd

*

d J (d ; b ; y)℄(  d )



Ž yŽ›x y DZ  d ℄ " ' Ž y DZ  d … dx #  d ; ( Ž yŽ›x y) -  " 'Žy

whi h is to hold for any Hamiltonian

*

*

*







 d ò Hd . This gives the expression (4.191a) for the distributed *



.

The boundary Hamiltonian (4.191b) an be determined just analogously, with the transformation



a ting on

in pla e of

.

Step 5. (Lo alization of the maximum prin iples.) The maximum prin iples (4.190a) and (4.190b) an be transformed respe tively into the forms (4.193a) and (4.193b) by means of Theorem 4.21. Let us treat the rst ase and verify the ondition (4.36). It is satised trivially when

Sd (x) are bounded independently of x. Also, by the oer ivity ondition (4.171a) on ' and by the growth ondition (4.171g) on , we an estimate the distributed Hamilto-

all

nian as follows

y; 

hd

*

(

x ; s) # "'(x ; y(x); s) %  (x) (x ; y(x); ›x y(x); s) % d (x) *

¢ "a(x) " bsp1 %  (x) a (x) % b y(x) 1 % sp1 *

1

1

/

1

q

*

 % d (x);

) and b1 ; 1 ò R% ome from (3.192b). This implies already the des ent p 2 an be estimated from above by

ondition (4.36) be ause the term   ( x ) 1  s  1 1 p 2 bs % C with C large enough depending on 2 , b and 1  (x) (re all that always 2   % sin e 2 ¡ 1) and then absorbed in the term "bsp1 . Having (4.36) at our

where

a1 ò L q

*

(

*

/

*

*

*

*

*

disposal, we an use Theorem 4.21(i) to get (4.193a). The boundary Hamiltonian an be estimated analogously by means of (4.171 ,e),

Å

and then (4.193b) immediately follows.

Remark 4.81 (Classi al maximum prin iples).

If it happens, by han e or under the

d ; b ) lives in the original spa es d # i Hd (ud ) and b # i Hb (ub ) for some ud ò Ud and ub ò Ub , then the

ondition (4.173), that the relaxed optimal ontrol ( in the sense that

maximum prin iples (4.193a) and (4.193b) take respe tively the form

y; 

:a.a. x ò :

hd

:a.a. x ò :

hb

y; 

*

(

*

(

y; 

x ; ud (x)) # max hd

*

sòSd (x)

y; 

x ; ub (x)) # max hb sòSb (x)

(

*

(

x ; s) ;

x ; s) ;

(4.196a)

(4.196b)

Ë 337

4.4 Ellipti optimal ontrol problems

while the adjoint boundary-value problem then looks as

"divŒa ›x  %  (y; ›x y; ud )   % r (x ; y ; ›x y) ; ud )  # ' r (x ; y; ud ) n - Œa ›x  %  (y; ›x y; ud )   %b r (x ; y; ub )  #  r (x ; y; ub ) 

*

*



*



*



on

;

/ 7 7 7 7 7

:

? 7 7 7 7 7 G

*



*



on

(4.197)

These are the  lassi al Pontryagin's-type maximum prin iples whi h one an expe t in the ellipti optimal ontrol problems.109 In parti ular, they take pla e when-

Sd (x), Sb (x), '(x ; r; -) (x ; r; -) are onvex and (x ; r; s ; -) and b(x ; r; -) are ane; f. also Example 4.57.

ever the problem has a linear/ onvex stru ture, whi h means and

Remark 4.82 (More general ost fun tionals). One ould also onsider ›x y involved J(ud ; ub ; y) # P '(x ; y; ›x y; ud ) dx % P (y; ub ) dS. The q q p1 m m , n , R n 1 ; R). The quali ation of ' is, instead of (4.189b), ' òCAR H d di ( , R , R in the ost fun tional, i.e.

*

;

;

;1

;

al ulus (4.195) would be then enhan ed as

´› y J (  d ;  b ; y ) ;

yµ # 

#

X (' r



Ž y Ž ›x y DZ d ) - y % ('  Ž y Ž ›x y DZ d ) - ›x y dx 





%

X Œ( ' r



X

(

 r Ž y DZ b ) - y dS 



Ž y Ž ›x y DZ d ) " div('  Ž y Ž ›x y DZ d ) - y dx

%X





Œ(  r 

Ž y DZ b ) % n ('  Ž y Ž ›x y DZ d ) - y dS : 



From this, we an read the orresponding adjoint boundary-value problem, whi h thus enhan es (4.192) as

"divŒa ›x  %[  Ž y Ž ›x y DZ d ℄   % [ r Ž yŽ›x y DZ d ℄  # ' r Ž y DZ d " div('  Ž y Ž ›x y DZ d ) 

*

*







n - Œa ›x  %[  Ž y Ž ›x y DZ d ℄  *



*

%

*

[

b r Ž y DZ b ℄  



Remark 4.83 (Multi riteria generalization).

;

/ 7 7 7 7 7

:

? 7 7 7 7 7 G

*

#  r Ž y DZ b % n ('  Ž y Ž ›x y DZ d ) 

on

on

(4.198)

Analogously to Remark 4.44, one an on-

' and  ve tor-valued or a multi riteria modi ation of (PELL OC ) to minimize (in a Pareto or Slater sense) both P ' ( x ; y; ›x y; u d ) d x and P  ( y; u b ) d S . A s alarization as

sider

in Remark 4.18 gives the s alar-valued ost fun tional in (4.169e) as

J(u ; y) # J(ud ; ub ; y) # F”X '(y; ud ) dx ; X (y; ub ) dS•

109

See Bonnans and Casas [131, 132, 181℄ for the ase of bounded ontrols.

338

Ë

4 Relaxation in Optimization Theory

for some (in general nonlinear) smooth (4.169e) would then read as

F :

R Ù R. The relaxed ost fun tional 2

J (d ; b ; y) # F( *





#

J d ; b ; y); y >, yields the boundary-value problem (4.205) if written in the lassi al formulation.116 This an be shown by testing the adjoint equation by  and using P ( y - ›x  ) -  d x # 0. Thus we an estimate

# P ›x z - ›x y % (
y)z dx # P (›x y - n )y dS # 0. Alternatively, the Neumann y - n # 0 and V # H 1 2 ( ) , L2 ( ) works equally. Hen e the uniform monotoni ity of A used before must be repla ed by other arguments. Here, we d d 1 1 2 2 have < y ; y > # dt dt P 2 ›x y % 2 z dx, whi h gives bounds for y # (y; z) in L (I; V) d provided the initial ondition y 0 ò V and, by omparison, also bounds for dt y. This have
# # 







g(t ; x) # g (t) and y(t ; x) # [y(t)℄(x). 

Remark 4.100 (Paraboli -equation interpretation of optimality onditions).

Also, the

abstra t adjoint problem (4.232) bears an interpretation as a lassi al terminalboundary-value problem for the (ba kward) paraboli equation when realizing the

f Ž y DZ  ℄  ò L (I; W 

*

meaning of the term [ r

´

2

*

1;2

(

; Rm )

*

) in (4.232) as

f rr Ž y DZ 
 ; yµ # ´  ; f r Ž y DZ 
yµ 

*

*

*







#

T * X X  0

(

r Ž y Ž ›x y DZ ) - y %  



*

- (

 Ž y Ž ›x y DZ ) : ›x y dxdt 



T %X X 0



*

- (

b r Ž y) - y dSdt ; 



4.5 Paraboli optimal ontrol problems

Ë 355

 # Q  is the restri tion of  with Q from Lemma 4.106 below,  (t ; x) # [  ( t )℄( x ) with  ò L2 (I; V)  W 1 2 (I; V ), and y (t ; x) # [y (t)℄(x) with y ò 2 1 2 m L (I; W ( ; R )). More spe i ally,  # Q  while y # Q y and  # Q with some  ò L2 (I; W 1 2 ( ; Rm ))  W 1 2 (I; W 1 2 ( ; Rm ) ), where Q from Lemma 4.106 below. 2 1 2 m Thus, ounting similarly that 'r Ž y DZ  ò L ( I ; W ( ; R ) ) bears interpretation as 2 2 m ' r Ž y DZ  ò L (I; L ( ; R )) in the sense that *

where

*

*

*

;

*



;



*

*

;

;



*



;

*



*



;

*



*



´ 'r 



with

y (t ; x) # [y (t)℄(x) 



T

Ž y DZ  ; yµ # X 

0

X (' r



for any test fun tion

Ž y DZ ) - y dxdt 

y ò L (I; V) with V # W 2

1;2



(

; Rm ), the

abstra t adjoint problem (4.232) means, at least formally,

 % divŒa ›x  % [  Ž y Ž ›x y DZ ℄   t %[ r Ž y DZ ℄  # ' r Ž y DZ  n - (a ›x  )%b r (y) # 0  (T; -) #  (y(T; -)) *

*



*



*

*

*

I , ; I, ; in :



in



*



on

The orresponding maximum prin iple (4.233) holding a.e. on

/ 7 7 7 7 7

(4.240)

? 7 7 7 7 7 G

I ,

an also be ob-

tained by transfer of the maximum prin iple from Proposition 4.91 on the oarser re-

. In parti ular, H from (4.261), must be of the form

laxation by Proposition 4.10 and further lo alization on nian (4.231), belonging now to

h y  (t ; s) # Qh y  

;

*

;


( t ; s )

*

#

X h y; 

(

*

the Hamilto-

t ; x ; s(x)) dx

h y  òCarp (I , ; R ). One an also modify the al ulus (4.84) with f (y ; s) " Ay in pla e of f (y ; s), obtaining that the Hamiltonian (4.231) augmented by the term <  ( t ) ; A y ( t )>, i.e.

with some

;

*

*

h y  ( t ; s) # ;

*



*

t ; f (y(t); s) " Ay(t) " ' (y(t); s) ;

( )



is onstant in time along optimal traje tories for autonomous systems, i.e. if and

a, b, ,

' do not depend on time.

4.5.b

An approa h through paraboli partial dierential equations

In the full generality, the previous onvex ompa ti ation is non-metrizable and thus requires non- ountable general-topologi al tools. Although being appli able to paraboli problems as outlined in Remarks 4.99 and 4.100, it is worth to exploit alternatively, instead of the Bana h-spa e-valued dynami al system, the hara ter of a paraboli equation on the spa e/time domain and allow fast os illations in ontrol not only in time but in time and spa e.

Ë

356

4 Relaxation in Optimization Theory

PAR We thus onsider dire tly the problem (POC ) from the viewpoint of the abstra t

optimal- ontrol problem (POC ) with the data

Y # L q (I; W 1 q ( ; Rm ))  W 1 q (I; W 1 q ( ; Rm ) ;

;

U # L (I , ; R ) ; Uad # p



(u ; y) ò X ò Y

*

;

 u ò U;

given, for every

*

)

;

(4.241a)

:a.a. (t ; x) ò I , : u(t ; x) ò S(t ; x)  ;

y ò Y;







T X X ' ( y; u ) d x d t 0

J(u ; y) # and

%

(4.241b)

by



T X X ( a ( y; ›x y ) : ›x y % ( y; ›x y; u ) 0

T % X X b(y) - y dSdt % X y(T) 0

y #

(u ; y);



X  ( y ( T )) d x

y dxdt t

y"y-





-

y (T) " y0 - y (0) dx ; 

(4.241 )



;

(4.241d)

B # 0, so that we have no state onstraints here. Again, the problem (PPAR OC ) need

not have any solution in general and its relaxation is desirable. We will pro eed by the routine way, taking here a linear subspa e

H ’ Carp (I , ; R ); supposing

H to be separable and C(I , )-invariant, and then putting

U ad # b lH B

where

(4.241b),

*

;

B i H ( U ad )

denotes the norm bornology on

U ad is onvex in H q" ; p;1

' òCAR H a òCAR

((

*

(

L p (I , ; R );



thanks to the spe ial form

 òCAR2 1 ( , Rm ; R) ; ;

,Rm ,Rm,n ; Rm,n ) ; b òCAR

q " ; q ; p ;( q ""  )

òCAR H (( I , )

(4.242)

. The natural data quali ation looks as

I , ) , Rm , R ; R) ;

q" " ; q; q

’ Y Hp (I , ; R ) ;

q # ;( q # "  )

((

I,

)

(4.243a)

, Rm ; Rm );

, Rm , Rm,n , R ; Rm )

(4.243b) (4.243 )

 ¡ 0; we again use Notation 3.105 on p. 233 with the exponents q" # and q from (1.79) with q instead of p . One an then dene the ontinuously extended ost fun tional J : H , Y Ù R by

with some small

*



J ( ; y) #

with

T X X [' 0

Ž y DZ ℄(dxdt) % X (y(T; -)) dx

(4.244)



' Ž y DZ  being understood, if needed, in the sense of measures on I , . Further : H , Y Ù X by

more, one an dene the extended state-equation mapping

 ( ; y);

y # 

*



T y X X a ( y; ›x y ) : ›x y " ( Ž y Ž›x y DZ  ) - y " y t 0

T % X X b(y) - y dSdt % X y(T) - y (T) " y0 0











dx dt -

y (0) dx ; 

(4.245)

4.5 Paraboli optimal ontrol problems

Ë 357

y ranges L q (I; W 1 q ( ; Rm ))  H 1 (I; L2 ( ; Rm )). We an see that  ( ; y) # 0 is q 1 q m 1 q 1 q m just equivalent to saying that y ò L ( I ; W ( ; R ))  W (I; W ( ; R ) ) is the ;

where





;



;

;

*

weak solution of the initial-boundary-value problem

y " div a(y; ›x y) % Ž yŽ›x y DZ  # 0 t n - a(y; ›x y) % b(y) # 0 y(0; -) # y0

in

I , ;

I, ; in : on

This leads us to the relaxed optimal ontrol problem (4.239) with stead of

Q U ad . *

/ 7 7

(4.246)

? 7 7 G

U ad

from (4.242) in-

Proposition 4.101 (Corre tness of the relaxation s heme). Let H be a separable C(I , )-invariant subspa e of Carp (I , ; R ), (4.219) together with (4.243) be satised, S admits a sele tion from L p (I , ; Rm ), and  be bounded from below. Then:

The relaxed problem (4.239) onsidered now with

(i)

has a solution. (ii) Every solution to (4.239) is

U ad from (4.242) instead of Q U ad *

p-non on entrating and an be attained by a minimizing PAR L1 -weakly

admissible sequen e for the original problem (POC ) whi h has relatively

ompa t energy.

(iii) Conversely, a limit of every minimizing admissible weakly* onverging sequen e for PAR

(POC )

(when embedded via i H ) solves (4.239).

’ L p (I , ; R )

Proof. First, let us take a bounded sequen e { u k } k òN

su h that

i H (u k ) Ù  weakly*. Furthermore, let y k denote the unique weak solution to the initialPAR boundary-value problem in (POC ) orresponding to u # u k , i.e. y k (0 ; -) # y 0 and, for 1 m any y ò C ( I , ; R ), it holds



T y dxdt X X a ( y k ; ›x y k ) : ›x y " ( y k ; ›x y k ; u k ) - y " y k t 0

T % X X b(y k ) - y dSdt % X y k (T) - y (T) " y0 - y (0) dx 0













u

f. Proposition 1.44. Thanks to the boundedness of { k } k òN in

y

thanks to (4.220a), also the sequen e { k } k òN

W 1 q (I; W 1 q ( ; Rm ) ;



;

*

# 0;

(4.247)

L p (I , ; R )

q 1; q is bounded in L ( I ; W ( ;

and

Rm  ))

); f. the estimates in the proof of Proposition 1.44. Thus, we

an suppose that, after taking possibly a subsequen e for a moment, it onverges

y. By the Aubin-Lions ompa t-embedding interpolated " L (I; L2 ( ; Rm )), it onverges strongly in L q " (I , ; Rm ). As the Nemytski " "

b q m q # "  ( I , ; R m ) due to mappings N and N are bounded into L ( I , ; R ) and L (4.243b, ), respe tively, we an improve the onvergen e of ›x y k Ù ›x y to be strong in L q (I , ; Rm,n ) when exploiting the uniform monotoni ity (1.65a) of a(x ; r; -), f. again

weakly in this spa e to some also with



(



)

(



)

Proposition 1.44. Then

(y k ; ›x y k ; u k ) Ù Ž y Ž ›x y DZ 

weakly in

L

(

q" ")



(

I , ; Rm )) ;

(4.248)

Ë

358

4 Relaxation in Optimization Theory

y(T) in L ( ; R 2

b(y k ) Ù b(y) strongly in L

R

q" ")

m )). Sin e also y ( T ) Ù (I , ; k m ), one an easily pass to the limit passage in (4.247) towards the inte-

f. Lemma 3.101. Also

(



gral identity:

T y dx dt X X a ( y; ›x y ) : ›x y " ( Ž y Ž ›x y DZ  ) - y " y t 0

T % X X b(y) - y dSdt % X y(T) - y (T) " y0 - y (0) dx 0









This shows that

y





# 0:

(4.249)

is a weak solution to the relaxed initial-boundary-value problem

(4.246). Exploiting (4.220b), the uniqueness of

y

for a given



an be proved by Gron-

wall's inequality similarly like in Proposition 1.44. Therefore the whole sequen e

y k }kòN onverges to y. We thus showed that the extended mapping  is single-valued (weak ,strong)- ontinuous extension of the original ontrol-to-state mapping  . {



*

Then one gets the points (i)(iii) by using Proposition 4.1, realizing that the state PAR

onstraints are not involved in (POC ) so that the relaxed problem is always feasible. The

non on entration of optimal relaxed ontrols an be proved by the analogous ontra-

(t ; x ; r; -) has p-growth of '(t ; x ; r; -), 1 and then the attainability by sequen es of lassi al ontrols with relatively L -weakly

di tion argument as used in the proof of Proposition 4.46, realizing that the growth no bigger than

p/q"



whi h is surely lesser than the

Å

ompa t energies thanks to Proposition 3.79.

Let us note that, in ontrast to Se t. 4.5.a, we have now also

p-non on entration

of optimal ontrol. Thus, as in Corollary 4.77, we an dene the relaxed problem in terms of

L p -Young measures, namely

Minimize subje t to

T '(t ; x ; y(t ; x); s) t ; x (ds)dxdt X X X 0

S(t ; x)

% X (y(T; x)) dx / 7 7

y " div a(y; ›x y) % Ž y Ž ›x y DZ # 0 t n - a(y; ›x y) % b(y) # 0 y(0; -) # y0 supp t x ’ S(t ; x) for a.a. (t ; x) ò I ,

y ò Y from (4.241a);  ò Yp (I , ; Rm )



in

I, ; in ;

on

;

with [

Ž y Ž ›x y DZ

℄(

t ; x) # PS

t x)

( ;

(t ; x ; y(t ; x); ›x y(t ; x); s)

I , ;

7 7 7 7 7 7 7 7 7 ? 7 7 7 7 7 7 7 7 7 7 7 G

(4.250)

t ; x (d s ). This representation

an be further used for the existen e result of the Filippov-Roxin type like in Proposition 4.74:

Proposition 4.102 (Existen e for (PPAR OC )).

Let (1.63a), (1.65a), (4.219), and (4.172) hold.

Let also the orientor eld

R% ; t ; x; r;  ; s  ò R %m ; s ò S t ; x ‡ ò I , is onvex for any r ò Rm and  ò Rm,n . Then the original

† ' ( t ; x ; r; s )% dened for a.a. ( t ;

x)

PAR

0

(

)

optimal- ontrol problem (POC ) possesses a solution.

1

(

)

4.5 Paraboli optimal ontrol problems

Ë 359

Let us ontinue further towards the optimality onditions. We again onne ourselves to a semilinear ase (4.189a) with

" 2 ; p;1

' òCAR H di ;

((

I , ) , Rm , R ; R) ;  òCAR2di1 ( , Rm ; R) ; ;

# #  2 ;(2 "  )

b òCARdi "

((

I,

)

p ;(2" "  )

òCAR H di 2 ;2;

q # 2 and strengthen (4.243) as:



((

;

, Rm ; Rm ) ;

(4.251a)

and

(4.251b)

I , ) , Rm , Rm,n , R ; Rm )

(4.251 )

 ¡ 0; we again use Notation 3.105 on p. 233 while for the exponents 2" and 2# see (1.79) with p # 2.

with some small

Proposition 4.103 (Maximum prin iple).130 Let q # 2, (4.189a), (4.251) be valid together

with all the assumptions of Proposition 4.101, and let (  ;

y) solves (4.226). Then the fol-

lowing integral maximum prin iple holds:

T X X [ h y;  0

*

T X X h y;  u (-)ò S (-) a.e. on I , 0

DZ ℄(dxdt) #

where the Hamiltonian

sup

(

*

t ; x ; u(t ; x)) dxdt ;

h y  ò H is given by the formula131 *

;

h y  (t ; x ; s) #  (t ; x) - (t ; x ; y(t ; x); s) " '(t ; x ; y(t ; x); s) ; ;

with

*

*

 ò X # Y # L2 (I; W 1 2 ( ; Rm ))  W 1 2 (I; W 1 2 ( ; Rm ) *

(4.252)

*

;

;

;

*

)

(4.253)

being the weak solution

to the adjoint paraboli terminal-boundary-value problem (4.240). Moreover, if the oer ivity (4.219a) is satised and

S is measurable losed-valued, then also the following L1 (I , ):

pointwise maximum prin iple holds as an equality in

h y  DZ 
(t ; x) # sup h y  (t ; x ; s) ;

*

;

sòS(t ; x)

for a.a. ( t ;

*

x) ò I , :

(4.254)

Proof. We will again apply Theorem 4.15 together with Theorem 4.21. Note that



is (weak*,weak*)- ontinuous, and therefore also (strong,weak*)- ontinuous. Tak-

ing into a

ount the

L (I; W 2

1;2

(

; R

ompa t embedding of

m )) into

L

" "

2

(

I , ; R

Y

W 1 2 (I; W 1 2 ( ; Rm ) )   is (strong,strong)-

#

;

m ), we an see that

;

*



ontinuous. Then we an use Theorem 4.15.132 Again, for larity we divide the proof into ve steps.

; y).) In  (-; y) and J ( ( ; y) ò L(H ; X ) with X # Y

Step 1. (Dierentials of ferentials › 

130



*

*

view of Lemma 3.103 the partial dif*

J  ; y ) ò L ( H ; R)

and ›  (

Maximum prin iples of this kind (i.e. holding for a.a.



*

are given

t ò I and x ò ) has been treated by Ahmed

and Teo [9℄, Casas [182℄, Chryssoverghi [211℄, Raymond and Zidani [634, 635℄, Sadigh-Esfandiari, Sloss, Bru h Jr., Sadek [712, 730℄, Zolezzi [819℄, et .

131

Let us note that

"

*ò L2 (I , ;

132

Rm

* - (

Ž y Ž ›x y) ò H

is ensured by the assumption (4.229) via (3.196) for any

).

In fa t, the highest-order terms, being linear and thus having onstant dierentials, are to be

treated with respe t to the original topology.

Ë

360

4 Relaxation in Optimization Theory

respe tively by the formulae:

#

´[›   (  ; y )℄(  ) ; y µ





J  ; y)
(  ) #

T X ['

› (

with

òH

*





y ò Y.

and



T X X ( Ž y Ž›x y DZ  ) 0

0

y dxdt

-



and



Ž y DZ  ℄(t) dt # ´  ; ' Ž yµ 



Moreover, they are (weak*,weak)- ontinuous as needed in

Theorem 4.15.

Step 2. (Dierentials of

 ( ; -)

J ( ; -).) In view of Lemma 3.103 the partial dier2 ( ) ) , L ( )) and › y J (  ; y ) ò L( Y; R) are given

and



entials › y  (  ; y ) ò L( Y; L ( I ; W 2



1;2

*



respe tively by the formulae:

´[› y  (  ; y )℄( y ) ;



T

yµ # X 

X ” a ›x y : ›x y



0

"y

T

J  ; y)℄( y ) # X

[› y (



0

y • dx dt t 



T %X X



b r (y) y - y dxdt % X y (T) - y (T) dx ; 











0

 r (y(T))
( y (T)) dx 

X







y (0) # 0. Let us note that ›y  ( ; y) a tually possesses a bounded inverse, as required in Theorem 4.15; indeed, (4.251 ) ensures via (3.197b) that r Ž " " " " " y Ž ›x y DZ  ò L2 2 "2 (I , ; Rm ),  Ž y Ž ›x y DZ  ò L2 2 2 2"2 "2 (I , ; Rm,n ), and

with

y; y ò Y





' r Ž y DZ 
y dxdt %

X



% Œ[ r ŽyŽ›x y DZ ℄ - y % [  ŽyŽ›x y DZ ℄ : ›x y - y 







with







/(

# /(2# "2)

b r (y) ò L2 

(



)

I , ; Rm ).

/(

)

Therefore, by the lassi al theory of linear paraboli equa-

   t y " div(a›x y ) % [ r Ž y Ž ›x y DZ ℄ y % [ 

tions, the initial-boundary-value problem







ŽyŽ

y DZ ℄›x y # f with n - a›x y % b r (y) y # 0 on I , and y (0; -) # r possesses a solution y , whi h denes the bounded linear operator (f; r) ÜÙ y : Y , L2 ( ) Ù Y whose "1 : X Ù Y . restri tion is just [› y  (  ; y )℄ 

›x









*







Step 3. (The adjoint problem.) The abstra t adjoint equation (4.23b) takes here the form

´ for all

; [›y  ( ; y)℄( y )µ #

*





[› y

J ( ; y)℄( y )

(4.255)





y ò Y and some  ò X # W 1 2 (I; L2 ( ; Rm )). By using the formulae from Step 2, *

*

;



the identity (4.255) an be rewritten as follows

T * *   * X X a ›x y : ›x  % Œ( r Ž y Ž›x y DZ  ) - y % (  Ž y Ž›x y DZ  ) : ›x y  -  " y dxdt t 0

T T % X X br (y) y - * dSdt # X X €'r Ž y DZ  - y dxdt % X r (y(T)) - y (T)) dx ; 0 0

















whi h is just the weak formulation of the adjoint terminal-boundary-value problem (4.240); spe i ally, (4.240) an be re overed from this integral identity when using Green formula on arbitrarily.

and by-part integration over I while taking the test fun tions y ò Y 

Ë 361

4.5 Paraboli optimal ontrol problems

Step 4. (The Hamiltonian.) The abstra t Hamiltonian (4.23d) an be now written in the

f

form y;  *

# h y  ;

*

with

h y  ò H determined with help of the formulae from Step 1 by ;

*

the identity

#  ; [›  ( ; y)℄(  ) " › J ( ; y);  #  ; ŽyŽ›x y DZ  "  ; 'Žy ;

 ; h y;  for any

*

*

*









òH

*







. This gives the expression (4.253) for the Hamiltonian



hy  ;

.

*

Step 5. (Lo alization of the maximum prin iple.) Eventually, the maximum prin iple (4.23 ) an be transformed into the form (4.254) by means of Theorem 4.21(i), the des ent ondition (4.36) being guaranteed as a onsequen e of the oer ivity (4.219a) the

t ; x ; r;  ; s) ¢

growth ondition and the growth  (

(

" t ; x) % r2 "1 % 2

"

/2

% sp

/2

"

ò L (I , ), f. (3.192b) together with the data quali ation (4.229). Indeed, " m taking  from (4.219a) and realizing that  ò L ( I , ; R ), one an estimate h y  from 2

with some

2

*

;

*

(4.253) as

h y  (t ; x ; s) ¢  (t ; x) ;

*

*

„

(

" t ; x) % r2 "1 % 2 "

¢ C " p n „ (t ; x) % (t ; x) *

;

for some

C"

;

p ; n large enough depending on

"

2

2

;

"

/2

"

…"

"

"2 … "

% sp

/2

"

% r %  2

2 2

"sp "

2

p

", p, and n. This gives (4.36), so that we

Å

an readily use Theorem 4.21(i) to get (4.254).

Remark 4.104 (State onstraints). the pointwise (i.e. in

s

 

Relying on a C ( I , )-regularity, one an onsider x and t) state onstraints on I , or I , , f. [634, 635℄. Compar

ing to the ellipti problem in Remark 4.84, the paraboli regularity is more involved, however. This renes the integral onstraints mentioned in Remark 4.93.

Remark 4.105 (Maximum prin iple for (PPAR OC )).

If some solution to the relaxed problem

PAR H POC ) is one-atomi (like in Remark 4.85), then Proposition 4.103 oers the maxiPAR mum prin iples for the original problem (POC ). We thus re over various results in lit(R

erature for bulk or for boundary ontrols proved often without any relation to relaxation methods, sometimes even for quasilinear equations, f. e.g. [9℄, Casas [182℄, Hu and Yong [395℄, Raymond and Zidani [634, 635℄, Sadigh-Esfandiari, Sloss, Bru h Jr., Sadek [712, 730℄, Zolezzi [819℄, et .

Let us ome ba k to the idea from Remarks 4.994.100 to exploit dire tly the results from Se t. 4.5.a. To this goal, we dene the linear operator

Carp (I , ; Rm ) Ù Carp (I; S) : h ÜÙ ”(t ; s) ÜÙ X h(t ; x ; s(x)) dx•

with

S # L p ( ; Rm ).

(4.256)

The following assertion will be found useful:

Lemma 4.106 (Coarsening of the onvex - ompa ti ation from Se t. 4.5.a). The liH ’ Carp (I , ; Rm )

near operator (4.256) is ontinuous. Considering a linear subspa e

Ë

362

4 Relaxation in Optimization Theory

and denoting the restri tion of (4.256) on

p Q* Y QH (I; S) Moreover, if

H by Q, it holds

# Y Hp (I , ; Rm ) :

(4.257)

H is C(I , )-invariant, QH is C(I)-invariant.

Proof. The norm of the linear operator (4.256) with respe t to the norms from Exam-

Carp (I , ; Rm ) and Carp (I; S) is 1. This an be seen from that h(t ; x ; s) ¢ p ( t ; x ) % C  s  with ò L (I , ) implies, for any s ò S, that

ple 3.76 on

1

!! ! !!X h ( t ; x ; s ( x )) d x !!! !! !! !

!

¢X

(



t ; x) % Cs(x)p dx # 

p

t % C s S

( )

t # P (t ; x) dx, and that obviously L1 I # L1 I , . H ’ Carp (I , ; Rm ), the mapping (4.256) embeds H into Carp (I; S). Then, Q : (QH) Ù H and, if  # w - lim ò i QH (u  ) ò (QH) for some bounded net {u  } ò p m in L ( I , ; R ), we have Q  # w - lim  ò  i ( u  ) ò H . Therefore we have (4.257). H Eventually, if gh ò H for h ò H and for any g ò C ( I , ), then also gQh ò QH for any g ò C(I) be ause gQh # Q(( g ‹ 1)h) where [( g ‹ 1)h℄(t ; x ; s) # g (t)h(t ; x ; s).

with



( )



( )

(

)

Thus, for

*

*

*

*

*

*

*

*















This onvex ompa ti ation enables us to perform a further lo alization of the maximum prin iple (4.233) onto parti ular spatial instan es. Indeed, the Hamiltonian

H from (4.261), must be of the form

(4.253), belonging now to

h y  (t ; s) # Qh y  ;



*

;


( t ; s )

*

#

X h y; 

(

*

t ; x ; s(x)) dx

(4.258)

h y  ò Carp (I , ; Rm ). For any  # Q  and h # Qh, we have always the identity [ h DZ  ℄( t ) # P [ h DZ  ℄( t ; x ) d x , whi h follows from

with some

*

*

;





ÄX [ h DZ  ℄(- ; x ) d x ;

gÅ #

T X X [ h DZ  ℄( t ; x ) g ( t ) d x d t 0

# h DZ  ; g #  ; g - h 

# Q  ; g - h #  ; Q(g - h) #  ; g - h # h DZ  ; g *



with



g(t ; x) # g(t) independent of x, whi h holds for any g ò C(I). It allows us to rewrite

(4.233) into the form

X [ h y; 

*

DZ ℄(t ; x) dx # sup

X h y;  ssò S ( t)

(

*

t ; x ; s(x)) dx

in the sense of

L1 (I) ;

 # Q  . Now we an apply again Theorem 4.21(i), whi h eventually gives the 1 maximum prin iple holding in the sense of L ( I , ):133 *

where

[

h y  DZ ℄(t ; x) # max h y  (t ; x ; s) : ;

*

sòS(t ; x)

;

*

(4.259)

 ; h> # # supuòUad PT P h y  (t ; u(t ; x)) dxdt, from whi h (4.259) dire tly follows if Theorem 4.21(i) is applied over I , .

133
subje t to 6 6 6 F

X [' M

Ž y DZ ℄(x) dx

y(x) # X a(x ; x )[f Ž y DZ ℄(x ) dx % y0 (x) M

 ò U ad ’

p YH (M;

Rn

)

; y ò L q (M; Rm ):

Lemma 4.115 (Corre tness of the extended state problem). (1 ; %),  ¢ q , and (4.285a-e) be satised. Then:

Let

:a.a. x ò M);

(

p; q ò [1; %℄,  ò

Ë 375

4.6 Optimal ontrol of integral equations

(i)

(ii)

 ò YH (M; Rn ), the extended state equation in (RH PINT OC ), i.e. y # PM a (- ; x )[ f Ž y DZ ℄(x) dx % y0 , admits just one solution y ò L q (M; Rm ). p n q m The mapping  : YH ( M ; R ) Ù L ( M ; R ) thus dened is (weak*,strong)- ontip

For any



nuous if restri ted on the losure of bounded sets.

 # i H (u) with u ò L p (M; Rn ), then y #  () solves the integral equation in (PINT OC ). p n q m In other words,  Ž i H #  where  : L ( M ; R ) Ù L ( M ; R ) denotes the original

(iii) If





ontrol-to-state mapping.

 ò YH (M; Rn ), we an take a sequen e {u k }kòN bounded in L p (M; Rn ) su h that i H ( u k ) Ù  weakly* in H . By Proposition 1.47, for ea h u k there is a solution y k ò L q (M; Rn ) to the equation p

Proof. For any

*

y k # A(N f (y k ; u k )) % y0

(4.288)

satisfying the estimate



y k L q

(

¢

M ;Rm )



a L q

(

M ; L  ( M ;R,n )) a 1 

1 " a L q

(

% u k p  L  M % y /

1

(

)

0

L q ( M ;Rm )

;

M ; L  ( M ;R,n )) ~ L q q" ( M ) 

/(

)

a1 and 1 ame from the assumed estimate f(x ; r; s) ¢ a1 (x) % b1 rq  % 1 sp  , q m

f. (4.285a) and (3.192b). Thus the sequen e { y k } k òN is bounded in L ( M ; R ). Theref   fore, the sequen e {N ( y k ; u k )} k òN is bounded in L ( M ; R ). By (4.285e), the linear inf tegral operator A is ompa t146 so that the sequen e { A (N ( y k ; u k ))} k òN (and thus also q m { y k } k òN ) is relatively ompa t in L ( M ; R ). Thus we an suppose ( onsidering a subq n sequen e, if ne essary) that { y k } onverges to some limit y in the norm of L ( M ; R ). /

where

/

,

By Lemma 3.101, we an use the (strong weak*,weak)- ontinuity of the extended Nemytski mapping

N

f



and pass to the limit in (4.288), whi h gives

f

y # A(N (y; )) % y0 # A(f Ž y DZ ) % y0 ;

(4.289)



f. (3.193). Therefore, we proved that

 (y; ) # 0 with  from (4.287) admits a solution.



Its uniqueness then follows by the estimate147

y " y2 L q

1

where

(

M ;Rm )

¢ a L q ¢ a L q

Ž y DZ  " f Ž y DZ  L  M R ~ L q q" M y " y L q M Rm

(

M ; L  ( M ;R,n )) f

(

M ; L  ( M ;R,n ))





1

/(

2

)

(

)

1

(

2

(

;

;

)

)

y1 ; y2 are two solutions to (4.289); obviously, by (4.285d) ne essarily y1 # y2 .

Å

The point (ii) is again an easy modi ation, while (iii) follows by Lemma 4.11.

146

See Zabre ko at al. [810, Se t. V.1.5℄ or also Krasnoselski  at al. [442, Se tions 6.45℄.

147

Here we used also (3.194a) and (3.194b) applied to the estimate (4.285 ) written in the form 

y1 " f Ž y2  ¢ (~y1 " y2 ) ‹ 1.

f

Ž

Ë

376

4 Relaxation in Optimization Theory

Proposition 4.116 (Corre tness of the relaxation s heme). Let p ò [1; %), q ò (1 ; %℄,  ò (1 ; %),  ¢ q , U ad be nonempty,148 and (4.285) be satised. Then: (i)

(R

INT H POC ) has a solution, and any solution is

p-non on entrating.

INT Every solution to (R H POC ) an be rea hed by a minimizing admissible sequen e for INT 1 (POC ) whose energy is relatively -weakly ompa t.

(ii)

L

(iii) Conversely, a limit of every minimizing admissible weakly* onverging sequen e for INT

(POC )

(when embedded via i H ) solves (R H POC ). INT

J : YH (M; Rn ) , L q (M; Rm ) Ù p

Proof. First, let us noti e that Lemma 3.101 ensures



,

R

dened by (4.286) to be (weakly* strong)- ontinuous extension of the original ost

J from (4.283e). By Lemma 4.115, the extended ost fun tional  # J Ž (idH ,  ) is weakly* ontinuous and, by (4.285f), also oer ive. Sin e Uad #Ö , one gets the

fun tional



*





points (i)(iii) by using Proposition 4.1. The non on entration laimed in (i) follows again by usual ontradi tion argument sin e otherwise the to the separability of

H

Ž p-non on entrating modi ation , whi h does exist thanks

(see Proposition 3.81), would drive the ontrolled system to

f x ; r; -) has the growth at most p/   p but would a hieve a '(x ; r; -) is oer ive as the power p; f.Lemma 4.22(ii). Å

the same state be ause  (

stri tly lower ost be ause

f x ; -; s)

To study the optimality onditions, we have to assume the smoothness of (

'(x ; -; s), i.e. to strengthen (4.285a) to

and

f òCARH di (M ,Rm ,Rn ; R ) q; p;  ;

and

Proposition 4.117 (Maximum prin iple).

' òCAR H di (M ,Rm ,Rn ; R):

Let

q; p;1 ;

(4.290)

p; q ò [1; %),  ò (1; %), 2 ¢ q, and y) ò Argmin(RH PINT OC ). Then the

(4.285b)(4.285d), and (4.290) be satised and let (  ; following integral maximum prin iple holds:

X [ h y;  M where the Hamiltonian

*

DZ ℄(dx) # sup

X h y;  u ò U ad M

(

*

x ; u(x)) dx ;

(4.291)

h y  ò H is given by the formula: ;

*

h y  (x ; s) # "'(y(x); s) " f(x ; y(x); s) - X a(x ; x)  (x ) dx % (x); ;

with

*

*

M

ò L1 (M) being arbitrary su h that ‹ 1 ò H

and with

(4.292)

 ò L q (M; Rm ) being the *



solution to the adjoint integral equation

 (x) " [f r Ž y DZ ℄ (x)X a(x ; x)  (x ) dx # [' r Ž y DZ ℄(x) *



*

M

148

Re all that this requirement means pre isely that

a measurable sele tion belonging to

L p (M;

Rn

).



(4.293)

S, whi h need not be even measurable, admits

Ë 377

4.6 Optimal ontrol of integral equations

x ò M . Moreover, if (4.285f) is satised, if S is measurable losed-valued, and a ò L  (M; L q (M; R,n )) with a (x ; x ) # a( x ; x), then the following pointwise

for a.a.



*

if also

*





maximum prin iple holds:

h y  DZ  # h Sy  ;

*

;

in

*

L1 (M)

h Sy  (x) # sup h y  (x ; s) :

with

*

;

(4.294)

*

;

sòS(x)

Proof. We will again apply Theorem 4.15 together with Theorem 4.21. Note that, by Lemma 4.115(ii),



is (weak*,strong)- ontinuous, and therefore also (strong,strong)-

ontinuous. Again, we divide the proof into ve steps.

; y).) In view of Lemma 3.103 the partial dieren (-; y) and J (q m  ; y) ò L(H ; R) are given respe tively by tials ›   (  ; y ) ò L( H ; L ( I ; R )) and ›  J ( the formulae (  ò H ): Step 1. (Dierentials of



*

*

*



›

 ( ; y)
(  ) # "A(f Ž y DZ  )

and







J  ; y)
(  ) #

› (

X [' M





Ž y DZ  ℄(dx) #  ; ' Ž y : 



Moreover, both dierentials are (weak*,weak)- ontinuous.149

Step 2. (Dierentials of

 ( ; -) and J ( ; -).) In view of Lemma 3.103 the partial dieren



q m q m q m tials › y  (  ; y ) ò L( L ( M ; R ) ; L ( M ; R )) and › y J (  ; y ) ò L( L ( M ; R ) ; R) are given

respe tively by the formulae:

›y

 ( ; y)
( y ) # y " A(f r Ž y DZ ) - y 



›y

with

and





J ( ; y)
( y ) #



X €' r M 



Ž y DZ  (x) - y (x) dx 

y ò L q (M; Rm ). Note that ›y  ( ; y) a tually possesses



a bounded inverse,150 as

required in Theorem 4.15.

Step 3. (The adjoint problem.) The abstra t adjoint equation (4.23b) with

B # 0

0 # 1 takes the form ´  ; [›y  ( ; y)℄( y )µ # ´›y J ( ; y); R  ò L q (M; Rm ). Using the formulae from Step 2, this identity results to *

y µ for all y ò L q (M;

*







some

X  M

(





*

*



and

m ) and

x) - ” y (x) " X a(x ; x )[f r Ž y DZ ℄(x ) y (x ) dx • dx # 





M

X [' r M 

Ž y DZ ℄(x) - y (x) dx : 

A is ompa t, they are even (weak*,strong)- ontinuous. f r Ž y DZ  ¢ ~ ò L q/(q") (M), and therefore, by (4.285d), the linear operator  y ÜÙ A((f r Ž y DZ ) - y) is a ontra tion on L q (M; m ), whi h implies that [›y  ( ; y)℄"1 does exists "1 . and its norm is not greater than (1 " a q ~ L q q" (M) ) L (M;L  (M;R,n ))

149

Sin e

150

Indeed, (4.285 ) ensures 

R



/(

)

378

Ë

4 Relaxation in Optimization Theory

By the Fubini theorem151 we an hange the order of integration in the kernel term,

x and x in this term, the identity

whi h gives, after repla ing the role of

X ” M

*

(

x) " X  (x )a(x ; x)[f r Ž y DZ ℄(x) dx " [' r Ž y DZ ℄(x)• - y (x) dx # 0 : *





(4.295)



M

y ò L q (M; Rm ),

As (4.295) is to be valid for arbitrary



(4.293) must hold. Note that,

L q (M; Rm ). As ›y  

thanks to (3.197b), the right-hand side of (4.293) belongs to



has

a bounded inverse as mentioned in Step 2, our adjoint integral equation possesses always a (unique) solution

 ò L q (M; Rm ), as required. 

*

Step 4. (The Hamiltonian.) The abstra t Hamiltonian (4.23d) with written in the form

f y  # h y  ;

*

;

with

*

hy  ò H *

;

0 # 1 an be now *

determined with help of the formulae

from Step 1 by the identity

 ; h y;  *



# " ´ ; [›  ( ; y)℄(  )µ % ´› J ( ; y);  µ *









# "X  (x)[A(f Ž y DZ  )℄(x) dx " X [' Ž y DZ  ℄(dx) *





M

M

# "X

X  M M

*

(

# "X

X  M

# "X

„ ( f

M M

whi h is to hold for any

òH

x)a(x ; x )[f Ž y DZ  ℄(x ) dx dx % X [' Ž y DZ  ℄(dx) 



M

*

(

x )a(x ; x)[f Ž y DZ  ℄(x) dx % ' Ž y DZ  ‘ (dx) 



Ž y) - X a(x ; -)  (x ) dx % ' Ž y … DZ  ‘ (dx) *



M

*



; of ourse, the Fubini theorem has been used on e

again. This gives the expression (4.292) for the Hamiltonian

hy  ;

.

*

Step 5. (Lo alization of the maximum prin iple.) Eventually, the maximum prin iple (4.23 ) an be transformed into the form (4.294) by means of Theorem 4.21(i). It su es to verify the des ent ondition (4.36): indeed, (4.285f) with (4.290) allow us to estimate

h y  (x ; s) ¢ a1 (x)% b1 y(x)q ;

where

*

a1 , b1



!

!

! ! % sp   !!!!X a(x ; x)  (x ) dx !!!! " a (x) " bsp ; /

1

*

0

!

! M

1 ome from (3.192b) whi h is ee tive due to (4.290). This gives a(x ; -)  (x ) dx L  M;R ¢ a L  M;L q M;R,n -  L q M;Rm   % M   P a ( x ; -)  ( x ) d x  ¢ b/(2 ) s  p % C with some C # C (  ; a ; b ; ) ò L 1 ( M ) 1 1 M and

*

(4.36) be ause P

p/ so that  s 

/



*

(

)



*

(

(



))

*

(

)

*

Å

large enough.

Example 4.118 (A spe ial Volterra equation).

If

M # I

and

a(t ; ) # 0

for

t   ,

the Hammerstein equation (1.88) is a spe ial nonlinear Volterra equation; in this

151

Let us note that usage of this theorem is legitimate be ause, by (3.197b),

L q/(q") (M;

R,n

) so that, for

y ò L q (M;

R

Rm

), (

y DZ ℄(x)y(x) dx lives in L q (M; m ) so that, y DZ ℄(x)y(x) dx is integrable as needed.

f r Ž y DZ )- y lives in L  (M;

for

*

ò L q (M; Rm ),

R

f r

Ž yDZ



eventually

*(-)

-

belongs to

P a (- ;  x)[f r Ž M P a (- ;  x)[f r Ž M

) and therefore

Ë 379

4.6 Optimal ontrol of integral equations

one-dimensional ase, it is more usual to use

hoi e

with

 # n,

a(t ; ) #



t (as time) in pla e of x. For the spe ial

I

if

0

I ò Rn,n denoting the unit matrix, and y

if 0(

t £ ; t   ;

(4.296)

t) # y0 ò Rm , the problem (PINT OC ) is equiv-

# 0 so that we meet the situation investigated in Se tion 4.3

ODE alent with (POC ) with

(disregarding the state onstraints) as a very spe ial ase of the integral-equation optimal ontrol problems but the fa t that the assumptions (4.285a)(4.285f) with the

q #  are stronger than the assumptions used in Se tion 4.3 tted pre isely for this spe ial Volterra ase. The adjoint integral equation (4.293) takes for a

natural hoi e

from (4.296) the form

T

 (t) " [f r Ž y DZ ℄ (t)X  () d # [' r Ž y DZ ℄(t) ; *



*



(4.297)

t

 0 # 1,  *

whi h is just the adjoint equation (4.105) for

 (t) *

T repla ed by " P t

*

# 0 and the adjoint state

 () d. Under this repla ement, the Hamiltonian (4.292) will 0 # 1, and also the maximum prin iples *

oin ide with the Hamiltonian (4.71) for

*

(4.106) and (4.294) will oin ide with ea h other.

Remark 4.119 (Bana h-spa e-valued generalization of Volterra equations).

The

in-

nite-dimensional dynami al system (4.221) an be also written as a Volterra-type integral equation

t

y(t) # X e "t A (f ( ; y(); u())) d % e"tA (y (

)

for

0)

0

t ò I;

(4.298)

where the integral is understood in the Bo hner sense and where the symbol

e"tA stands for a semi-group of mappings generated by A in the sense that limtÙ % (e"tA "id)(y )/t # A(y); a tually, one has the exponential formula for e"tA gen"tA (y) # lim "k eralizing the lassi al al ulus, namely e k Ù (id% tA/ k ) (y ). The integral 0



identity (4.298) is also onsidered as a denition of a so- alled mild solution to (4.221). For

A # 0, the exponential e"tA be omes the identity, and (4.298) takes the form from

Example 4.118.

Remark 4.120 (The ase q # % and spa e onstraints).

Let us onsider the state

onstraints (

where

ò C(M , Rm ; R

û

the olle tion {

x ; y(x)) ¢ 0

:a.a. x ò M);

(4.299)

(

) has a ontinuous partial derivative

r ( x ; -)} x ò M is equi-uniformly ontinuous, and





r

R

û

#

/

r

su h that

is ordered by a one

D . Then we an put  # L (M; R ) ordered by a one D # { ò L (M; R ); :a.a. x ò M : (x) ò D } whi h has a nonempty interior152 provided int( D ) #Ö . It for es us to 

û



û





152

Let us remind that non-emptyness of int(



D)

is essential to derive the optimality onditions, f.

Proposition 4.8, and also for a stability with respe t to a onstraint perturbation, f. Proposition 4.5.

Ë

380

4 Relaxation in Optimization Theory

q # % to ensure the ontinuity and even the Fré het dierentiability of the exB : H , Y Ù  dened by B ( ; y) # N (y). It is a routine INT exer ise to verify that for any optimal solution (  ; y ) to the relaxed problem (R POC ) H enhan ed by the onstraint (4.299) there are some multipliers (  0 ;  ) ò R , vba( I ; R ) not vanishing simultaneously and satisfying  0 £ 0 and  £ 0, the omplementarity

ondition P ( x ; y ( x ))  (d x ) # 0, and the maximum prin iple M take

*

tended onstraint mapping





*

*

*

*

û

*

*

X [ h y;  0 ;  M *

where the Hamiltonian

DZ ℄(dx) # sup

X h y;  0 ;  u ò U ad M

*

h y 0 ;

*

;



*

òH

*

x ; u(x)) dx ;

(

*

(4.300)

is given, up to an integrable onstant, by the

formula153

h y 0

*

;

with

;



(

*

x ; s) # "0 '(t ; y(x); s) " f(x ; y(x); s)- X a(x ; x)  (dx ) *

*

 òvba(M; Rm ) being the solution to the adjoint integral equation154 *

 " [f r Ž y DZ ℄ X a(x ; -)  (dx) # 0 ' r Ž y DZ  % [ *



*

*





r

M

Ž y℄  : *

Let us nally remark that further data quali ation may ensure

L M; R



spa e not only repla e

L



(

û

dard measures

(

M; R

) and

C(M; R

m ) but even

vba(M; R

û

r a(M; R

û

1

X

Minimize

0

) with the separable spa e

C(M; R

û



0











1" x )xI x (1"x)I

(





154

R





? 7 7 7 7 7 7 7 7 7 G

(4.303)

for for

x¢ x; x¡ x; 

(4.304)



R

 P a ( x ; -)  *(d x ) # A *  * ò L  ( M ;  ) be ause the adjoint operator A * maps M  m )* to L  ( M ;  )* Ê L  ( M ;  ). Thanks to (3.196), whi h is ee tive due to (4.290), the Hamil-

Let us note that

M;



a is now symmetri , namely: a(x ; x ) # 

(



%X x(1" x )f( x ; y( x ); u( x )) d x ;



L

/ 7 7 7 7 7 7 7 7 7

1

x

tonian

Let us onsider the fol-

x X (1" x ) xf ( x ; y ( x ) ; u ( x )) d x

u(x) ¢ S(x) :a.a. x ò [0; 1℄; m y ò L (0; 1; R ) ; u ò L2 (0; 1; Rn ) :



) and more stan-

'(x ; y(x); u(x)) dx



The kernel

A to have as a target

), respe tively.

y(x) # r0 (1"x) % r1 x %

subje t to

(4.302)

m ), whi h would eventually enable us to

Example 4.121 (Relation to ellipti optimal- ontrol problems). INT lowing spe ial ase of (POC ) with M # [0 ; 1℄ and  # n :

153

(4.301)

M

h y;0 ; *

*

R

a tually lives in

H.

R

Of ourse, this integral equation is to be understood in a suitably generalized sense. In fa t, (4.302)

just means that

* # (*0 ['r Ž y DZ ℄ % [



r

Ž y℄  ) Ž (›y  ( ; y))" *

1

.

Ë 381

4.6 Optimal ontrol of integral equations

while

y0 (x) # r0 (1" x) % r1 x. Let us note that A from (4.283d) is then the Green operator x

orresponding to the linear part. The Lips hitz onstant ~( ) from (4.285 ) should satisfy ~

L  (0 ; 1)

 



a L

1 

(0 ; 1;

L



(4.305)

R,m ))

(0 ; 1;

in order to fulll the ontra tion ondition (4.285d). Using monotoni ity argument as in Se t. 4.4 instead of the Bana h-xed-point one, we ould alternatively admit

f x ; -; s) su h that r ÜÙ r " f(x ; -; s) : Rm Ù Rm oer ive and stri tly mono2 tone with less than the onstant from the inequality y 2 ¢ ddx y 2L2 0 1;Rm L 0 1;Rm for y (0) # 0 # y (1). The integral equation is then equivalent to the two-point arbitrary (

(

;

)

(

;

)

boundary-value problem for the 2nd-order linear ellipti ordinary equation:

d y # f(x ; y; u) dx 2

"

0; 1); y(0) # r ; y(1) # r ;

on (

2

0

(4.306)

1

whi h an be he ked just by dire t al ulation.155 The adjoint equation (4.293) now looks as

1

 (x) # "f r (x ; y(x); u(x))X a(x ; x )  (d x ) "  (x) *



*

*



(4.307)



0

a(x ; x ) dened by (4.304) Let us further dene  by for



and for some



*

0 %  vba 0 1;Rm ¡ 0. *

non-negative,

*

(

;

)

* 

1

 (x) # X a( x ; x)  (d x ) *

*





x ò [0; 1℄:

for a.a.



0

From (4.307) with al ulation of

d2 * dt2  

(4.308)

like used for (4.306), we an see that this



* 

solves, in fa t, the adjoint two-point boundary-value problem

"

d  % f r (x ; y; u)  # " dx 2

*





*

*



2

the Hamiltonian (4.292) takes the form

 (0) #  (1) # 0;

0 ; 1) ;

*

on (

*



(4.309)



h y  (x ; s) #  (x)f(x ; y(x); s) " '(x ; y(x); s). *

;

*



Eventually, let us remark that multidimensional ellipti problems from Se t. 4.4.b an be treated similarly when admitting the kernel

a( x ; x) to be singular; in fa t, it is the 

Green fun tion of the linear ellipti operator forming the main part of the ellipti equation.

A to a t not only to L (M; Rn ) but even to C ( M ; R L (M; R ) and vba(M; R ) in Remark 4.120 with more standard separable C(M; R ) and metrizable r a(M; R ), respe tively.

Remark 4.122.



Further data quali ation may ensure

m ), whi h would eventually enable us to repla e

û



û

û

û

155

If the short-hand notation

w # f(x ; y; u) is used, one an al ulate:

x 1 d y d # 2 œr0 (1"x) % r1 x % X (1"x)x w(x) dx % X x(1"x)w(x) dx 2 dx dx 0 x x 1 d # x w(x) dx % X (1"x)w(x) dx # "xw(x) " (1"x)w(x) # "w(x): œr " r " X  dx 1 0 0 x 2

2

5 Relaxation in Variational Cal ulus: S alar Case ... after his and

seeing

own

renamed

tions.

Lagrange's work, Euler dropped

method, the

espoused subje t

the

that

of

Lagrange,

al ulus of varia-

[360, p.110℄

Herman Heine Goldstine

(19132004)

... dis overy by Weierstraÿ in 1879 marks a turning point in the history of the Cal ulus of Variations

Os ar Bolza The

maximum

prin iple

has

also

(1857-1942)

found

appli-

ation in the al ulus of variations, sin e all the results of the latter an be obtained from it.

Lev Semyonovi h Pontryagin

(1908-1988)

This hapter deals with variational problems in the form of a minimization problem on the Sobolev spa e

W 1 p ( ). Due to os illation ee ts ( aused by a possible non;

onvexity of an energy density involved in the problem) and/or on entration ee ts ( aused by a possible la k of oer ivity on

W 1 p ( ) for any p ¡ 1) the solution of the ;

original variational problem may fail to exists and the need of extension (= relaxation) of this problem immediately appears. We will make it basi ally as a ontinuous exten-

- ompa ti ation of the Sobolev W 1 p ( ). It is just the benet of the s alar problems that the lass of onvex -

sion of the original problem to a suitable onvex spa e

;

ompa ti ations is su iently ri h so that it brings no substantial restri tions on the data of the original problem, ontrary to the ve torial variational problems whi h will be treated in Chapter 6. After developing a wide lass of onvex

- ompa ti ations of W 1 p ( ) ;

in Se -

tion 5.1, we will onstru t a relaxation of the variational problem and investigate the stability of the relaxed problem in the reexive ase, i.e. for

p ¡ 1. An important phe-

nomenon of oer ive problems is that on entration ee ts are essentially ex luded so that only os illation ee ts may appear. In solid-phase ontinuum me hani s, su h problems may serve as a model of a mi rostru ture appearing in a so- alled anti-plane shear problem.1 This will be performed in Se tion 5.2, while in Se tion 5.3 we will formulate optimality onditions for the relaxed problems. It turns out that the onveniently sele tive and informative optimality onditions ombine one part from the Euler-Lagrange equation (extended by ontinuity, of ourse) with one part from the Weierstrass ondition, playing the role of the Pontryagin-type maximum prin iple.

1

We refer to Antman [21℄ or Gurtin and Temam [366℄, or also to Bauman and Phillips [91℄, Friese ke

[336℄, Raymond [632℄, et . See Horgan [393℄ for a thorough bibliography.

https://doi.org/10.1515/9783110590852-005

5.1 Convex ompa ti ations of Sobolev spa es

In Se tion 5.4 we will also mention the nonreexive ase

Ë 383

p # 1, overing problems

of the nonparametri minimal hypersurfa es.2 Yet, in addition to on entration ee ts in gradients, whi h appear typi ally in minimal-surfa e problems, we will admit simultaneously os illation ee ts. However, mu h less an be said about su h problems

p ¡ 1. p ¡ 1, Se tion

in omparison with the ase Conning again to

5.5 develops a numeri al approximation the-

ory for the relaxed problems by using inner onvex approximations of the extended Sobolev spa e. A nontrivial example of s alar variational arises in mi romagneti s, whi h eventually illustrates the previous results in Se tion 5.6.

5.1

Convex ompa ti ations of Sobolev spa es

- ompa ti-

The aim of this se tion is to develop a su iently wide lass of onvex  ations of the Sobolev spa es

W 1 p ( ) and W0

1;

;

p

(

) with ’

Rn being a Lips hitz

domain, whi h we will use later as a tool for relaxation of variational problems dened as minimization problems on these Sobolev spa es. Conning ourselves to we will use readily the theory of onvex

- ompa ti ations of

L p ( ; Rn ), developed already in Se tion 3.4.

Therefore, throughout the whole se tion we will onsider the norm bornology

p ò [1; %),

the Lebesgue spa e

W 1 p ( ) ;

endowed by

B generated, for example, by the norm3

y

W 1; p ( )

:# y L1

(

% ›x y L p

)

(

;Rn )

:

(5.1)

- ompa ti ation of W 1 p ( ) will rely n on a C ( )-invariant linear subspa e H ’ Car ( ; R ) equipped with a norm indu ing p n a ner topology than the natural lo ally onvex topology oming from Car ( ; R ). p 1 q n 1 p We dene : L ( ) , Car ( ; R ) Ù C ( W ( )B ) by ;

A su iently general onstru tion of a onvex

p





[

1 (g; h)℄(y) :#

and suppose that

ompa t

;

g; y

% [ h℄(›x y) # X g(x)y(x) % h(x ; ›x y(x)) dx :

1   q   p

embedding

of

W

1;

(5.2)



*

with

p ( ) into

p from (1.42), whi h L q ( ); f. (1.41). Then *

guarantees the the

fun tional

1 (g; h)℄(-) : W 1 p ( ) Ù R is ontinuous be ause of the ontinuity of the Nemytski p n 1 q q mapping N h : L ( ; R ) Ù L ( ) and be ause g ò L ( ) Ê L ( ) ’ W 1 p ( ) . Taking some H , we put ;

[



FH1

2

# L q ( ) , H  % 1



 onstants on

*

W 1 p ( ) : ;

;

*

(5.3)

We refer, e.g., to the monographs by Giaquinta and Hildebrandt [353℄ or Giusti [356℄ for a survey of

su h kind of problems.

3

This norm is more suitable for the proof of Theorem 5.1 than the standard norm introdu ed in Se -

tion 1.4. Of ourse, both norms are equivalent with ea h other so that they generate the same bornology on

W 1;p ( ).

Ë

384

5 Relaxation in Variational Cal ulus: S alar Case

Furthermore, we dene the embedding 1

FH

e1H : W 1 p ( ) Ù (FH1 ) ;

as the restri tion on

of the evaluation mapping. The following result is of a fundamental importan e.

Theorem 5.1 (Convex - ompa ti ation of W p n subspa e of Car ( ; R ). Then: (i)

*

FH1

FH1 W 1 p ( )

and

0

p

p ( )). Let 1

B - onvexifying

form

;

1;

 q p

subspa es

*

and H be a linear

C(W 1 p ( )B ) ;

of

and

C(W0 ( )B ), respe tively. In parti ular, (M(FH1 B ); (FH1 ) ; e1H ) is a 1 p

onvex  - ompa ti ation of W ( ) in its anoni al form, and similarly 1 p 1 1 1 B ) ; (F  1 p ) ; e ) is a onvex  - ompa ti ation of W ( ). (M(F  1 p 0 H W H W H



1;

of

*

;

0

(ii) If

;

(

)

0

;

;

*

(

)

H is separable and ontains a oer ive integrand, i.e. h(x ; s) £ a(x) % sp ;

;h ò H ;a ò L ( ) : 1

then the onvex

- ompa ti ations

FH1 B )

M(

1

and M(FH 

oer ive and sequentially lo ally ompa t.

(5.4) 1; p

W0

(

) B ) are both

B-

p ¡ 1 and (5.4) holds with equality, or if there is M dense in L p ( ; Rn ) su h that h u ò H with h u (x ; s) # s " u(x)p for u ò M , then these onvex - ompa ti ations

(iii) If

are norm- onsistent. The proof will be based on the following essential assertion by Ekeland and Temam, presented here without proof whi h is onstru tive and rather te hni al.

Lemma 5.2 (I. Ekeland and R. Temam [283℄).4 Let " ¡ 0, y ; y ò W p ( ), h ; :::; h m òCarp ( ; Rn ), m ò N, be given. Then there exists y ò W p ( ) so that 1

and let

1;

1

" " " " "y "

" 12 y " 12 y 1

2

!! !!X h ( x ; › y ( x )) x !!! i for any

1;

2

" " " " " "L 1 ( )

¢ "

and

(5.5a)

!!

" 21 h i (x ; ›x y (x)) " 12 h i (x ; ›x y (x)) dx !!!! ¢ " 1

2

1 ¢ i ¢ m . Moreover, this assertion also holds if W

Proof of Theorem 5.1. It su es to show that F

1;

p ( ) is repla ed by W 1 ; p ( ).

H is B - onvexifying for H 1

(5.5b)

!

0

# Car ( ; Rn ); p

B omposed from the balls B % in the norm (5.1).  # N , {nite subsets of Carp ( ; Rn )}, ordered by the relation £ , “, whi h % makes  a dire ted set so that we an use it to index nets. Let % ò R be given and 1 p y1 ; y2 ò W ( ) satisfy y i W 1 p ¢ % for i # 1; 2. For  # (l ; {h1 ; :::; h k }) ò , let us put y  # y with y from Lemma 5.2 with " # 1/ l . Let us note that, in parti ular, for any Æ ¡ 0 we have y  W 1 p ¢ max( y1 W 1 p ; y2 W 1 p ) % Æ if  # (l ; {h1 ; :::; h k }) is su iently large with respe t to the ordering of  ; namely if the set of integrands in  p

ontains the integrand h ( x ; s ) #  s  and l ò N is large enough. for (2.16) we want to use the base of

We put

;

;

;

4

(

)

(

)

;

(

)

;

(

)

In fa t, Ekeland and Temam [283, Chap.X, Proposition 2.11℄ proved even a bit stronger result but, on

the other hand, only for

W0

1;

p

(

). However, the modi ation for W 1;p ( ) is straightforward: it su es

0 “ and to get (5.5) by restri tion on .

to apply the original zero-tra e version to a larger domain

5.1 Convex ompa ti ations of Sobolev spa es

Ë 385

f # 1 (g; h)% with some g ò L q ( ) and h òCar p ( ; Rn ) 1 1 and ò R, we want to show that the net { f ( y  )}  ò  onverges to 2 f(y1 ) % 2 f(y1 ) and 1 p y  belongs eventually to the ball B % of the radius % in W ( ), whi h just means that 1 FCar p ;Rn is B - onvexifying with respe t to the base { B % } % òN of B ; f. (2.16). Let us rst suppose that y i W 1 p   % , i # 1 ; 2. Let us rst put q # 1 so that g ò L ( ). Then, using (5.5), for every  £ (2 max(1 ; g L

)/ " ; { h }) we an estimate: 

Considering now arbitrary

;

(

)

;

(

)





!! !! f ( y )  !!

(

)

" 12 f(y ) " 12 f(y 1

!! !! !!

2)

¢ %

!! !! 1 1 !! ! !!X g ( x )Œ y  ( x ) " y 1 ( x ) " y 2 ( x ) d x !!! 2 2 !

! ! !! 1 1 !![ h ℄(› y ) " [ h ℄(› y 1 ) " [ h ℄(› y 2 )!!! x x x  !! !! 2 2

¢

"

2

%

"

¢ ":

2

" ¡ 0 is arbitrary, the desired onvergen e of f(y  ) to 12 i#1 2 f(y i ) is thus proved and, moreover, it is evident that y  W 1 p   % provided the index k ò N is su iently

As

;

;

(

)

large. If

y1 W 1 p y ;

(

)

# % or y W 1 p # %, we must additionally use sequen es {y il }lòN   % and y il onverges to y i for l Ù  strongly in W p ( ) for 2

;

(

)

1;

su h that il W 1; p ( )

i # 1; 2. Then we exploit ontinuity of the fun tions from FCarp ;Rn and onstru t y For 1   q we an pro eed similarly, using in addition the fa t that y  onverges 1 1 q 1 p to 2 y1 % 2 y2 in L ( ), whi h follows from the boundedness of {y  }kòN in W ( ), 1 p q the ompa tness of the embedding W ( ) ’ L ( ), and the onvergen e of y  to 1 1 1 2 y1 % 2 y2 in L ( ). 1 1 p Be ause F is obviously a linear subspa e of C ( W ( )B ) ontaining onstants, H 1 1 p ( ) due to Theorem 2.22(i). M(F B ) is a onvex  - ompa ti ation of W H 1

(

)

arefully a net { l ;  } l òN ; k òN (details are omitted).

;

;

;

;

1

As to the point (ii), (5.4) allows to use Theorem 2.22(iv) to show that M(F B ) is

H

then

B - oer ive

and lo ally ompa t; namely the ondition (2.18) is to be exploited

k # 3 and f1 # 1 (1; 0), f2 # 1 ("1; 0), and f3 # 1 (0; h) with h from (5.4). The separability of H further yields the sequential lo al ompa tness. 1 p It remains to show (iii). As u ÜÙ (P h ( x ; u ( x )) d x ) with h from (5.4) holding

p n with equality denes (up to a onstant) just the norm of L ( ; R ), the weak* on1 vergen e of a sequen e { e ( y k )} k òN implies, beside the weak onvergen e of { y k } k òN H p 1 p in L ( ), also the onvergen e of { y k W 1 p } k òN . As the Sobolev spa e W ( ) is uniformly onvex for p ¡ 1, it implies the onvergen e of { y k } k òN in the norm of W 1 p ( ) by the Fan-Gli ksberg theorem. It proves that the ontinuous embedding is 1 p even (strong,weak*)-homeomorphi al. The same applies for W 0 ( ). for

/

;

;

(

)

;

;

In the alternative ase, one an use the se ond ondition in Theorem 3.39(iii) to see

e1H (y k ) Ù e1H (y) just means that y k Ù y weakly in L q ( ) and ›x y k Ù ›x y strongly p n ). Thus again y Ù y strongly in W 1 ; p ( ), respe tively in W 1 ; p ( ). in L ( ; Å k 0 that

R

Alternatively, the elements of the onvex

- ompa ti ations introdu ed in its y; ) omposed from a

anoni al form in Theorem 5.1 an be re ognized as ouples (

Ë

386

5 Relaxation in Variational Cal ulus: S alar Case

y and a generalized Young fun tional  re ording in the limit the y 1 1 p q ding i : W ( ) Ù L ( ) , H by H

usual fun tion

possible os illation/ on entration ee ts in the gradient ›x . Let us dene the embed;

*

i1H (y) :# (y; i H (›x y)) ; i : L p ( ; Rn ) Ù H

where H

YH

1;

*

p

(

has been dened by (3.84). Furthermore, let us put

) :# b lL q

p Y0; H ( ) 1;

Theorem 5.3. W 1)

of (

(5.6)

), H ; B i H „W 1

(

:# b lL q

), H ; B i H„ W 0 1

(

1;

*

p

1;

(

p

*

)… ;

(5.7a)

)… :

(5.7b)

(

p

1 q * ( ) ; L ( ) , H ; i ) forms a onvex  - ompa ti ations H p ( ) whi h is equivalent with (M(F 1 B ) ; (F 1 )* ; e 1 ) via the adjoint mapping H H H : (FH1 )* Ù L q ( ) , H * , i.e. the following diagram ommutes:

The triple ( Y H

1;

1;

*

W 1 p ( ) iH e1 ✠ ❅ ❅ ❘H 1 p Y H ( ) ✛ ✲ M(FH1 B ) 1 ( ) ;

1

;

*

The same holds true for

YH

1;

p

(

W0

1;

p

(

), FH1 W 1 p 0

), respe tively.

Proof. First, let us note that

;

(

1; p 1 1; p ( ), F , and H

) , and Y 0 ; H ( ) in pla e of W

1 is obviously linear and, thanks to suitable topologies,

also ontinuous. Moreover, (

1)

*

xes

W 1 p ( ) in the sense ( 1 ) Ž e1H # i1H , whi h follows from ;

*

the identity

´(

1 *

)

Ž e H (y); (g; h)µ # ´e H (y); (g; h)µ 1

1

1

# g; y % [ h℄(›x y) # ´i H (y); (g; h)µ 1

holding for any

y ò W 1 p ( ), g ò L q ( ), and h ò H . 

;

Furthermore, we want to prove that the adjoint operator ( ) : (F H ) Ù L q ( ) , H * is inje tive and has a weakly* ontinuous inverse if restri ted on the ane 1

1 *

M # { ò (FH1 ) ; # 1}. Indeed, for any 1 ; 2 ò (FH1 ) 1 # ( 1 ) 2 and # 1 # we have the identity *

manifold (



1 *

)

*

*

su h that

*

1 ;

1

(

g; h) %

#

(

1 *

1 ; (g; h) % 1 ; 1

#

(

1 *

2 ; (g; h) % 2 ; 1 # 2 ; 1 (g; h) %

)

)

g ò L q ( ), h ò H , and ò R. Due to the denition (5.2) of FH1 , it just means that  1 #  2 . Let us now suppose that we have a sequen e {  k } k òN in M and  ò M 1 1 q su h that {( )  k } k òN onverges to ( )  weakly* in L ( ) , H . This implies 

valid for any

*

*

*

5.1 Convex ompa ti ations of Sobolev spa es

lim  k ; (g; h) %

# lim 1

k Ù

k Ù

#

(

(

1 *

)

Ë 387

 k ; (g; h) %  k ; 1

1 *

)

 ; (g; h) %  ; 1 #  ; 1 (g; h) %

g ò L q ( ), h ò H , and ò R, whi h means pre isely that  k Ù  weakly* in 1 1 1 1 1 (F ) . As M(F B ) ’ M by the very denition of M(F B ) and ( ) Ž e H # i H has H H H 1 p 1 been shown, we an on lude that M(F B ) Ê Y H H ( ), the ane homeomorphism 1 p 1 1 ( ) being just ( ) . M(F B ) Û Y H H 1 p 1 p The modi ation for the ase W 0 ( ) in pla e of W ( ) is trivial. Å 

for any 1

*

*

;

;

*

;

;

YH

1;

To give a more detailed des ription of the elements of

p

(

p

) and of Y0 H ( ), we 1; ;

introdu e the set of the so- alled gradient generalized Young fun tionals, dened by

GH ( ; Rn ) :# p

† òH

*

; ;{y k }kòN bounded in W

1;

G0 H ( ; R ) :# †  ò H ; ;{y k }kòN bounded in W0 p

n

1;

*

p p

;

#  ‡;

(5.8)

-lim k òN H (›x k )

i

y

#  ‡:

(5.9)

G0 H ( ; Rn ) ’ GH ( ; Rn ) ’ YH ( ; Rn ). Unfortunately, we have to onne n Ù Rn to p ¡ 1. There is no loss of generality to suppose, for id : R p

Obviously, ourselves

y

): w

i

(

-lim k òN H (›x k )

*

): w

*

(

p

p

;

denoting the identity, that

1 ‹ id) ò H n :

(5.10)

(

Proposition 5.4. YH

1;

p

(

) #

p Y0; H ( ) 1;

#

Let

p ¡ 1 and (5.10) be valid. Then5

† ( y;  ) ò W

1;

† ( y;  ) ò W 0

p

1;

p

(

) , GH ( ; Rn ); (1 ‹ id) DZ  # ›x y ‡ ; p

and

) , G0 H ( ; R ); (1 ‹ id) DZ  # ›x y ‡ :

(

p

n

;

p

(5.11) (5.12)

) ò Y H ( ) there is a sequen e {y k }kòN bounded in W 1 p ( ) p n su h that y k Ù y and i H (›x y k ) Ù  weakly*. Then, by (5.8),  ò G ( ; R ). Moreover, H we obtain (1 ‹ id) DZ  # ›x y by passing to the limit in the obvious identity Proof. By (5.7a), for any ( y;

1;

;

1 ‹ id) DZ i H (›x y k ) #

(

›x

yk :

(5.13)

y; ) ò W 1 p ( ) , GH ( ; Rn ) su h that (1 ‹id) DZ  # ›x y. By 1 p (5.8) there is a bounded sequen e { y k } k òN in W ( ) su h that i H (›x y k ) Ù  weakly*. 1 p 1 p As W ( ) is reexive for p ¡ 1, there is also some y ò W ( ) su h that (after taking 1 p possibly a subsequen e) y k Ù y weakly in W ( ). Passing to the limit in (5.13) then yields ›x y # (1 ‹ id) DZ  # ›x y , so that y " y # is a onstant on . Then obviously i H (›x (y k % )) # i H (›x y k ) Ù  and also (y k % ) Ù ( y % ) # y. Altogether, i1H (y k % ) Ù p

;

Conversely, let us take some (

;

;

;



;









5

p ¡ 1, (1 ‹ id) DZ  lives in L p ( ; Rn ) due to Proposition 3.43(iii), so that the 1 ‹ id) DZ  # ›x y has a good sense in L p ( ; Rn ).

Let us note that, sin e

equality (

Ë

388

5 Relaxation in Variational Cal ulus: S alar Case

y; ), whi h shows that (y; ) ò Y H is trivial; note that even # 0.

1;

(

p

(

). The modi ation for the zero-tra e ase (5.12)

Å

y ÜÙ y : W 1 p ( ) Ù - ompa ti ation of W 1 p ( ). Let us remind ;

It will be also useful to extend the tra e operator

W "1/p; p ( 1

) on the parti ular onvex

;

that we suppose the Lips hitz boundary

.

Proposition 5.5 (Extension of the tra e operator). 1 ¢ q   pŽ

with

1

pŽ

If

p ¡ 1 and

from (1.40)

:

(5.14)

y ÜÙ y admits a (weak*,strong)- ontinuous p # 1, the boundary is of the C 1 - lass,6 and

then the tra e operator

YH

1;

p

(

) Ù L q1 (

).

If

(

ane extension

)

C( ) ‹ (Rn ) ’ H ; *

(5.15)



then the tra e operator extension

YH

1;

p

(

W 1 p ( ) Ù L1 (

) Ù r a(

;

)

admits a (weak*,weak*)- ontinuous ane

).

p ¡ 1 follows simply from the ompa tness ) for q 1 satisfying (5.14); f. (1.39) and (1.41). W Ù Let us go on to the ase p # 1. First, let us show that

Proof. The ase 1;

p ( )

of the tra e operator

L q1 (

:g ò C( ) ;g ò C( ; Rn ) : div g ò L q ( ) & n - g 

1

As

is a

C1 (

)

1



1

# g:

(5.16)

-boundary, we an re tify it lo ally by a dieomorphism7 so that lo ally

like a semi-spa e and an ane manifold of o-dimension 1. Then,

(x ) with x ò g ò C( ) we an take g1 ò C( ; Rn ) dened by g1 (x) # ng being the orthogonal proje tion of x , i.e. x " x has the dire tion n . As the ve tor eld

) # g, so that g1 is parallel, we have surely div g1 # 0. Moreover, n - g1  # n - (ng (5.16) folds for the re tied ase. The original ase will yield (5.16) with div g 1 ò C ( )

we an treat



for a given









possibly nonvanishing.

r a(

- ompa ti ation of L1 ( ), see Ex1 1 p 1 ample 3.73. Then we an use Proposition 2.32 for U 1 # W ( ), F1 # F , U 2 # L ( ), H 1 F2 # FH2 with H2 # C( ) ‹ R ’ Car ( ; R), and  being the tra e operator y ÜÙ y . The boundedness (2.24) of  (with respe t to the anoni al norm bornologies) is obvious. Therefore it remains to verify (2.25), whi h means here that, for any g ò C ( ), the 1 mapping y ÜÙ P g ( x ) y ( x ) d S belongs to F . In view of (5.3) with (5.15), this means that H Now we are to realize that

) is a onvex

;

*

:y ò W

6

This means that

1;

p

(

) :

X

g(x)y(x) dS # X g0 (x)y(x) % g1 (x) - ›x y(x) dx



an be divided into a nite number of overlapping parts, ea h of them being a

graph of a s alar ontinuously dierentiable fun tion on an open subset of

7

This means a ontinuously dierentiable homeomorphism.

Rn"

1

.

5.1 Convex ompa ti ations of Sobolev spa es

Ë 389

g0 ò L q ( ) and g1 ò C( ; Rn ). Using Green's theorem, we an see that this a tually holds be ause of (5.16): it su es to put g 0 # "div g 1 . Å 

for some



- ompa ti ations

Let us end this se tion by several examples of onvex

W 1 p ( ).

of

;

Example 5.6 (Coarse onvex - ompa ti ations).

The hoi e

H # {0} p

(5.17)

Y H ( ); L q ( ) , H ; i1H ) Ê (W 1 p ( ); (L q ( ); weak); i) with i being the em1 p q bedding W ( ) ’ L ( ). The ane homeomorphism is just ( y; 0) Û y . A less trivial example for the reexive ase (i.e. p ¡ 1) takes

reates (

1;

*

;

;

H # L p ( ) ‹ (Rn ) ; 

*

(5.18)

p

- ompa ti ation (Y H ( ); L q ( ) , H ; i1H ) equivalent with 1 p 1 p (W ( ); (W ( ) ; weak) ; id) via ane homeomorphism ( y; ›x y ) Û y . Both onvex  - ompa ti ations are lo ally equivalent with ea h other, the former one being 1;

whi h reates the onvex ;

*

;

oarser than the latter one; f. also Example 2.27.

Example 5.7 (Fun tions of bounded variations).

In the nonreexive ase

p # 1,

a

reasonable hoi e is8

H # C0 ( ) ‹ (Rn ) : *

(5.19)

This hoi e reates basi ally the spa e of fun tions with bounded variation, dened standardly as9

BV( ) :# where ›x

y ò D( )

*

– yòL

1

(

);

X ›x y (d x )

  %— ;

(5.20)

is the distributional gradient whi h is supposed to be simultane-

and ›x y means the variation of this measure.10 Realizing that n ) is for the hoi e (5.19) homeomorphi just with r a( ; n ) Ê C ( ; n )* Ê 0

ously a measure on

YH ( ; R R R H , the ane homeomorphism between BV( ) and Y H1 1 ( ) is via the mapping y ÜÙ ( y; ›x y ) with ›x y understood in the sense of measures on , or equivalently as a fun tional on H with H from (5.19). Let us also note that H from (5.19) does not satisfy (5.15) 1 1 and a tually the tra e operator y ÜÙ y  annot be extended ontinuously on Y H ( ); 1 1 indeed, one an easily onstru t a bounded sequen e { y k } k òN ’ W ( ) su h that y k  # 1 and y(x) # 0 for dist(x ; ) £ 1/k, and see that w*-limkÙ y k # 0 in BV( ). 1

;

*

*

;

;



C0 ( ) is the losure in C( ) of the spa e of fun tions with a ompa t support in .

8

Re all that

9

See Attou h et al. [33, Chap.10℄, Giusti [356, Denition 1.3℄, or Ziemer [817, Se t. 5.1℄.

10

More pre isely,

P

›x

ydx # sup{P y(x) div g(x) dx; g ò C(01) ( ;

Denition 1.1℄ or Ziemer [817, Denition 5.1.1℄.

Rn ;

)

g C0 ( )

¢ 1}, see Giusti [356,

Ë

390

5 Relaxation in Variational Cal ulus: S alar Case

In other words,

BV( ) is too oarse onvex - ompa ti ation of W

1;1

(

) so that the

tra es of BV-fun tions has not a good sense.11

Example 5.8 (Extension of W

1;1

(

) by J. Sou£ek).

Another hoi e in the ase

p # 1 is

H # C( ) ‹ (Rn ) ; *

(5.21)



Y H1 1 ( ), whi h is equivalent with the extension12 W 1  ( ) of W 1 1 ( ) devel1 oped by J. Sou£ek [733℄ and onsisting of fun tions y ò L ( ) whose gradient is a measure on . More pre isely, ;

reating

;

;





W 1  ( ) :# ;



r a( ; Rn ); ; a sequen e {y k }kòN : w*-lim (y k ; ›x y k ) # (y; ) ‡

† ( y;  ) òr a( ) ,



(5.22)

k Ù

W 1 1 ( ) embedded into W 1  ( ) via the mapping i : y ÜÙ (y; ›x y). Realizing the q n embedding L ( ) ’ r a( ) and the equivalen e H Ê r a( ; R ) ( f. Examples 3.50 1 1 1  1% n and 3.73), we get13 the equivalen e ( Y ( ) ; r a( ) ; i) via the H ( ); H ; i H ) Ê (W q n ane homeomorphism being just the embedding L ( ) , H Ù r a( ) , r a( ; R ). 1  Let us also note that W ( ) is a stri tly ner onvex  - ompa ti ation of W 1 1 ( ) than BV( ). For example, the weak* limits of the sequen es {y k }kòN and 1 1 {" y k } k òN bounded in W ( ), with y k su h that y k  # 1 and y ( x ) # 0 for dist( x ; ) £ 1/k, an be distinguished from ea h other in W 1  ( ), but not in BV( ). 1 If the boundary is of the lass C , it was proved by Sou£ek [733, Thm. 1(iii)℄ 1  that, having ( y;  ) ò W ( ), there exists a unique measure  1 ò r a( ) su h that 1 <  1 ; n j g  > # < y; g / x j >% holds for any g ò C ( ) and any 1 ¢ j ¢ n . The mea1  sure  1 is alled the tra e of ( y;  ) and the operator ( y;  ) ÜÙ  1 : W ( ) Ù r a( ),

with

;

;



*





;

*

;





*





;



;

;

;



(

)

;



(

)



;



being (weak*,weak*)- ontinuous (see [733, Thm. 2(ii)℄), is obviously the extension of the usual tra e operator

y ÜÙ y : W 1 1 ( ) Ù L1 ( ;

). It is evident that it oin ides

with the extension stated in Proposition 5.5.

Example 5.9 (The bi-dual of W whi h means here

11

1;1

(

)).

H # L

Considering again



(

p # 1

and taking (5.18),

) ‹ (Rn ) ; *

In Giusti [356, Se t. 2℄ the tra es of BV-fun tions are dened as essential limits from

L1 (

(5.23)

, living in

); f. also the so- alled inner tra es by Sou£ek [733℄. However, su h a tra e operator is not

weakly* ontinuous.

W 1 ( ).

12

In [733℄, a slightly dierent notation is used, namely

13

We must use also the Bana h-Steinhaus prin iple together with the oin iden e of the bornology

L1 ( ) with the relativized bornology of r a( ), whi h implies that any sequen e {y k }kòN ò W 1;1 ( ) 1% n 1;1 su h that { i ( y k )}kòN is weakly* onvergent in r a( ) is inevitably bounded in W ( ). Also, we 1% n must use the metrizability of the weak* topology relativized on bounded subsets of r a( ) . This of

implies the oin iden e of the bounded losure used in (5.7a) and the sequential losure used in (5.22).

5.1 Convex ompa ti ations of Sobolev spa es

Ë 391

Y H1 1 ( ) Ê W 1 1 ( ) . The ane homeomorphism is just made by the adq n q joint operator to ( g 0 ; g 1 ) ÜÙ ( g 0 ; g 1 ‹ id) : L ( ) , L ( ; R ) Ù L ( ) , H . It follows from the fa ts that the fun tionals of the form y ÜÙ < g 1 ; ›x y > % < g 0 ; y > with q n 1 1 (g0 ; g1 ) ò L ( ) , L ( ; R ) over14 all ontinuous linear fun tionals on W ( ) 1 1 1 (therefore the topology indu ed on W ( ) via the embedding i is just the weak H 1 1 1 1 topology) and that Y ( ) H ( ) is, in fa t, the ompletion of parti ular balls of W ;

we obtain

;

**











;

;

;

;

with respe t to the weak uniformity, whi h is however just the bi-dual spa e by the

y; ) of the bi-dual W 1 1 ( ) an be understood as fun tions y whose gradient is a nitely-additive measure  ò H Ê vba( ; Rn ). Let us note that this onvex - ompa ti ation of 1 1 W ( ) is stri tly ner than Sou£ek's extension from Example 5.8 be ause of analogous reasons as vba( ) ± r a( ); f. Example 3.50. Goldstine theorem. Likewise in Sou£ek's extension, the elements ( ;

spa e of

*

;



Example 5.10 (Extension of W H #

–h

1;1

(

) by I. Fonse a).

For

p # 1 and

: , Rn Ù R; ;h ò C ( ; ) : h(x ; s) # h Œx ; 0

0

0

s s

 

 s —

(5.24)

( f. also (3.105)), we obtain the extension proposed by Irene Fonse a [316℄.15 Elements

Y H1 1 ( ) an be identied in this ase as ertain ouples (y;  ) ò L1 ( ) , F( ; Rn ). n p Obviously, C 0 ( ) ‹ (R ) ’ H ’ l( C ( ) ‹ Ô (R )) provided R is a greater ring than that ;

of



*



from (3.45) so that Fonse a's extension is (even stri tly) ner than the

BV-fun tions on

and (even stri tly) oarser than the extension Y H1 1 ( ) with H from (5.26) below. ;

Example 5.11 (Young-measure extension).

Taking

H # L1 ( ; C0 (Rn )) ; we are in position that

YH ( ; Rn ) Ê Yp ( ; Rn ), p

(5.25)

is (up to this equivalen e) omposed from the ouples (

y;

) with

p

Y H ( ) y ò L q ( ) and

f. also Example 3.69. Then

1;

ò Yp ( ; Rn ). This is basi ally the original on ept for relaxation of non onvex varia-

tional problems developed16 by Young [805, 807℄; the ouple (

alized surfa e (or, if

y;

) was alled a

gener-

n # 1, a generalized urve) while the omponent y was addressed

as a tra k.

Example 5.12 (Finer extensions).

The previous extensions still lost some (or all) desir-

able properties like lo al ompa tness, B - oer ivity, and norm- onsisten y. Therefore

14

See, e.g., Adams [4, Thm. 3.8℄.

15

However, the ve torial ase is treated in [316℄ for the purpose of a proof of lower semi ontinuity of

the fun tional of the type

y

ÜÙ

P

h(x ; ›x y(x)) dx with h(x ; -) positively homogeneous and quasi on-

vex.

16

Of ourse, more than half a entury ago, Young ould not be familiar with all those ne te h-

niques needed for problems with

L ( ;

Rn

).

L p -stru ture so that he had to suppose gradients apriori bounded in

392

Ë

5 Relaxation in Variational Cal ulus: S alar Case

it is worth to treat also ner extensions whi h would satisfy the ondition (5.4). An example for it might be the hoi e

H # C ( ) ‹ Ô p (R )

(5.26)



with

R being a suitable ring of ontinuous bounded fun tions on Rn . Then Y H

1;

p

(

) is

(up to an equivalen e) omposed from the ouples where the se ond omponent is a DiPerna-Majda measure; f. also Example 3.70. Furthermore, taking

H # G ‹ Ôp (R ) with some linear subspa e

C( ) ’ G ’ L

amples 3.48 and 3.71. Another of



(

(5.27)

) would yield still a ner extension; f. Ex-

B - oer ive lo ally ompa t onvex - ompa ti ation

W 1 1 ( ) an be obtained by a slight renement of Fonse a's extension, repla ing ;

H from (5.24) by ( f. also Example 3.72): H # PSfrag repla ements PSfrag repla ements

Summary 5.13.

–h

For

) : h(x ; s) # h Œx ; : , Rn Ù R; ;h ò C( , 0

p ò (1; )

0

s s

 

 s — :

(5.28)

the relations among the above examples an be dis-

played by the diagram

DiPerna,Majda

Young

W 1; p ( )

(5.26)

while for

(5.25)

W 1; p ( )

(5.18)

(5.17)

p # 1 the relevant diagram looks as:

if DiPerna,Majda (refined) (5.27) G#L (3.45)

and

bidual of W 1 1 ( )

Soucek W 1  ( )

(5.23)

(5.21)

;



(

)



R ontains (3.45)

Fonseca (refined)

DiPerna,Majda (5.26) if

(5.28)

BV( ) (5.19)

R ontains (3.45)

Fonseca

Young

(5.24)

where ea h arrow goes from a ner onvex properties of the above onvex

W 1;1 ( ) (5.17)

(5.25)

ing table:

;

- ompa ti ation to a oarser one. Some

- ompa ti ations an be summarized in the follow-

Ë 393

5.1 Convex ompa ti ations of Sobolev spa es

Convex H  - ompa ti ation: from: W 1 p ( ) (5.17) W 1 p ( ), p ¡ 1 (5.18) BV( ), p # 1 (5.19) W 1  ( ), p # 1 (5.21) W 1 1 ( )** , p # 1 (5.23) Young (5.25) Fonse a, p # 1 (5.24) DiPerna-Majda (5.26)

B - oer ive

no no no no no no yes yes

; ;

; ;

;

Tab. 5.1. Properties of onvex

Properties sequentially lo ally norm linear B - oer ive ompa t onsistent manifold no no no yes yes no no yes yes no no yes yes no no yes yes no no yes no no no no yes yes no no yes yes yes no

- ompa ti ations of the Sobolev spa e W 1;p ( ).

Remark 5.14 (Extreme gradient Young measures).

The analog of Proposition 3.9, i.e.

that all the extreme points of the onvex set of gradient Young measures are Dira s a.e. on

,

holds for

n # 1

only. Quite surprisingly, ounterexamples for

ist17 with even weak* ontinuous non-Dira gradient Young measures for

n £ 2 exn £ 3, see

V. ’verák onstru tion [700, Prop. 4.2℄.

Example 5.15 (Fun tions of bounded deformation). In the ve torial ase of fun tions

Ù Rm , the BV-extension of W ( ; Rm ) from Example 5.7 or the Sou£ek's extension m , n ) or H # C ( ) ‹ (R m , n ) , respe tively. For from Example 5.8 take H # C ( ) ‹ (R appli ations in small-strain elasti ity when m # n , one an onsider smaller spa es 1;1

*

0

n,n H # C0 ( ) ‹ (Rsym )

*

H # C( ) ‹ (Rsym ) n,n



*



*

or with

(5.29a)

Rsym :# n,n

A ò

R

n,n

; A#

A  :

(5.29b)

The former hoi e (5.29a) reates basi ally the spa e of fun tions with bounded deformation, dened standardly as18

BD( ; Rn ) :# † y ò L ( ; Rn ); 1

s y :# 1 (› y)% 1 › y òr a( ; 2 x 2 x

›x

n,n Rsym

)‡

;

(5.30)

s y stands for the symmetri distributional gradient. Let us also note that BD ( ;

i.e. ›x

is a stri tly oarser onvex

- ompa ti ation of W

1;1

(

; R

n ) than

W

1;

 ( ;

R

n ).

Rn

)

n £ 2 # Æ u x % Æ"u x if x ò K ’ and x # Æ if x ò \ K with an arbitrary u ò L ( ; Rn ) and K # K , ::: , K with some Cantor sets K i ’ R of positive Lebesgue measure (also alled SmithVolterraCantor sets), i # 1 ; :::; n , f. [700, Prop. 3.1℄.

17

The onstru tion of an extreme non-Dira gradient Young measure by V. ’verák [700℄ for

leads to a highly dis ontinuous Young measures: 

1

x

2

1

( )

1

2

( )

0

2

The rst momentum is obviously zero so that it is indeed a gradient Young measure with an underlying

y ò W 1; ( ) onstant. 18

These spa es have been invented essentially by P.-M. Suquet [740℄, f. also [755℄. Often, the notation

BD( ) is used instead of BD( ; Rn ).

394

Ë

5.2

Relaxation of variational problems;

5 Relaxation in Variational Cal ulus: S alar Case

p ¡ 1.

In this se tion we will deal with a problem of the variational al ulus

(PVC )

. Minimize > subje t to F

(y) # X '(x ; y(x); ›x y(x)) dx % X (x ; y(x)) dS

y ò W 1 p ( ) : ;

Su h problems appear in various bran hes of ontinuum me hani s, being related with stati ongurations governed by variational prin iples of the minimum-energy type; in the solid-phase me hani s, a typi al example is the anti-plane shear problem.19 Here

' : ,R,Rn Ù R represents a potential-energy density and  : ,R Ù R

is an energy density oming from external for es a ting on the boundary. The boundary term orresponds to (possibly nonlinear) boundary onditions of the Robin type. The ase of Diri hlet boundary onditions would require only quite straightforward modi ations; f. Remark 5.27 below. We will suppose the p -polynomial growth of the potential density ' with p ò 1; %) but we will admit '(x ; r; -) non onvex. The problem (PVC ) need not have any

(

y

solution due to possible os illation ee ts, i.e. every minimizing sequen e { k } k òN for (PVC ) inevitably exhibits faster and faster os illations in ›x

y k ; in ontinuum-me hani s

interpretation, this ee t is usually alled a ne stru ture. Therefore we are for ed to make a suitable relaxation of the problem. Of ourse, we will again prefer an extension by ontinuity, whi h is more natural than a lower semi ontinuous extension20 , though some fundamental results for lower-semi ontinuously extended problems ( f. Theorem 5.18 below) will be employed here too. The ontinuous extension has, on

ontrary to the mere lower-semi ontinuous extension, also the advantage that the solutions to the relaxed problem hold enough relevant limit information about the ne stru ture in gradients of minimizing sequen es, representable typi ally by a Young measure; let us agree that, having in mind ontinuum-me hani s interpretation, we will o

asionally address this limit ne stru ture as a mi rostru ture. Basi ally, we will extend (PVC ) on a suitable onvex

- ompa ti ation of W 1 p ( ) ;

onstru ted in the previous se tion by means of a suitable (i.e. su iently ri h but still separable)

C( )-invariant linear subspa e H ’ Carp ( ; Rn ). It leads us naturally

to impose the following data quali ation:

q; p;1

' òCAR H

19

(

, R , Rn ; R)

and

 òCarq1 ( ; R) ;

(5.31a)

We refer, e.g., to Friese ke [336℄, Gurtin and Temam [366℄, Horgan [393℄ and Raymond [632℄. For

n # 1 see also Brandon and Rogers [148℄ or Müller [557℄.

20

For extension of variational problems by lower semi ontinuity we refer, e.g., to Buttazzo [161℄, Da-

orogna [242℄, Ekeland and Temam [283℄, et .

p¡1

5.2 Relaxation of variational problems;

with

1 q p

*

and

.

Ë 395

q1 satisfying (5.14). Besides, we will assume the oer ivity of our

data in the sense

'(x ; r; s) £ a(x) % br % sp for some

a ò L1 ( ), a1 ò L1 (

),

(x ; r) £ a1 (x)

and

(5.31b)

b; ò R% and ò (0; q℄. Based on Proposition 5.4, we an

onstru t the following relaxed problem

 (y; ) # X

. Minimize 6 6

(R H PVC )





> subje t to 6 6 F

[

' Ž y DZ ℄(dx) % X  Ž y dS

1 ‹ id) DZ  # ›x y ; p y ò W 1 p ( ) ;  ò GH ( ; Rn ) :

(

;

Of ourse, (5.10) is to be assumed and the integrals are to be understood in the sense of measures respe tively on

and on

, if needed. Let us note that, for a given

', (5.31a)

H to be su iently large or, in other words, the used onvex - ompa ti a-

requires

tion to be su iently ne; e.g., following (4.122) with (5.10), it su es to take

H # g -(' Ž y) % g0 ‹id; g ò C( ); g0 ò L p ( ; Rn ); y ò L q ( ) : 





(5.32)

The following assertion justies (R H PVC ) as a orre t relaxation of (PVC ).

Proposition 5.16 (Corre tness of the relaxation s heme).

Let (5.10), (5.31) be valid.

Then: The relaxed problem (R H PVC ) has a solution and

(i)

(ii) Every solution (

min(RH PVC ) # inf (PVC ).

y; ) of (RH PVC ) an be attained by a minimizing sequen e {y k }kòN 1 # (y; ).

for (PVC ) in the sense w*-lim k Ù i H ( y k )

(iii) Conversely, every minimizing sequen e for (PVC ) has (after being embedded via

i1H )

a weak* luster point and ea h su h a luster point solves (R H PVC ). Proof. First, let us noti e that the embedding tional

:

L q ( )

,

p YH ( ;

R Ù R dened in n)

W 1 p ( ) ’ L q ( ) is ompa t, the fun ;

,weak*)- ontinuous y ÜÙ y :

(R H PVC ), is (strong

as a onsequen e of Lemma 3.101 and of the ontinuity of the tra e operator

W 1 p ( ) Ù L q1 (

 in (5.31a). Moreover,  is the extension of the original potential in (PVC ) in the sense  ( y; i H (›x y )) #  ( y ). Note that (5.31b) ensures the oer ivity in the sense lim y 1 p Ù (y) # % W

;

) together with the growth restri tion of







;

(



)

be ause of the estimate

p

(y) £ bX y dx % ›x y L p

(

;R n )

" a L1 " a (

)

1

L1 (

)

;

a, a1 , b, , and ome from (5.31b). y 1 1 p quen e must be bounded in W ( ) and thus { i ( y k )} k òN has a weak* luster point H p 1 p n ( y;  ) in W ( ) , H . Obviously,  ò G ( ; R ) and also (1 ‹ id) DZ  # ›x y an be H obtained by the limit passage in (5.13), so that ( y;  ) is admissible for (R H PVC ). By ontinuity, we have also  ( y;  ) # inf (PVC ).

where

Let us take a minimizing sequen e { k } k òN for (PVC ). As (PVC ) is oer ive, this se;

;

*



396

Ë

5 Relaxation in Variational Cal ulus: S alar Case

y; ) does not solve (RH PVC ). Then there is  (y0 ; 0 )    (y; ). By Theorem 5.3 and 1 p Proposition 5.4, ( y 0 ;  0 ) is attainable by a sequen e { y k } k òN ’ W ( ) (embedded 1 via i ). By the ontinuity of  , lim k Ù  (y k ; i H (›x y k )) #  (y0 ; 0 )   inf (PVC ), a onH Let us suppose, for a moment, that (

y ; 0 )

( 0

admissible for (R H PVC ) su h that





;









Å

tradi tion.

It is now ertainly worth investigating a stability of the relaxed problem under a

ertain data perturbations.21 Namely, we will onsider a perturbed problem

(PVC ; " )

. Minimize > subje t to F

with the perturbed data

with some

'"

 " (y) # X ' " (x ; y(x); ›x y(x)) dx % X  " (x ; y(x)) dS

y ò W 1 p ( ) : ;

and

 " satisfying



' " (x ; r; s) " '(x ; r; s) ¢ (a(x) % brq % sp )" ;

(5.33a)



 (x ; r) "  (x ; r) ¢ (a(x) % br

(5.33b)

"

a ò L1 ( ) and 1   q   p

*

and

)

"

q1 satisfying (5.14).

Proposition 5.17 (Stability of relaxed problem). the perturbed data satisfy (5.31) with

q1

Let (5.10) hold, both the original and

a, b, , and independent of ", and also (5.33) be "

valid. Then the relaxed perturbed problem (R H PVC ) always possesses a solution and

lim min(RH PVC" ) # min(RH PVC ) ;

(5.34a)

Limsup Argmin(RH PVC" ) ’ Argmin(RH PVC ) :

(5.34b)

" Ù0

" Ù0

"

Proof. We an use again Proposition 5.16 to show that (R H PVC ) has a solution. Then the stability (5.34a) and (5.34b) will follow from Proposition 4.5 (for the ase

"

# 0) if one veries the assumptions (4.7b) and (4.7 ).

R " # R and

However, (4.7b) follows immediately from (5.33). As the oer ivity ensured by (5.31b) is assumed uniform and of

"

 (0; i H (0))

is bounded from above independently

" ¡ 0 thanks to (5.33), we an see that also (4.7 ) is valid in this ase.

Å

- om1 p Y H ( ) of W 1 p ( ) with H # L p ( ) ‹ (Rn ) , f. Example 5.6. Then 1 p Y H ( ) Ê W 1 p ( ) equipped with the weak topology. Yet, the potential  annot be extended ontinuously unless ' ( x ; r; -) is ane, whi h is not an interesting ase, An alternative relaxation s heme for (PVC ) relies on the oarse onvex ;

pa ti ation ;



;

*

;

however. Instead of the ontinuous extension, one ould think about lower semi ontinuous extension, whi h would require

'(x ; r; -) onvex. In general, we an rely only  of , dened by

on a lower semi ontinuous regularization

21

We refer also to Dont hev and Zolezzi [272, Chap. 8℄ or Zolezzi [820℄.

5.2 Relaxation of variational problems;

(y) #

p¡1

.

lim inf ( y ) :

y ò W 1; p ( ) yÙ y weakly

The advantage of working on the original spa e

(5.35)



W 1 p ( ) ;

Ë 397

is deteriorated, beside the

loss of information in omparison with the ne relaxation (R H PVC ), by a ne essity to

, whi h is not entirely trivial. For v ò C(Rn ) bounded from below, we need to dene the onvex envelope v ò C(Rn ) by the formula

evaluate

**

epi(v

**

)

# oepi(v) :

(5.36)

Then the following lassi al result is at our disposal:

Theorem 5.18 (Integral formula for ).22 Let (5.31) be valid. Then (y) # where

'

**

(

X '

**

(

x ; y(x); ›x y(x)) dx % X  (x ; y(x)) dS ;

(5.37)

x ; r; -) : Rn Ù R is the onvex envelope of '(x ; r; -).

The nontrivial formula (5.37) enables us to pose the oarsely relaxed problem

X '

Minimize subje t to

**

yòW

1;

(

x ; y(x); ›x y(x)) dx % X (x ; y(x)) dS

p ( ) :

/ ? G

(5.38)

Then Theorem 5.18 ensures that (5.38) is a orre t relaxation for (PVC ) and Proposition 4.2(ii) immediately enables us to establish relations with the nely relaxed problem (R H PVC ) summarized in the following assertion; note that the oer ivity of the problem together with the metrizability of the weak topology relativized on bounded subsets of

W 1 p ( ) enables us to formulate it in terms of sequen es. ;

Corollary 5.19 (Relations between (RH PVC ) and (5.38)).23 Let (5.10) and (5.31) be valid. Then: (i)

The oarsely relaxed problem (5.38) has a solution and every its solution an be attained (weakly in

22

W 1 p ( )) by a minimizing sequen e for (PVC ).

The role of onvexity in the

;

s-variable

has been re ognized already in 20ties sin e the work by

n-dimensional '(x ; r; s) is lo ally Lips hitz

Tonelli [764℄. For the relaxation in the 1-dimensional ase see Raymond [628℄ and in the

ase but with

'(x ; r; s) independent of r

see Buttazzo [161, Se t. 4.4℄. If

ontinuous in a ertain sense, we refer to Da orogna [242, Chap. 5, Cor. 2.3℄. The full statement is basi ally by Ekeland and Temam [283, Chap. X, Cor. 3.8℄, realizing also that, thanks to the growth restri tion on

 in (5.31a), the boundary term is even ontinuous. Let us also note that, due to the oer ivity

(5.31b), the sequential weak lower semi ontinuity, whi h is often onsidered, basi ally oin ides with the mere lower semi ontinuity thanks to the Eberlain-’muljan theorem.

23

As to (i), see also Ekeland and Temam [283, Chap.X, Corollaries 2.16 and 3.2℄. The relation between

PVC ) has been also investigated by Pedregal [598, Corol-

(5.38) and the Young-measure relaxation of ( lary 3.3℄.

Ë

398

5 Relaxation in Variational Cal ulus: S alar Case

(ii) Conversely, every minimizing sequen e for (PVC ) possesses a weakly onverging sub-

sequen e whose limit inevitably solves (5.38). (iii) For every solution

y to (5.38), there exists  ò YH ( ; Rn ) su h that (y; ) solves the p

nely relaxed problem (R H PVC ). (iv) Conversely, if (

y; ) solves (RH PVC ), then y is a solution to (5.38).

Let us ome ba k to the ontinuous relaxation s heme (R H PVC ). It should be emphasized that it may be not entirely satisfa tory, espe ially as far as the optimality

onditions on ern, see Remark 5.28. However, it is possible to modify this relaxation, using the following assertion whi h basi ally gives an ee tive hara terization of

p-

non on entrating gradient generalized Young fun tionals.

Lemma 5.20. Let H be separable and satisfy (5.10),  ò YHp ( ; Rn ) be p-non on entrap ( ). Then  ò G p ( ; R n ). ting, and (1 ‹ id) DZ  # ›x y for some y ò W H 1;

Proof.24 First, let us noti e that, if being true, the assertion in question holds also for a

H . In other words, it su es to prove the assertion for some larger spa e H ontains, together with h , also its  oer ive modi ation h l dened by subspa e of

of the test integrands. Therefore, we may and will assume that

h l (x ; s) # max h(x ; s); sp " l : Supposing

lò

N, we may additionally assume the enlarged spa e again separable.

p H an be extended on the enlarged spa e, remaining again a (p-non on entra-

Note that every ( -non on entrating) generalized Young fun tional on the original spa e

ting) generalized Young fun tional. Let us put

K y #  ò GH ( ; Rn ); (1 ‹ id) DZ  # ›x y : p

p

K y is an image of the set Y H ( )  1 p ({ y } , H ) via the anoni al proje tion of W ( ) , H Ù H . Sin e, by Theorem 5.3, 1 p Y H ( ) is onvex, K y is onvex too. Let us take h ò H and put I h # inf {<  ; h >;  ò K y }. For a xed ª ò N, we an apply Theorem 5.18 to the fun tional y ÜÙ P h l ( x ; ›x y ( x )) d x , whi h guarantees an exis

1 p ten e of a sequen e25 { y k } k òN su h that y k Ù y weakly in W ( ) and simultaneously P h l ( x ; ›x y k ( x )) d x Ù P h l ( x ; ›x y ( x )) d x where h l ( x ; -) denotes again the onvex



envelope of h l ( x ; -); here p ¡ 1 has been employed. As h l is oer ive, the sequen e p n {›x y k } k òN must be bounded in L ( ; R ) so that { i H (›x y k )} k òN onverges (after taking p n possibly a ner subsequen e) to some  ò Y H ( ; R ). Obviously,  belongs also to K y so that I h l ¢ <  ; h l > # lim k Ù P h l ( x ; ›x y k ( x )) d x # P h l ( x ; ›x y ( x )) d x .



1;

Taking (5.11) into a

ount, one an easily see that *

;

*

*

;

;

**

**





**





24

The proof uses some ideas by Kinderlehrer and Pedregal [424, 426℄.

25

In fa t, the oer ivity of

with sequen es.

hl

together with the separability of

W 1;p ( ) eventually allows us to work

5.2 Relaxation of variational problems;

As

p¡1

.

Ë 399

 is supposed p-non on entrating and H is separable, by Proposition 3.78(i) it ò Yp ( ; Rn ). Then one obtains

admits a representation by means of a Young measure

 ; h l

#X



X

Rn

h l (x ; s)

x (d s ) d x

# X hl

**

Œx ; X

s

Rn



x (d s ) d x

# X h l (x ; ›x y(x)) dx : **



I h l ¢ . I Žh # inf {;  ò K y p-non on entrating}. As h l is ertainly Ž # I as a onsequen e of Lemma 4.22(i). Therefore, bounded from below, we have I hl hl Ž we have I ¢ <  ; h l >. hl p n For arbitrary p -non on entrating  ò Y ( ; R ), it holds lim l Ù <  ; h l > # <  ; h >. H Indeed, suppose the ontrary, i.e. <  ; h l " h > £ Æ ¡ 0 for all l ò N, and take a sequen e p n p { u k } k òN ’ L ( ; R ) su h that i H ( u k ) Ù  and the set { u k  ; k ò N} is relatively weakly 1 p

ompa t in L ( ). From the estimate  h ( x ; s ) ¢ a ( x ) % b  s  we get also 0 ¢ [ h l " h℄(x ; s) ¢ a(x) % (b%1)sp and [h l " h℄(x ; s) # 0 whenever s ¢ ((l " a(x))/(b% 1))1 p #:

l (x). Then we have the estimate Altogether, we showed that Let us abbreviate



/

X [h l

" h℄(x ; u k (x)) dx ¢

X {

#

X {

a(x) % (b%1)u k (x)p dx

x ò ;  u k ( x )£ l ( x )}

a(x) % (b%1)u k (x)p dx :

x ò ; a ( x )%( b %1) u k ( x )p £ l }

Æ/2 independently of k ò N when taking l large enough be ause the set { a % ( b %1) u k k ò N}, being relatively weakly ompa t in L1 ( ), is

This an be made smaller than

p  ;

also uniformly integrable thanks to the Dunford-Pettis theorem 1.28(ii). Passing to the

 ; h l " h> ¢ Æ/2, a ontradi tion. I Žh l ¢ . Indeed, for any " ¡ 0 we an take some p -non on entrating  " ò K y su h that <  " ; h l > ¢ <  ; h l > % " . As both  " and  are p-non on entrating we an pass to the limit, whi h gives 6 subje t to F

y ò W0

1;1

X ' ( x ; y ( x ) ; ›x y ( x ))

(

) :

dx

5.4 Relaxation of variational problems;

p#1

.

Ë 411

Su h problem is related with the lassi al nonparametri minimal-hypersurfa e prob-

lem, also alled the Plateau problem: Minimize subje t to If

n # 2,

X y1 %

›x

y(x)2 dx

/

y ò W 1 1 ( ) ; y # y 0 : ;



(5.64)

? G

this problem has a straightforward interpretation: it minimizes the 3-

dimensional surfa e in

,

R that is the graph of the fun tion y

the side ondition that this graph is xed on the ontour

: Ù

by a pres ribed

R under

y 0 ò L1 ( 

).

1

Obviously, su h problem an be overed by (PVC ) if one takes

'(x ; r; s) #

y1 %



s % u(x)2

(5.65)

u ò L1 ( ; Rn ) su h that u # ›x y0 for some y0 ò W 1 1 ( ) having the 1 pres ribed tra e y 0 on ; then y solves (PVC ) if and only if y % y 0 solves (5.64). 3 Though a soap bubble spanned on the ontour {( x ; a ) ò R ; x ò ; a # y 0 ( x )} an ;

with a suitable



solve readily45 the minimum surfa e problem (if one negle ts minor ee ts as, e.g., a gravitational for e), it is known that (5.64) need not have any solution. The point is that the shape, whi h the soap bubble takes, need not be the graph of any fun tion

Ù

R. Thus (5.64) orresponds rather to the soap-bubble problem with the admissible shapes restri ted to be inside the ylinder , R so that the soap bubble an only tou h but not penetrate the surfa e , R. To speak rigorously, (5.64) need not have a solution be ause the minimized fun tional (though being onvex and oer ive) is dened on a non-reexive spa e. More in detail, minimizing sequen e of (5.64) will quite typi ally

on entrate gradient on the boundary

, whi h just orresponds to the situation when

the soap bubble is in onta t with the ylinder

, R.

1

The general problem (PVC ) does not suppose any onvexity of

'(x ; r; -) so that on-

entration ee ts an be a

ompanied by the os illation ones. Unfortunately, mu h less an be said about su h problem in omparison with the ase

p ¡ 1.

1

Again, (PVC ) will have to be relaxed and, of ourse, we will prefer a ontinuous

C( )-invariant

extension, taking some su iently ri h



subspa e

H ’ Car1 ( ; R)

normed by a norm generating a ner topology than the (relativized) natural lo ally

onvex topology of

Car ( ; R). The reader an ertainly anti ipate that then the natu1

ral data quali ation for

' will be:

q;1;1

' òCARH with

1 q p

*

, R , R n ; R)

(5.66)

. It allows us to dene the relaxed problem

(R

45

(

1

H PVC )

. Minimize > subje t to F

 (y; ) #

(

X



[

' Ž y DZ ℄(dx)

y; ) ò Y0 H ( ) :

See the experimental results by Hildebrandt [387℄.

1;1 ;

Ë

412

5 Relaxation in Variational Cal ulus: S alar Case

Of ourse, the integral is to be understood in the sense of measures on

, if needed.

1

Sin e (R H PVC ) does not show any detailed stru ture of the problem, a justi ation 1

of it as a orre t relaxation of (PVC ) is quite simple.

Proposition 5.33 (Corre tness of the relaxation s heme).

oer ive in the sense that, for some a ò L ( ) and b ¡ 0,

Let (5.66) be valid and

' be

1

'(x ; r; s) £ a(x) % bs :

(5.67)

Then: (i)

1

The relaxed problem (R H PVC ) has a solution and

(ii) Every solution ( {

y; )

min(RH PVC ) # inf (PVC ). 1

1

1

of (R H PVC ) an be attained by some minimizing sequen e

y k }kòN for (PVC ) in the sense w*-limkÙ i1H (y k ) # (y; ). 1



1

(iii) Conversely, every minimizing sequen e for (PVC ) has (after being embedded via

i1H )

1

a weak* luster point and ea h su h a luster point solves (R H PVC ). Proof. It su es to realize that, thanks to (5.66) and Lemma 3.101, extension of 1



is a ontinuous

 (i.e.  is ontinuous and  Ž i H # ), and that the admissible domain Y01 H1 ( )) is just the bounded losure of the admissible domain

;

for (R PVC ) (this means

H



1



1

for (PVC ) (this means

;

W0

1;1

(

)) embedded via i1H . Then the assertion is obvious; f. also

Å

Proposition 4.1.

Again it is sensible to ask about a stability of the relaxed problem with respe t to data perturbations, reating the problem

1

(PVC ; " )

Minimize

 " (y) # X ' " (x ; y(x); ›x y(x)) dx

subje t to

y ò W01 1 ( ) : ;

We an immediately obtain the following result whose the proof is analogous as that one of Proposition 5.17.

Proposition 5.34 (Stability of the relaxed problem). (5.67) be satised both for

p # 1 and 1   q   p

*

'

and

'"

Let

independently of

q;1;1

' " ; ' òCAR H

(

,R,Rn ; R),

" ¡ 0, and (5.33a) be valid for 1

. Then the relaxed perturbed problem (R H PVC ; " ) always possesses

a solution and

lim min(RH PVC " ) # min(RH PVC ) 1

" Ù0

1

;

and

(5.68a)

Limsup Argmin(RH PVC " ) ’ Argmin(RH PVC ) : 1

" Ù0

1

;

(5.68b)

Unfortunately, in general ase we have not at our disposal any deeper results like

y; ) ò L q ( ) , YH1 ( ; Rn ) ›x y # (1 ‹ id) DZ  ( y;  ) # 0 (

/ 7 ? 7 G

âá (y; ) ò Y

H ( ) ;

1;1 0;

(5.69)

5.4 Relaxation of variational problems;

with (

y; ) òr a(

Ë 413

p#1

.

y; ) due to Proposition 5.5. Therefore, onp ¡ 1 where (5.69) holds true at least if  is p-non on entra-

) denoting the tra e of (

trary to the previous ase

ting, we are in general not able to onstru t any relaxation s heme showing a more detailed stru ture and making thus possible to obtain more informative results. Nevertheless, let us at least briey mention on a few examples how one an handle the spe ial ase

:(x ; r) ò , R : '(x ; r; -) : Rn Ù R

is onvex

:

(5.70)

Example 5.35 (Extension on the bi-dual of W ( )). It is known46 that  W ( ) Ù R from (PVC ) is weakly lower semi ontinuous provided (5.67)

:

1;1

0

1;1

1

0

(5.70) hold. It allows us to extend

and

 on the bi-dual spa e of W01 1 ( ) by the formula47 ;

(y; ) #

lim inf

( y ) ;

(5.71)



i 1H (y)Ù( y;  ) weakly* yò W 01 1 ( ) ;

H # L

where



(

) ‹ (Rn )

*

and

i1H : W01 1 ( ) Ù Y01 H1 ( ), f. (5.23). As  ;

;

W01 1 ( );

is

;

) oin ides with the weak topology on W 0 ( ),  is a tually an extension of  in the 1 sense  Ž i H #  . As  is oer ive if (5.67) is assumed,  is oer ive as well, whi h

weakly lower semi ontinuous and the (relativized) weak* topology on

W0

1;1

**

(

1;1

yields existen e and well-posedness of the problem relaxed on the bi-dual spa e.

) ‹ (Rn ) , by Proposition 4.2(ii) the relation between (RH1 P1VC ) and 1 1 n n (R PVC ) is lear: if ( y;  1 ) solves (R H H 1 PVC ), then ( y;  ) with  òvba( ; R ) Ê L ( ; R ) n n dened by the restri tion of  1 on L ( ) ‹ (R ) Ê L ( ; R ) solves the problem 1 1 extended on the bi-dual of W ( ). If

H1 “ L



*

(





*

*



;

Example 5.36 (Extension on W

1; 0



(

)).

The generalized solutions obtained in Exam-

ple 5.35 involve the nitely-additive measures in pla e of a gradient of

y, whi h is still

fairly abstra t on ept. To get rid of this drawba k, Ka£ur and Sou£ek [409, 410℄48 developed a oarser relaxation s heme reated, in fa t, as a lower-semi ontinuous ex1;



 like (5.71) but on the oarser onvex - ompa ti ation W0 ( ) whi h 1  1 1 denotes Sou£ek's extension of W 0 ( ) like in Example 5.8, i.e. W 0 ( ) is the sequen1 1 q n tial losure of W 0 ( ) embedded via y ÜÙ ( y; ›x y ) into L ( ),r a( ; R ). It is known49 tension of



;

;



;



46

We refer, e.g., to Buttazzo [161, Thm. 4.1.1℄ or Da orogna [242, Chap. 3, Thm. 3.4℄. Thanks to the oer-

ivity (5.67), the weak lower semi ontinuity oin ides with the sequential one thanks to the Eberlein’muljan theorem.

47

More pre isely, using the

 #  with # (y; )}; note that  (y; ) # % if

-regularization introdu ed on p. 59, one should write

0 # inf ( Ž (i1H )"1 ) where (i1H )"1 (y; ) 1 "1 (i ) ( y;  ) # . H

# {y ò W

1;1 0

(

); i1H (y)

0

0

48

Essentially the same approa h has been also used by Giaquinta, Modi a, Sou£ek [354℄.

49

See Sou£ek [733, Thm. 4(ii)℄; denoting by

(

) ) , our understanding of ›x

Rn

P

* *

R

 ò r a( ; n ) the measure orresponding to  ò ( C ( ) ‹  y # (1‹id) DZ  in the sense of r a( ; n ) orresponds to the identity

 - g d x # 0 to be valid for any g ò C y(x)div g(x) dx % X 





(1)

(

;

R

R

n ), used in

[733, Denition 2℄.

414

Ë

5 Relaxation in Variational Cal ulus: S alar Case

that (5.69) holds true in this spe ial ase. This allows us to write the relaxed problem in the form Minimize subje t to

where



(y; ) 1 q n ( y;  ) ò L ( ) , Y ( ; R ) H ›x y # (1 ‹ id) DZ  ( y;  ) # 0

H # C( ) ‹ (Rn ) ; n in the sense of r a( ; R ) ; in the sense of r a( ) ; *

with





/ 7 7 7 ? 7 7 7 G

(5.72)

i1 y Ù (y; ) weakly* in W 1  ( ).50 Using ;

is dened as in (5.71) but with H ( ) 

the onvex stru ture of the onstraints in (5.72), we ould derive the optimality onditions similar to that in Proposition 5.24 ( f. also Remark 5.27) but with the maximum prin iple only in the integral form. The essential dieren e now is that we have not the des ent ondition guaranteed so that the integral maximum prin iple annot be lo alized, whi h is related with the fa t that on entration ee ts annot be ex luded.

Example 5.37 (Extension on W

1;



)).51 By the hoi e (5.19) and by the formula (5.71) BV( )), one an onstru t 1 1 the relaxed problem by extension on the weak* losure of W 0 ( ) in BV( ). Anyhow, BV( ) is too oarse onvex - ompa ti ation of W 1 1 ( ) so that (5.71) denes only lower semi ontinuous regularization of  ; in other words, one annot re kon with  Ž i1H # . It an easily be seen from the fa t that the weak* losures in BV( ) of W01 1 ( ) and of W 1 1 ( ) are the same. It is the reason why, in general, an expli it formula for  ontains a orre tion boundary term. E.g., if ' ( x ; r; s ) # x1 %  s % u ( x )2 0

(

(with the weak* onvergen e understood in the sense of

;

;

;

;

as in (5.65), then52

(y) # with ›x

y ò r a( )

X ' ( x ; y; ›x y ) d x

being the gradient of

denoting the inner tra e of

y,

y

% X y  dS

in the distributional sense and

y ò L1 ( )

see Giusti [356, Chap. 2℄. The relation with the Ex-



y2 ò W0 ( ) obtained by the extension from Example 5.36, we an get a solution y 1 ò BV( ) by putting simply y 1 # y 2 q n in the sense of L ( ) but restri ting the gradient ›x y 2 ò r a( ; R ) to so that n ›x y 1 # ›x y 2  òr a( ; R ). ample 5.36 is obvious: Having a solution

1;





Remark 5.38 (In orre t relaxation s hemes).

One ould formally onsider a relax-

ation s heme like (5.72) even without having the ondition (5.69) guaranteed. Then, however, the minimum of (5.72) might be stri tly lesser than the inmum of the origi1

nal problem (PVC ) and (5.72) ould be interpreted as a relaxed problem to a variational

50

51

'(x ; r; s) # '(s), a formula for  an be found in Ka£ur and Sou£ek  is the extension of  as shown in [410, Se t. 4℄.

In the ase

spe ial ase,

[410℄. In this

The extension of minimal-surfa e type problems in terms of BV-fun tions has been studied by a

lot of authors, e.g. Bou hitté and Valadier [142℄, Giusti [356℄, Lea i [483℄, Morrey [552℄, and essentially also by Temam [752℄, see also Ekeland and Temam [283, Chap. 5℄, et .

52

See Giusti [356, Chap. 14℄.

5.4 Relaxation of variational problems;

p#1

.

Ë 415

u # ›x y as well as the tra e onstraint y # 0 are to be fullled only with a ertain toleran e (related with the weak* topoln 1 n 1 ogy of r a( ; R ) and r a( ) relativized respe tively on L ( ; R ) and L ( ), i.e. with problem where the variational onstraint



the weak topology). This reminds the toleran e whi h we admitted in onstrained optimization problems in Se tion 4.1. Nevertheless, su h interpretation is hardly a

eptable for variational problems where it would hange essentially the nature of the whole problem. As a result, relaxation s hemes like (5.72) may be a

epted only if (5.69) is guaranteed to avoid the relaxation gap.

Example 5.39 (Convex ve torial problems witn symmetri gradients).

An

interesting

L1 - hara ter arises for Rn -valued fun tions when only the 1 s  1 symmetri gradient ›x y :# (›x y ) % ›x y is involved. Then fun tions of bounded de2 2

variational problem of an

formation from Example 5.15 are to be exploited. Inspired by perfe t plasti ity at small strains in the stati variant, one is to think about a nonsmooth variational problem:

with

s y"p2

Minimize

(y; p) # X

subje t to

y ò W 1 1 ( ; Rn )

yd ò L1 ( ; Rn )

(5.29b). The variable



;

given and with

p

›x

{

Aò

n , n ; tr A # Rsym

is in the position of a plasti strain while

/ ? G

(5.73)

0} with

n , n from Rsym

n,n p ò L1 ( ; Rdev )

and

n,n Rdev :#

% p dx % X y " yd  dS

y

is a displa ement

n , n ) is an elasti strain.53 The problem (5.73) is oer ive only " p ò L ( ; Rsym

sy

2

and ›x

on non-reexive spa es as mentioned there, and thus needs a relaxation to the spa e

n,n BD( ; Rn ) , r a( ; Rdev ). The relaxed problem is then

s

Minimize

 (y; ) #

subje t to

y òBD( ; R



/ X ›x y "   d x % X   (d x ) % X  y " y d  d S

? n ) ;  òr a( ; n , n ) ; ›s y "  ò L 2 ( ; n , n ) ; x sym G dev 2



R

R

(5.74)

s y is

where  -  denotes the total variation of the involved measures. Let us remind that ›x

r a( ; R

valued in



n,n sym ) for

y òBD( ; R

n ). Moreover, the tra e

the last integral is a uniquely determined measure from

y

of

r a( ; R

y on

o

urring in

n ), f. [465, Thm. 2.3℄.

The problem (5.74) is oer ive and one gets easily existen e of its solution. On the other hand,



is now nonsmooth and optimality onditions are thus more involved. The

latter hoi e (5.29b) would lead to a bit ner extension than natural as far as denition of tra es in

53

BD( ; Rn ) and is more

r a( ; Rn ), f. [465℄.54 Another variant of (5.73)

It should be emphasized that, in ontinuum me hani s, a dierent notation is used:

dardly for displa ement while

y

u stands stan-

is sometimes denoting a deformation under large strains. Here we

used the notation onsistent with the previous general theory. The onstraint

tr p # 0 is motivated

from a so- alled iso hori -plasti ity model.

54

It was shown by R. Temam and G. Strang [756℄ that tra es of BD-fun tions ( alled BD-tra es in [465℄)

are in

L1 ( ;

Rn

). This is analogous to inner tra es in Example 5.7.

Ë

416

5 Relaxation in Variational Cal ulus: S alar Case

might involve Diri hlet boundary ondition:

s y"p2

Minimize

(y; p) # X

subje t to

y ò W 1 1 ( ; Rn ); y # yd ;

with some



›x

% p"pd  dx

;

and

/ n,n p ò L1 ( ; Rdev )

n,n pd ò L1 ( ; Rdev ) given, whi h would lead to the relaxed problem55

 (y; ) #

Minimize

s

dx % X "pd (dx)

( y " y d ) ‹ n % n ‹ ( y " y d )  %  # 0 on ; n,n s n,n y òBD( ; Rn );  òr a( ; Rdev ); ›x y "  ò L ( ; Rsym ) :

subje t to

1 2

X ›x y "  

2

2



/ 7 7 ? 7 7 G

(5.76)

n,n n,n 1 pd an be onsidered in r a( ; Rdev ) instead of L ( ; R dev ) here and, if the P - d x would be understood as P - (d x ), in (5.75) too. Although (5.73) and



A tually, integral

(5.75)

? G



(5.75) are ve torial problems rather than s alar, their relaxation is onvex and they indeed t to this hapter rather than the next Chapter 6.

5.5

Convex approximations of relaxed problems

We end this hapter by a pie e of numeri s. Without any ambitions to develop a general theory, we will onne ourselves to onvex inner approximations of the extended

YH

p

) reated by the nite-element method. More on retely, we will y and element-wise homogeneous approximation of its generalized-Young-fun tional-valued gradient  . Besides, we will onne ourselves to the ase p ¡ 1 so that our task will be to approximate the relaxed problem

Sobolev spa e

1;

(

use element-wise ane approximation of

(R H PVC ).

We suppose, for simpli ity,



a polyhedral domain triangulated, for ea h mesh

d ¡ 0, by a triangulation Td onsisting of elements of the diameter not d. Ea h element E ò Td is therefore a simplex with n%1 verti es. For d1 £ d2 ¡ 0, we suppose that Td1 ’ Td2 , this means Td2 is a renement of Td1 . Besides, we will also suppose that the family {T d } d ¡0 is regular56 in the sense that there is " ¡ 0 su h that ea h element of ea h T d ontains a ball of the radius "d . p n We will employ the inner onvex approximation of Y ( ; R ) onstru ted by H p p n n means of a mapping P : Car ( ; R ) Ù Car ( ; R ) dened by d parameter ex eeding

Pd h
(x ; s) :#

55

1 

E

X h( x ; s) d x E 

if

x ò E ò Td ;

(5.77)

The boundary ondition in (5.76) is to be understood as a relaxation of the Diri hlet boundary on-

dition in (5.75) in the sense of measures and (-) dor measure on

56



means multipli ation by an (

, f. [247, 548℄ for nontrivial details.

For this (standard) notion of regularity we refer to Ciarlet [217℄.

n"1)-dimensional Haus-

Ë 417

5.5 Convex approximations of relaxed problems

f. also Se t. 3.5.b. On this rather abstra t level, we will assume that there are some linear subspa e

V ’ C p (Rn ) and a linear subspa e G su h that

G ‹ V ’ H ’ l(G ‹ V) ; G0 ’ G ’ L where  l refers to the natural topology57 of Also, we will assume that the sense that58

H



(

) ; H is G-invariant;

Carp ( ; Rn ) and G

0

is dened by (3.164).

as well as its norm is ompatible with

Pd : H Ù H

is a ontinuous proje tor

(5.78a)

Pd

from (5.77) in

:

(5.78b)

H d # Pd H ’ H . By Propositions 3.83(i) and 3.86(i), Pd YH ( ; Rn ) p n is a onvex, weakly*  - ompa t subset of Y ( ; R ). Also, we have here obviously H p P d1 Ž P d2 # P d1 whenever d1 £ d2 ¡ 0, so that the onvex approximations Pd YH ( ; Rn ) in reases for d Ù 0; f. Proposition 3.83(iii). p

*

Then we will denote

*



In view of the relaxed problem in the form (R

H PVC ), the above onstru tion sug-

gests the following approximate problem

(R

d H PVC )

Minimize . 6 6 6 6

> 6 6 6 6

subje t to

X



[

' Ž y DZ ℄(dx) % X  Ž y dS

1 ‹ id) DZ  # ›x y ; y ò W 1 p ( ) element-wise ane on Td ; p  ò Pd YH ( ; Rn ):

(

;

*

F

Let us note that, for simpli ity, we assumed that the ost fun tional needs no numeri al approximation to be evaluated ee tively. Also note that the set of admissible pairs (

y; ) for the approximate relaxed problem is again onvex in W 1 p ( ) , H ;

Proposition 5.40 (Convergen e of numeri al approximations). (5.77), and (5.78) be satised, and (i)

Let

*

.

(5.10),

(5.31),

H be separable. Then: d

The approximate relaxed problem (R H PVC ) has a solution.

(ii) If (

y; ) solves (RdH PVC ), then  is p-non on entrating.

(iii) The following onvergen e is valid:

lim min(RdH PVC ) # min(RH PVC ) ;

(5.79a)

d Ù0

Limsup Argmin(RdH PVC ) ’ Argmin(RH PVC ) :

(5.79b)

d Ù0

(iv) If, in addition, the assumptions of Corollary 5.26(i) are valid, then even:

Lim Argmin(RdH PVC ) # Argmin(RH PVC ) :

(5.80)

d Ù0

57

Of ourse, it would su e to onsider any ner topology, e.g. a strong topology of

58

In fa t, it su es to suppose



Pd L(H;H)

Pd H

’ H

be ause (5.77) always ensures

¢ 1 if - H is the norm (3.141), whi h an be always supposed.

Pd h H

H.

¢ h H

so that

Ë

418

5 Relaxation in Variational Cal ulus: S alar Case

Let us rst prove a rather te hni al assertion:

Lemma 5.41.  ation

If

H is separable and  ò Pd YH ( ; Rn ), then its p-non on entrating modi*

p

Ž p  lives in Pd YH ( ; Rn ), as well. *

Proof. Let us take

1 ò YH ( ; Rn ) su h that  # Pd 1 and its p-non on entrating modp

*

Ž p n i ation  1 ò Y ( ; R ) whi h does exist by Proposition 3.81 be ause H is separable. H Therefore we an take a sequen e { u k } k òN ’ L p ( ; Rn ) su h that the set Ž p 1 { u k  ; k ò N} is relatively weakly ompa t in L ( ) and i H ( u k ) Ù  1 weakly*, and 

onsider a net { u }   ò  ,  onstru ted in the proof of Proposition 3.86; in parti uk Ž lar, it holds lim k Ù limò i H (u k ) # Pd 1 . By de la Vallée-Poussin riterion (Theo p rem 1.28(iv)), we an easily see that also the set { u  ;  ò  ; k ò N} is relatively weakly k Ž 1

ompa t in L ( ), whi h shows that P  1 is p -non on entrating. d In view of (5.77), we an easily see that if an integrand h ò H has a growth stri tly less than p , so has also P h ò H . Therefore, d ( ;

)

*



*

Pd

*

It shows that

Ž Ž 1 ; h # 1 ; Pd h # 1 ; Pd h # Pd 1 ; h #  ; h : *

Ž Pd 1 is the p-non on entrating modi ation of .

Å

*

d

Proof of Proposition 5.40. First, let us note that the set of admissible pairs for (R H PVC )

0; i H (0)). Then (i) an be - ompa tness of W 1 p ( ) , Y p ( ; Rn ), by the ontinuity of both the ost fun tional  and the onstraint mapping  (dened in (AP)), and by the oer ivity of the problem.

is always nonempty be ause it ontains, e.g., the pair (

;

shown by the standard arguments, namely by the weak*





Using Lemma 5.41, the point (ii) an be demonstrated by the same ontradi tion argument as used for Proposition 5.21.

d

Let us go on to (iii). Realize that the set of admissible pairs for (R PVC ) is erH 

tainly ontained in that one for (R PVC ), so that learly H

min(RdH PVC ) £ min(RH PVC ) # 

min(RH PVC ); here we used also the separability of H required in Proposition 5.21. p ( ) , Y p ( ; R n ) satisfying  ( y;  ) # Now we want to prove that every ( y;  ) ò W H d (1 ‹ id) DZ  " ›x y # 0 an be approximated by suitable admissible pairs for (R PVC ) H when d Ù 0. By Proposition 5.21(i), we may onsider that  is p -non on entrap ( ) ting, so that by Lemma 5.20 we an get a bounded sequen e59 { y k } k òN ò W su h that i H (›x y k ) Ù  weakly* in H . Moreover, we an also suppose y k Ù y 1;



1;

*

W 1 p ( ); f. the proof of Proposition 5.4. Eventually, by mollifying (if ne 1 essary) this sequen e, we an even suppose y k ò C ( ) so that we an dene the element-wise ane interpolant of y k on the triangulation T d , denoted by  d y k ò W 1 p ( ). For k xed and d Ù 0, we have  d y k Ù y k strongly in W 1 p ( ) be-

weakly in

;

(

)



;

59

Let us remind that

;

H is supposed separable so that we an work in terms of sequen es as far as the H * on erns.

weak* topology on bounded subsets of

5.5 Convex approximations of relaxed problems

Ë 419

y k . Therefore w*-limdÙ0 i H (›x  d y k ) # i H (›x y k ) be ause h(x ; ›x  d y k (x)) " h(x ; ›x y k (x))
dx # 0 for any h ò H as a onp n 1 sequen e of the ontinuity of the Nemytski mapping N h : L ( ; R ) Ù L ( ). At the 1 d same time, the pair i (  d y k ) # (  d y k ; i H (›x  d y k )) is admissible for (R PVC ) be ause H H p n (1 ‹ id) DZ i H ( u ) # u for any u ò L ( ; R ) and, in parti ular, also for u # ›x  d y k , and be ause  d y k is element-wise ane of T d hen e i H (›x  d y k ) is element-wise homop n geneous, belonging thus to P Y ( ; R ). Sin e limk Ù limdÙ0 i1H ( d y k ) # (y; ), we d H 1 p

an sele t a subsequen e su h that  d y k Ù y weakly in W ( ) and i H (›x  d y k ) Ù  weakly* in H if both d Ù 0 and k Ù . Let us suppose that ( y;  ) has been a solud tion to (R H PVC ). We have always  (  d y k ; i H (›x  d y k )) £ min(R PVC ) £ min(R H PVC ). H As  is weakly,weakly* ontinuous, we obtain  (  d y k ; i H (›x  d y k )) Ù  ( y;  ) #

ause of the regularity of obviously

limdÙ

0

P

*



;

*









min(RH PVC ), whi h shows (5.79a).

Then (5.79b) follows by the standard ompa tness arguments, taking into a

ount the oer ivity of the problem. If ately.

If

V

Argmin(RH PVC ) is a singleton, (5.80) results immediÅ

from (5.78a) is nite-dimensional, the set of admissible pairs for (R

d H PVC ) is,

in fa t, a onvex subset of a nite-dimensional linear manifold60 and as su h, it an be implemented dire tly on omputers; then we fa e the approximation of Type I (see the lassi ation in Se tion 3.5). In the opposite ase, we have got the approximation of Type II and a further theoreti al work is needed to implement the semi-dis retised

d H PVC ) on omputers. Namely, we have to pose and analyze proper optimality d d

onditions for (R PVC ). Of ourse, thanks to the onvex stru ture of (R PVC ), we are able H H

problem (R



to perform it in an entirely parallel way how it was done for (R PVC ) in Se tion 5.3. Now,

H

the maximum prin iple will in lude the dis rete Hamiltonian

h dy  ò H dened by ;

*

h dy  #  ‹ id " Pd (' Ž y) : *

;

Let us note that (5.31a), (5.42) and (5.78b) ensures

y ò L q ( ) and



*

ò Lp



(

; R

(5.81)

*

h dy  ;

*

to live in

n ) is element-wise onstant.

H d # Pd H

provided

Proposition 5.42 (Maximum prin iple for approximate problems). Let q £ 2 satisfy also 1   q   p , q £ 2 satisfy (5.14), and (5.31), (5.42), (5.43), (5.77)(5.78b) be valid. d n If ( y;  ) solves (R H PVC ), then there is  ò L ( ; R ) element-wise onstant su h that the *

1

*



integral identity

X (x)

is fullled for any prin iple holds:

- ›x

y (x) " ' r Ž y DZ 
(x) y (x) dx # 





y ò W 1 p ( ) ;



h dy  DZ  # maxn h dy  ;

60

Let us note that

element-wise ane on

*

Pd (G ‹ V) # {g : Ù

s òR

R

;

*

(-

; s)

X

 r (x ; y(x)) y (x) dS 



Td in

(5.82)

and the following maximum

L1 ( ) :

element-wise onstant on

Td } ‹ V

(5.83)

is nite-dimensional.

Ë

420

5 Relaxation in Variational Cal ulus: S alar Case

d

Proof. Let us note that (R H PVC ) has again the stru ture of the auxiliary problem (AP) but

Y , K # Y d , Pd YH ( ; Rn ) and with  : Y d , H d Ù L d , where 1 p p n we abbreviated Y d # { y ò W ( ) element-wise ane on T d } and L d # { u ò L ( ; R ) p n element-wise onstant on T d }. Also we used the fa t that P Y ( ; R ) an be equally d H

onsidered as a subset of H ; f. Lemma 3.82. Moreover, being nite-dimensional, the d p n dual to L d an be identied with L d itself. As for any u ò L d , i H d ( u ) ò Y H d ( ; R ) Ê p Pd YH ( ; Rn ) and  (0; i H d (u)) # u, we an see that the onstraint quali ation (5.40) p

*

with the onvex set

*



;

*

*

*



is satised. It allows us to employ the optimality onditions (5.41), whi h results here respe tively to (5.82) and to the inequality

:  ò Pd YHp ( ; Rn ) :



*



"  ; hy  ;



*

£0

 ò L d Ê L d . Sin e Pd  #  Pd ( ‹ id) #  ‹ id provided  ò L d , we an obviously write *

*

 Sin e

;

*

# Pd  " Pd  ; h y  *

*

p YH ( ;

*

;



Pd h dy  # h dy  ;

*

and

Pd  #  *





and also

*

" ; hy  

*

*

to be valid for some multiplier

(5.84)

;

R

*

 ò Ld , *

for

*

#  "  ; Pd h y  ;



we have


*

# ´ "  ; h dy 



# <  ; Pd h dy  ;



*

>

;

*

:

µ

#

n ). As a result, the inequality (5.84) with h d for any  ò y;  in pla e of h y;  p n ), whi h gives the maximum prin iple <  ; h d > # holds even for every  ò Y ( ; H y;  max òYHp ( ;Rn ) <  ; h dy;  >. This maximum prin iple an be equally written in the form 

R



*

*

*

*



d X h y; 

*

DZ  dx #

sup X h dy;  u ò L p ( ;Rn )

*

(

x ; u(x)) dx :

Then one an just use Theorem 4.21(i) to lo alize this integral maximum prin iple to get eventually (5.83); note that the des ent ondition (4.36), being fullled for thanks to (5.49), is fullled for the dis rete Hamiltonian

Corollary 5.43 (Chattering solutions). and

h dy  ;

*

too.

hy  ;

*

Å

Let the assumptions of Proposition 5.42 be valid

H be separable. Then:

) to (RdH PVC ) with  being (n%1)-atomi . d * (ii) If ( y;  ) solves (R PVC ),  ò L d is the orresponding multiplier and, for a.a. x ò , the H d n Ù fun tion h a hieves its maximum at no more than k points, then y;  ( x ; -) :

(i)

There exists a solution ( y;

R

*

 is k-atomi .

R

Proof. Likewise in the proof of Corollary 5.25, we an show that any that satises (

n%1)

suitable onditions involving the integrands

 ò Pd YH ( ; Rn ), h # h dy  and p

*

;

*

h l (x ; s) # s l for 1 ¢ l ¢ n, solves (RdH PVC ); the fa t that  (y0 ; 0 ) #  (y0 ; ) for p h DZ 0 # maxsòRn h(-; s) # h DZ  and h l DZ 0 # h l DZ  with 0 ;  ò Pd YH ( ; Rn ) follows



*

here from the identity

 ; '

Žy

0

# #

Pd

*

 ; ' Ž y0 #  ; Pd (' Ž y0 )

 ; 

*

‹ id " h dy0  ;

*

#  ;  ‹ id " h dy0  *

0

;

*

Ë 421

5.5 Convex approximations of relaxed problems

#

 0 ; Pd ( '

Žy

0)

# Pd  ; ' Ž y *

0

0

#  ; ' Ž y :

0

0

Then the point (i) follows from Proposition 4.28 modied for our pie ewise homogeneous ase (details are omitted).

Å

The point (ii) again follows from Proposition 4.27.

By means of Corollary 5.43(i) we an eventually implement the semi-dis retised

d

 only  is (n%1)-

relaxed problem (R PVC ) on omputers: we an minimize the ost fun tional H for (

y; ) ò W

1;

p ( )

,

p YH ( ;

R

n ) su h that

y

is element-wise ane and



atomi and element-wise homogeneous, whi h forms a nite-dimensional61 (non onvex) manifold. Then Corollary 5.43(i) ensures that this manifold ontains at least one

d

solution to (R PVC ). H The last task we want to pursue on erns an error analysis. However, the error estimates we are able to establish here dier from the usual results for, e.g., ellipti equations mainly be ause the relaxed problem (R H PVC ) is not uniformly onvex. This fa t implies that we are able to establish only error estimates on erning 

min(RH PVC ) " min(RdH PVC ), whi h an be onsidered however as a signi ant indi a-

tor of the e ien y of the parti ular method.62

min(RdH PVC ) Ù min(RH PVC ) of the order O (d  ) with some  ¡ 0. We will onsider a subspa e H ’ H equipped with a norm stronger than the norm of H , so that the embedding H Ù H is ontinuous. Thus H “ H p n and we an estimate the generalized Young measures from Y ( ; R ) in the standard H Our aim is to get the onvergen e



*



dual norm of

H

*



*



. Our error analysis will rely on the following general pattern (just for

notational simpli ity we will not handle the boundary potential, putting

 # 0):

Proposition 5.44 (Error-estimation s heme). Let (5.31) be valid,  # 0, the mapping y ÜÙ ' Ž y map every B bounded in W p ( ) into a bounded subset of H , i.e. 1;



:B ò B ;C B ò R% :y ò B : and let this mapping be ( W



' Ž y p H ¢ CB ;

" ; p ( ); H)-Lips hitz

1

:B ò B ;~B ò R% :y; y ò B : 



(5.85a)

ontinuous on ea h

' Ž y"' Ž y H ¢ ~B y" y W 1" p 



;

(

B ò B , i.e.

) :

(5.85b)

R

p ) ò W 1; p ( ) , YH ( ; n ) and every d ¡ 0, there exist an add 1; p missible pair ( y d ;  d ) for (R H PVC ) su h that the sequen e { y d } d ¡0 is bounded in W ( ) Eventually, let, for every ( y;

and the sequen e {( y d ;

 d )}d¡0 has the following approximation property

61

Its dimension is

y " y d W 1" p ;

(

)

¢ C d  y W 1 p 1

;

(

)

;

(5.86a)

m d % (n%1) - n2 - n d where n d and m d denotes the number of elements and of mesh

points, respe tively.

62

One an o

asionally nd error estimates also for the solutions (in appropriate dual norms on

H*

weaker than - H * ) but it is always for spe ial problems possessing a unique solution; f. [230,232,502℄.

Ë

422

5 Relaxation in Variational Cal ulus: S alar Case



  "  d p H ¢ C2 d  H :

(5.86b)

*

*

min(RH PVC ) " min(RdH PVC ) # O (d  ).

Then 

 # 0 make  (y; ) # subje t to F

d

(PVC )

(y) # X '(x ; y(x); ›x y(x)) dx % X  (x ; y(x)) dS

y ò W 1 p ( ) ;

element-wise ane on

Td :

This, in fa t, oin ides with an approximation of the relaxed problem by its restri -

ally non onvex

p

- ompa ti ation Y H ( ) to the nite-dimensional but gener1 1 p manifold i { y ò W ( ) element-wise ane on T d ; in a

ord with H 1;

tion from the onvex

;

the lassi ation from Se tion 3.5, su h s heme is of Type III.

d

The straightforward omparison of the dire t dis retisation (PVC ) with the dis reti-

d sation (R PVC ) an be based on the observation that ertainly H

min(RH PVC ) ¢ min(RdH PVC ) ¢ min(PVCd ) ;

67

(5.101)

Su h kind of approximation has appeared quite frequently in the literature ( ontrary to the previ-

d PVC ) in multidimensional ase under data quali ation similarly

ous dis retisation). The analysis of (

weak as those used here was performed in works by Chipot, Collins, and Li, see [200, 202, 205℄ and

'(x ; r; s) # ' (s) % ' (r) with # 0 in [200, 205℄ or some spe ial ' having the growth at most O (r ) in [202, 205℄, the error d

% ) if 0     1. A similar result, ) if £ 1 or O ( d  min(RH PVC ) " min(PVC ) is of the order either O( d namely the rate O ( d ) for ' # 0, was obtained also by Brighi and Chipot [153℄ in a bit dierent onCarstensen and Ple hᣠ[175℄. For Diri hlet boundary onditions, and either

'0

1

0

0

1/2

1/2

/(

1)

0

text. Let us remind that, in ontrast to [153, 200, 202℄, we admitted

O ( d  ) with 0   

' to be spatially dependent and our

¢ 1 depending on the spatial smoothness of ' and, if n £ 2, also % p -regularity of at least one solution to (R P ). Moreover, there are several results on erning on W H VC the one-dimensional ase n # 1 by Brighi, Chipot, Chen, Collins, Kinderlehrer, Luskin, Ni olaides and Walkington, see [152, 197, 230, 232, 234, 502, 575℄. For ' ( x ; r; s ) # ' ( s ) % ' ( x ; r ) with a spe ial ' and guaranteed rate was 1

;

'1

having two wells (= lo al minima) of the same value and a quadrati growth in neighbour-

1

with

hoods of these wells, the rate of 

0

0

d 2 min(RH PVC ) " min(PVC ) was shown to be O ( d ). This is better than

the guaranteed general rate derived here, but (5.101) ensures in this spe ial ase the order also at least

O ( d2 ). Moreover, the problems in [230, 232, 234, 502℄ are so spe ial that even min(RdH PVC ) # min(RH PVC ) be ause (RH PVC ) possesses in these spe ial ases the solution ( y;  ) with y ane and  homogeneous, d whi h is obviously admissible also for our approximate problems min(R PVC ) for any d ¡ 0; therefore, H  in fa t, even the rate O ( d ) was a hieved. For implementation of su h algorithms see also Ma and Walkington [504℄.

5.5 Convex approximations of relaxed problems

Ë 427

d ) is i1H (y) with y admissible for (PVC d d admissible for (R H PVC ) too. Therefore the previous approximation (R H PVC ) annot be d worse than (P ) as far as the rate of onvergen e of the minimum of the dis rete

whi h obviously follows from the fa t that every

VC

problem towards the minimum of the original problem on erns. Both methods lead to non onvex mathemati al-programming problems, the former one having slightly greater dimensionality but of the same order

O(d"n ) as the latter one provided Corol-

lary 5.43 an be applied to implement it. The theoreti al eort behind the implemen-

d

tation and slightly greater dimensionality of (R H PVC ) are ompensated by the fa t that

d

sometimes this method an give mu h faster onvergen e than (PVC ). It appears espe ially if (R H PVC ) admits a (pie e-wise) homogeneous solution but does not admit a

d

d

one-atomi solution; then (R H PVC ) may give dire tly the solution of (R H PVC ) (so that it a hieves the onvergen e rate of minima O (

d



d

)) while (PVC ) onverges only with some

nite rate.

Example 5.47 (Tartar's broken extremal [575℄).68

An

interesting,

even

only

one-

dimensional double-well problem (PVC ) takes n # 1, # (0; 1), p # 4, 3 1 5 3 '(x ; r; s) # (s2 " 1)2 % (r " g(x))2 with g(x) # " 128 (x " xb ) " ( x " x b ) , but with the 3 25 1 1 Diri hlet boundary onditions y (0) # g (0) and y (1) # y 1 # ( " 12 xb % 24 x2b )(1 " xb ). 24 Altogether, we thus onsider the problem 1

minimize

X €( y



(

0

subje t to

x)2 " 1)2 % (y(x) " g(x))2  dx ;

y ò W 1 4 (0; 1); y(0) # g(0); y(1) # y1 : ;

/ 7 ? 7 G

(5.102)

the relaxed problem (RP) has the unique solution

g(x) y(x) # ® 1 3 24 (x " xb ) % (x " xb ) x

#

®

1  2 (1 " u (x)) Æ"1

Æu



(

%

1  2 (1 % u (x)) Æ1

for for

x ò (xb ; 1) ;

for

x ò (0; xb ) ;

for

x)

x ò (0; xb ) ;

x ò ( x b ; 1) :

Example 5.48 (Two-dimensional broken extremal [175, 200, 202, 575℄).

A two-dimen-

sional variant of (5.102) has been used for testing various numeri al approa hes using the original non-relaxed problem as in [200, 202℄, or a oarse relaxation (knowing the onvex envelope of

'(x ; r; -))

as in [175℄, or a relaxation by ontinuous ex-

tension employing generalized Young fun tionals [536℄ or onventional Young measures [84, 86, 89, 172, 173℄. To be more spe i , let us onsider

68

n # 2, p # 4, # (0; 1)2

In [179℄, this one-dimensional problem, after a numeri al approximation of Type I ( onvex/nite

dimensional), has been used to devise and to test omputationally the adaptive support-estimation strategy exploiting the maximum prin iple from Proposition 5.42, f. Remark 4.64. The dis rete problems have been solved by a linear-quadrati programming routine. A similar one-dimensional problem but with distributed on entration ee ts swit hed to just one-atomi measure was devised in [459, Example 1.4℄.

Ë

428

5 Relaxation in Variational Cal ulus: S alar Case

and

'(x ; r; s) :# s"a2 s% a2 % (r " g(a- x))2 for

a # ( os π6 ; sin π6 )

 b # 21 .

and for

namely

The relaxed problem has a unique solution,

. g(a- x) y(x) # > 1 3 (a- x " b ) % (a- x " b ) F

x

#

. > F

24 1 (1 " a -›x y ( x )) Æ " a 2

Æ

% 21 (1 % a-›x y(x)) Æ a

for

a- x ò (0; b );

for

a- x ò (b ; $2);

for

a- x ò (0; b );

yD :# y

(5.103a)

(5.103b)

a- x ò (b ; $2);

for

x y(x)

›

provided the boundary data

3 1 5 3 ( " b ) " ( " b ) ; 128 3

g() :# "

with

y just from (5.103a). Using a nu-

are hosen with

meri al approximation of Type I ( onvex/nite dimensional) with P1-FEM dis retisation of

, the resulted linear-quadrati problem was in [89℄ solved iteratively by using

linearization of the lower-order term by linear programming. In addition, the adaptive support-estimation strategy (devised in [179℄ by exploiting the maximum prin iple from Proposition 5.42) was used and ombined with an adaptive renement of the dis retisation of

.

bounded area69) of

On Fig. 5.2, one an see a very ne dis retisation (of a sele ted

R

2

from whi h only few points are sele ted as a tive within some

iteration of the above mentioned algorithm. This qualitatively a

elerates the linearprogramming algorithm.

0.9

2

0.8

1.5

0.7

1

0.6

0.5

0. o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

0.476 x

o

o

o

o

0.55

oooooooooooooooo ooooooooooooooooooxooooooooooooo 0.5

0.5

0

0.4

oooooooo ooooooooooooooooooxooooooooooooooooo

−0.5

0.3

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

oo

0. 5

−1 0.

0.2

−1.5

0.1

1

−2 −2

−1.5

−0.5

−1

0

0.5

1.5

1

0. 5

2

0.

0. 5

0.

0. 5

1

0.8

0 0.6 −0.1 0

0.4 0.2

Fig. 5.2:

0.2

0.4

0.6

0.8

The approximation of

le ted element where

x

y from (5.103a) (left) and of the support of

x

from (5.103b) on one se-

from (5.103b) is two-atomi , approximated by using only few a tive nodes out

"2; 2℄

of the ne dis retisation of [ tion: ourtesy of

0

1

Sören Bartels

2

(middle) with a detailed zoom (right). Cal ulation and visualiza-

(Univ. Freiburg).

The linearization with an iterative treatment does not rely on quadrati lower-order term. An analogous approximation nonquadrati lower-order term by quadrati fun -

69

More spe i ally, the sele ted area in

R

2

was a square

S

# ["2; 2℄

2

dis retised on 4761 nodes, as

displayed on Fig. 5.2(middle). In fa t, still ner dis retisations have been a hieved in [89℄, the nest having 48841 nodes on

S

# ["2; 2℄

2

for ea h of 177 elements of

, but always only few dozens have

been a tivated for the optimization routine. The adaptive Young-measure implementation in multidimensional problems was used also in [172, 178℄.

Ë 429

5.6 Example: Mi rostru ture in ferromagneti materials

tionals leads to a sequential-quadrati programming (SQP) as used for this test problem in [505℄. The same 2-dimensional problem was used in [536℄ for testing e ien y of oarse relaxation from Se t. 3.3.d by

exploiting the semidenite programming

(SDP).70

5.6

Example: Mi rostru ture in ferromagneti materials

Another physi ally well-motivated example of variational problem whi h naturally allows for a onvex relaxation arises in modelling of ferromagneti materials. It is also a ni e example when both singularly-perturbed and the orresponding limit relaxed problem have a good interpretation, ree ting the multi-s ale hara ter of the problem and the mi ros opi al and the mesos opi al views, respe tively.

’ Rn n

an be des ribed, from an ele tromagneti viewpoint, by a magnetization u : Ù R A onguration of rigid ferromagneti body o

upying a bounded domain

having a given magnitude here normalized to 

u(x) # 1

1 for notational simpli ity:71

for almost all

xò :

(5.104)

S n"1 the unit sphere in Rn . Here we use the notation onsistent with the other parts of this book, while standardly the letter u is used for magnetisation.

We will denote by

On the mi ros opi al level, a generally a

epted model for steady-state onguration is due to Landau and Lifshitz [473, 474℄, relying on a minimum-of-Gibbs'-energy prin iple, i.e.72

(PMG )

. minimize 6 6 6 6 > 6 6 6 6

subje t to

the set

70



u # 1 div(›x y "  u ) # 0

y ò W 1 2 (Rn ); u ò L

in



;

F where

(y; u) # X '(u) " hext - u dx %

;

1 X › y  d x 2 Rn x 2

Rn ;

; Rn ;

in 

(

)

y is a potential of the demagnetising eld and  is a hara teristi fun tion of

. The parti ular terms in  are respe tively an anisotropy energy,73 Zeeman's

For using of relaxation by algebrai moments in a one-dimensional ase see also [279, 534℄ or, in a

nonlo al variant, [24℄.

71

A tually,

1 state

in a position of a so- alled saturation magnetisation and (5.104) is a so- alled

Heisenberg onstraint, whi h is a ertain idealization used often mainly in mathemati al literature. A more realisti model allows for the magnetisation valued in the vi inity of the sphere

S n"1

whi h is

then treated only as energeti ally preferred but not as a hard onstraint.

72

Su h model, its relaxation, and its numeri al approximation have been investigated, e.g., by

Carstensen and Prohl [177℄, Choksi and Kohn [206℄, DeSimone [264℄, James and Kinderlehrer [404℄, James and Müller [405℄, Kruºík and Prohl [455, 456℄, Li and Wu [490, 491℄, Pedregal [597℄, Prohl [622℄, Rogers [650℄, Tartar [747℄ and also in [461, 694℄.

73

The density

Rn Ù R

' : ,

is supposed to be an even nonnegative fun tion depending on mate-

rial properties and exhibiting rystallographi symmetry. Two important ases are the uniaxial ase,

Ë

430

5 Relaxation in Variational Cal ulus: S alar Case

hext ), and the energy of demagnetisy2 dx # 12 P u-›x y dx. The dierential onstraint div(›x y "  u) # 0

energy (involving a given external magneti eld ing eld

1 2 PRn

›x

in the position of a state equation, alled shortly the Maxwell equation, is the rest from

the Maxwell's ele tromagneti dynami system with the va uum permeability normalized to 1 in the stati ase where all ele tri al ee ts and time derivatives vanish.74

u # 1. Therefore,

This problem is always non onvex be ause of the onstraint  

the problem (PMG ), in general, does not need to have any solution and its relaxation is desirable. We take a separable linear subspa e

H ’ Car( ; S n"1 );

H is C( )-invariant,

ontaining the integrands (

x ; s) ÜÙ '(x ; s) and 1 ‹ id.

(5.105)

Then one an onstru t a ne relaxation by ontinuous extension of (PMG ) as

(R

Minimize . 6 6

H PMMG ) > subje t to 6 6 F

 (y; ) # X ' DZ  " hext - (id DZ ) dx %



y "  (id DZ ) # 0 in Rn ; y ò W (Rn );  ò Y H ( ; S n"1 ) :

div›x

1 X › y  d x ; 2 Rn x 2



1;2

Proposition 5.49 (Corre tness of the relaxed problem (RH PMMG )).75 Let ' òCar( ; S n" n and h ext ò L ( ; R ), and let a separable linear spa e H omply with (5.105). Then: (i) The relaxed problem (R PMMG ) has a solution and inf (PMG ) # min(R PMMG ). H H

1

)

2

y; ) of (RH PMMG ) an be attained by some minimizing y k ; u k )}kòN for (PMG ) in the sense w*-limkÙ (y k ; i H (u k )) # (y; ).

(ii) Every solution ( {(

sequen e



(iii) Conversely, every minimizing sequen e for (PMG ) has, after being embedded into

W 1 2 (Rn ) , Y H ( ; S n"1 ), a weak* luster point and ea h su h a luster point solves ;



the relaxed problem (R H PMMG ). Proof. The relaxed problem (R H PMMG ) an be understood as a ontinuous extension (i.e. a relaxation) of the original problem (PMG ).

 determines y in a unique and ontinuous way, being u ÜÙ y o

urring impli itly in 2 (PMG ). Indeed, testing the Maxwell equation by y gives P n ›x y  d x # P (id DZ  ) - ›x y d x , R

1 2 from whi h we get the bound ›x y L 2 Rn ;Rn ¢ (id DZ  ) L 2 ;Rn ¢   . It follows from The important fa t is that

a ontinuous extension of the orresponding mapping

/

(

where

)

(

)

' attains its minimum along one axis, and the ubi ase when it attains its minimum along

three or four axes.

div b # 0 and url h # 0 b # 0 (h% u) the magneti indu tion and h the intensity of magneti eld, putting the va uum permeability  0 equal 1. From the equation url h # 0, one an see existen e of a magneti potential y su h that h # "›x y so that, altogether, div(›x y "  u) # 0 follows. 75 As  's are here supported on a bounded set , they have always a Young-measure representation.

74

More in detail, the magnetostati rest from the full Maxwell system is

with

For relaxation results as in Proposition 5.49 we then refer to [404, 456, 597, 650℄.

Ë 431

5.6 Example: Mi rostru ture in ferromagneti materials

the Lax-Milgram lemma that the Maxwell equation has, for any unique weak solution

yòW

1;2

R

(

n ) and that the mapping 

 ò Y H ( ; S n"1 ), the 

ÜÙ ›y is ane and weakly*

ontinuous. Obviously, it is also a weakly* ontinuous extension of the mentioned

u ÜÙ y.

mapping

In parti ular, the magnetostati energy

ontinuous.

 ÜÙ 21 PRn ›x y2 dx is weakly* lower semi-

The assertion then follows by the standard argumentation. Of ourse, universally one an onsider (

L ( ) ‹

id!!!!S n"1 ) and then ea h

2



H

L

as a linear hull of



(

; C(S n"1 )) €

has a uniquely determined Young measure. Su h

Young measures des ribe the relevant mesos opi al hara ter of the ne stru ture

u. We all this limit a mi rostru ture.

of the magnetisation

Let us note that the problem (R PMMG ) has a onvex stru ture. Therefore, its soluH tion (

y; ) an be fully hara terized by the rst-order optimality onditions of the type

(5.41), whi h leads again to a orresponding Weierstrass-type maximum prin iple.

Proposition 5.50 (Maximum prin iple [461℄). :

a.a.

xò :

Any solution ( y;

h y DZ 
(x) # max h y (x ; s)

with

s #1

 

) to (RH PMMG ) satises

h y # hext "›x y‹id " ' :

(5.106)

) satises the Maxwell equation div(›x y "  (id DZ )) # 0 and the max) solves (RH PMMG ). Moreover, there exists at least one hattering minimizer: more spe i ally, su h  has a Young-measure representation su h that, for a.a. x ò , x no more than n %1 atoms. Conversely, if ( y;

imum prin iple (5.106) holds, then ( y;

Proof. We will use the abstra t optimality onditions (5.41). We rst spe ify the linear onstraint mapping

 : Y H ( ; S n"1 ) , W 1 2 (Rn ) Ù W 1 2 (Rn ) 

;

;

*



 ( y;  ) ;

y # X 



›x

by the formula

y - ›x y dx "  ;  ›x y ‹ id : 



As the Maxwell equation has a weak solution even for a non-zero right-hand side ranging

W "1 2 (Rn ), the operator  is surje tive and, in parti ular, the onstraint quali a;

tion (5.40) is satised. As

 (-; y) is ane, ›  (y; ) ò H



›   ( y;  ) ;

while

is determined by the formula

 #  ; ' " hext ‹ id ; 



 y (y; ) ò W "1 2 (Rn ) is determined by the formula ;



› y  ( y;  ) ;

y # X 

Rn

›x

y - ›x y dx : 

 (y; ) #    %  y y. The "1 2 (Rn ) Ê W 1 2 (Rn ), i.e. equation › y  ( y;  ) #  y  with some multiplier  ò W n (5.41a), reads as div(›  " › y ) # 0 on R in the weak sense, whi h gives simply  # y 1 2 n when taking  ; y ò W (R ). The onstraint mapping *



*





is linear and in the form *





;

*



;



*

*

;

*

432

Ë

5 Relaxation in Variational Cal ulus: S alar Case

Realizing

   " ›  (y; ) #   y " ›  (y; ) # hext ‹ id " ' "  ›x y ‹ idò H , the *

*

*





in lusion (5.41b) reads as the inequality

0 £    " ›  (y; );  " #  " ; hext ‹ id " ' "  ›x y ‹ id : *

*







This implies the integral maximum prin iple, and then (5.106) by usual lo alization arguments. As

 is involved in n- onditions id DZ  and one other ondition with h y DZ , in view

of Proposition 4.28,

Corollary 5.51 (Uniqueness).76 For a given hext ò L ( ; Rn ), all the solutions (y; ) of (R PMMG ) have the same demagnetising eld y . Moreover, if ' ( s ) # s % - - - % s n , (i.e. the H 2

2

2

2

uniaxial magnet with the x 1 as an easy-magnetisation axis), then also the ma ros opi al

magnetisation

id DZ  is determined uniquely.

Sket h of the proof. 77 The relaxed problem (R H PMMG ) is onvex and stri tly onvex in

y. Therefore the potential of the demagnetising eld y is determined uniquely. also id DZ  is determined uniquely from the Maxwell equation div(  (id DZ  )) # div(›x y ). The oarse relaxation, f. Remark 5.53 below, has a 2 2 potential [ ' % Æ S n"1 ℄ ( s ) # s % - - - % s n % Æ S n"1 ( s ), f. [264℄, whi h is stri tly onvex 2 in the last n "1 variables. The possible nonuniqueness might therefore be only in (1 ‹ s 1 ) DZ  . This an be ex luded by using the maximum prin iple (5.106) similarly as

terms of

Then

**

we did in Corollary 5.26. Cf. also [451, Prop. 4.5℄. Another way how to re over existen e of solutions for (PMG ) is to regularize the problem by adding a term onvex in some derivatives of

u. Considering just the rst

derivative78 leads to

(PMG ; " )

. Minimize 6 6 6 6 6 > 6 6 6 6 6 F

subje t to

The additional term

 " (y; u) # X '(u) %

"

2

›x

u2 " hext - u dx %

u # 1 in ; y "  u) # 0 in Rn ; 1 2 y ò W (Rn ); u ò W 1 2 ( ; Rn ) :

1 X › y  d x 2 Rn x 2



div(›x

;

"

2 ›x u

2

;

has a physi al meaning as a so- alled ex hange energy.

This additional ex hange energy guarantees that the problem (PMG ; " ) has a (possibly not unique) solution (

m " ; u " ). However, for " ¡ 0 small, m "

may exhibit fast spatial

os illations, so- alled ne magneti -domain stru ture.

76

A. DeSimone [264, Prop. 4.4℄ proved that even the average over

tion, i.e.

P



of the ma ros opi al magnetisa-

id DZ  dx, is always determined uniquely. On the other hand, the uniqueness of id DZ  itself

does not hold in general, e.g. for ubi magnets.

77 78 ›x

See also [264, 461℄. In fa t, even a nonlo al ex hange energy (like a fra tional derivative instead of the full gradient

u) would lead to the desired regularizing ee t, f. Rogers [650℄.

5.6 Example: Mi rostru ture in ferromagneti materials

Sin e

" ¡ 0 is typi ally

Ë 433

very small in usual bulk ferromagnets,79 it is natural to

onsider it only as a singular perturbation and to investigate the asymptoti s when

" Ù 0. In the limit, it is expe ted to lead to the relaxed problem (RH PMMG ). Indeed:

Proposition 5.52 (Singular perturbations towards the relaxed problem [264℄). Let ' òCar( ; S n" ) and hext ò L ( ; Rn ). Then: (i) The perturbed problem (PMG " ) has a solution and lim " Ù min(PMG " ) # min(R PMMG ). H n (ii) Every sequen e of solutions to (PMG " ) has, after being embedded into W (R ) , n " Y H ( ; S ), a weak* luster point (y; ) for " Ù 0 and ea h su h a luster point 1

2

0

;

;

1;2

;



1

solves (R H PMMG ). Sket h of the proof. In fa t,

"

- onverges to

 for " Ù 0. For the onstru tion of a

re overy sequen e see [264℄. The previously exposed theory for numeri al approximation an straightforwardly by applied to the relaxed problem (R PMMG ). Let us illustrate the usage of the H dis retisation of Type I (nite-dimensional onvex) as in Se ts. 3.5.b and 3.5. .80 The

'(s) # K sin2È(s ; e1 ) where È(s ; e1 ) denotes the angle between the ve s and the unit ve tor e1 # (1; 0; 0). Figure 5.3 shows results of numeri al experi-

stored energy tor

ment with a ylindri al-shape spe imen of a uniaxial ferromagnet in a homogeneous external magneti eld applied in the dire tion of the axis of the spe imen whi h is also the easy-magnetisation axis of the ferromagneti material. Due to this overall axial symmetry, the three-dimensional problem redu es to a two-dimensional and the

S1 , whi h was dis retised by 8 points where the Young S measure was supported (used for the proje tor P from (3.167)) on ea h of 128 re tangu

lar elements used for the dis retisation of the domain (and for the proje tor P from sphere

S2

is a tually a ir le

Se t. 3.5.b), while the demagnetising eld on the whole spa e

R

3

was omputed by an

exa t formula and the overall linear-quadrati mathemati al programming problem has been solved by the K. S hittkowski [716℄ routine, f. [694℄ for details.81

79

The rigorous proof of this s aling is by A. DeSimone [264℄. It is not relevant in very small or at

magnets as nanodots or thin lms.

80

The onstru tion by two proje tors responsible for dis retisation over

and over

has been used

in [455, 490, 491, 694℄.

81

An adaptive strategy to redu e number of avoid non-a tive atoms of Young measure for ompu-

tation, based on the Weierstrass maximum prin iple as in Remark 4.64, has been implemented by M. Kruºík [450℄ and A. Prohl [455℄ while another approa h based on that at least some minimizers of the spa e (semi)dis retised problem (i.e. approximation of Type II) are of a hattering hara ter with apriori estimated number of atoms has been used in Z.-P. Li and X. Wu [490, 491℄, namely 4-atomi solutions have been implemented in 2-dimensional problems, whi h is ompatible (although pessimisti ) with Proposition 5.50.

Ë

434

5 Relaxation in Variational Cal ulus: S alar Case

PSfrag repla ements easy-magnetisation axis

outer magneti eld

gray s ale

Fig. 5.3:

0

gray s ale

Ms

"Ms

axis

outer magneti eld

Ms

0

easy-magnetisation

"Ms

PSfrag repla ements

axis

easy-magnetisation axis

easy-magnetisation

Left: a omputational simulation, showing a ross-se tion of a three-dimensional uniaxial

ylindri al ferromagnet in an outer homogeneous magneti eld, depi ting the magnetisation inside and the demagnetising eld outside the magnet. Right: a magneti domain mi rostrure (rening near the boundary) as observed experimentally on an Ni2 MnGa-ferromagnet through a polarized opti al mi ros ope. Courtesy of

Martin Kruºík

and

Remark 5.53 (Coarse relaxation).

Oleg He zko

(Cze h A ademy of S ien es, Prague).

Let us also remark that id

DZ

appearing in (R PMMG )

H

represents a ma ros opi al magnetization. It was shown by DeSimone [264℄, f. also Pedregal [597℄, that this magnetization solves the  oarsely relaxed problem involving a onvexi ation of the energy (

' % Æ S n"1 )

**

.

' augmented by the indi ator fun tion of S n"1 , i.e.

6 Relaxation in Variational Cal ulus: Ve torial Case Be ause

of

the

su

ess

of

the

`dire t

methods'

in the Cal ulus of Variations, many writers have shown that ertain integrals are lower semi ontinuous.

Charles Bradfield Morrey Jr.

(19071984)

Up to the present time, ... . transitions between dierent

rystal

modi ations

...

has

not

been

fully laried. ... the energy hanges ontinuously even although the symmetry hanges dis ontinuously. [472℄

Lev Davidovi h Landau

(19081968)

In the previous hapter we studied minimum-energy-type variational problems

y : Ù Rm with ’ Rn and m # 1. Su h problems were addressed as the s alar ones. If m ¡ 1, we will speak about ve torial variational involving an unknown fun tion

problems. The purpose of this hapter is just to expose a relaxation and a omprehensive approximation theory for ve torial (i.e.

m ¡ 1) multidimensional (i.e. n ¡ 1)

variational problems in a full generality, without any fo us to spe i problems. Despite of a formal similarity of the s alar and the ve torial variational problems, the dieren e between them is essential. This phenomenon is related with the fa t that the de isive property for existen e or non-existen e of a solution to the original problem is quasi onvexity, see (6.1) below. This property has a non-lo al hara ter and our understanding of it is so far very poor. Nevertheless, if

m # 1 or n # 1, the qua-

si onvexity oin ides with usual onvexity whi h has a lo al hara ter and admits a

omplete understanding. This makes the s alar problems nearly trivial in omparison with the ve torial ones. After introdu ing some auxiliary results about quasi onvexity and gradient Young fun tionals in Se tions 6.1 and 6.2 respe tively, we shall present in Se tion 6.3 a orre t well-posed relaxation of general ve torial problems (= the problem (R PVVC )) and, H

d

in Se tion 6.4, its nite-element approximation (= the problem (R PVVC )). This approxiH mation is still rather theoreti al only and, be ause of the outlined di ulty of general ve torial problems, one must unfortunately give up ambitions for a further omprehensive treatment unless one onnes oneself to spe ial ases only. Instead of this, Se tions 6.5 and 6.6 deal with further approximations onsisting respe tively in restri tion or enlargement of the admissible domain of the already onstru ted approximate relaxed problems  this reates respe tively the problems (R

d;k d H PVVC ) and (R H ; X PVVC ). The

situation is summarized on the following hart, where also a dire t approximation, i.e. the problem (6.63), of the original problem is in luded:

https://doi.org/10.1515/9783110590852-006

Ë

436

6 Relaxation in Variational Cal ulus: Ve torial Case

PSfrag repla ements

Original vectorial variational problem

(= extension by continuity)

PVVC )

(

FE

M

−a

pp

ro

relaxed Relaxed variational FEM− approximation Approximate problem problem

relaxation

xim

ati

on

(

d or

Ù

RdH PVVC )

(

ou ter theoretical on ati ap convergence convergence m i pr x for d Ù 0 o ox for X Ù X conditional r p im ap convergence r ati e for k Ù  on inn The problem The problem

e

c en

rg

ve

n co

RH PVVC )

0



f

Approximate original problem (6.63)

d;k (R H PVVC ) Nicolaides,Walkington

Chipot, Collins, Luskin, etc.

RdH X PVVC )

(

;

These approximations are eventually ee tive and an be implemented on omputers, whi h enables us in Se tion 6.7 to demonstrate on omputational simulations a ertain appli ability of them on modelling of mi rostru ture arising in ferro-elasti materials1 Be ause of spa e restri tions, we will have to present a lot of basi fa ts on erning quasi onvexity without proofs.2

6.1

Prerequisites around quasi onvexity

We shall rst dene various weakened modes of onvexity of a fun tion on (

v : Rm,n Ù R v(F) in

m , n)-matri es. As now the variable s ranges R v(s) to emphasize that F ò Rm,n is a matrix; the notation F is motivated from m , n , we shall better write

pla e of

nonlinear elasti ity where it usually stands as a pla eholder for deformation gradient.

Rm,n Ù R is alled poly onvex3 if there is a onvex fun tion R Ù R su h that v F # ! F; adj F; :::; adjmin m n F , where min m ; n

A fun tion

min(m ; n)

! : A #1

v :



(

)

(

2

(

;

)

)

(

)

min(m ; n),  :# (m )(n ) #  mm"  nn" is the number of subdeterminants  of the order  , and adj  F ò R denotes the ve tor omposed from all subdeterminants

abbreviates

!

!

!(

)!

!(

)!

; as usual, by a subdeterminant of the order  we understand the determinant of any (  ,  )-submatrix of the matrix F . Let us note that adj1 F # F . If n # m ,

of the order

1

Other purely me hani al appli ations are in a hair-like mi rostru ture arising within ommuni a-

tion of biologi al ells in a ollagen network [361℄.

2

At this point, the reader is referred for more details to the monographs by Attou h et al. [33℄, Buttazzo

[161℄, Ciarlet [216℄, Da orogna [241, 242℄, Evans [287℄, Morrey [553℄, Müller [559℄ or Zeidler [813℄. There is also a great, ever-growing amount of papers; let us mention in parti ular the works by A erbi and Fus o [3℄, Alibert and Da orogna [16℄, Ball [59, 60℄, Ball and James [63, 64℄, Ball and Knowles [65℄, Ball and Murat [66℄, Bhatta harya [116119℄ et al. [120℄, Chipot [201℄ and Kinderlehrer [204℄, Firoozye [312℄ and Kohn [313℄, Fonse a [315, 316℄, Fonse a and Müller [322℄, James [401℄, Kinderlehrer and Pedregal [424, 426℄, Kohn [430℄ and Strang [433℄, Kohn and Müller [431, 432℄, Kristensen [445℄, Luskin [502℄, Malý [514℄, Mar ellini [517℄, Matos [521℄, Morrey [552℄, Müller [557, 558℄, Pedregal [596, 598600℄, Reshetnyak [636℄, Serre [726℄, ’verák [741744℄, et .

3

The notion of poly onvexity has been invented by Ball [60℄.

Ë 437

6.1 Prerequisites around quasi onvexity

det F instead of adjn F , while instead of adjn" sometimes

of F dened as [ of F℄ij :# ("1)i%j [adjn" F℄ij is used. m , n The fun tion v : R Ù R is alled quasi onvex4 if

then we shall write briey

1

the so- alled ofa tor matrix

:F ò Rm,n :y ò W

1

1 ; 0

(

; Rm ) :

v(F) ¢

1 



X vF

% ›x y(x) dx :

(6.1)

v : Rm,n Ù R will be alled rank-one onvex5 if the fun m , n whi h are tion t ÜÙ v ( F 1 % t ( F 2 "F 1 )) : R Ù R is onvex for any matri es F 1 ; F 2 ò R rank-one onne ted, whi h means that Rank( F 1 " F 2 ) ¢ 1, where Rank F denotes the n m : x ÜÙ Fx . Equivalently, v is dimension of the range of the linear operator R Ù R m , n , a ò R m , and b ò R n . rank-one onvex if t ÜÙ v ( F % ta ‹ b ) is onvex for any F ò R Eventually, the fun tion

These modes of onvexity are related by the following hart:

onvexity

âá

poly onvexity

âá

quasi onvexity

âá

rank-one

onvexity

(6.2)

:

min(m ; n) # 1, while for min(m ; n) £ 2 they are a tually dierent ex ept possibly the rank-one onvexity and the quasi onvexity for m # n # 2.6 The de isive mode of onvexity from the variational-

All these notions oin ide with ea h other provided

al ulus view-point is quasi onvexity whi h is, unfortunately, extremely di ult to be ee tively handled. Therefore, the other modes of onvexity (i.e. the poly onvexity and the rank-one onvexity) be ome important be ause they yield both-side, omparatively ee tive estimates of the quasi onvexity. If both

v

and

"v

are quasi- onvex, then

v

is alled quasi-ane. In fa t, ea h

quasi-ane fun tion is (up to a onstant) a linear ombination of subdeterminants,7 i.e. it has the form

v(F) #

4

min(m ; n)

H a  ; adj  F %  #1

for some

a  ò R ; ò R :

(6.3)

This notion of quasi onvexity has been introdu ed by C.B. Morrey Jr. [552℄. In fa t, this deni-

tion is independent of the shape of

.

Later, J. Ball and F. Murat [66℄ introdu ed a so- alled

quasi onvexity whi h is equivalent to (6.1) provided

"C ¢ v(F) ¢ C(1 %

p F )

for some

Cò

R

W 1; p -

% ; f. [66,

Proposition 2.4(i)℄.

5

The notion of rank-one onvexity has been introdu ed by Morrey [553℄.

6

Any subdeterminant of the order at most 2 an serve as an example of poly onvex but non- onvex

v : R Ù R have been stated by Da# F " 2 F det F for ¡ 1 small enough (see also Alib-

fun tion. Examples of a quasi- but not poly- onvex fun tion

orogna and Mar ellini [245℄, namely

v(F)

4

2

2

2

ert and Da orogna [16℄), ’verák [741℄, Serre [726℄, and Terpstra [759℄. The question whether rank-one

onvexity implies quasi onvexity remained open for many de ades but eventually it was answered for

n £ 2 and m £ 3 negatively by ’verák [741℄ (see also Pedregal [600℄), while the ase n # m # 2 remains still open.

7

For this result see Ball [60℄, Morrey [553℄, or Da orogna [241, 242℄, Giaquinta and Hildebrandt [353℄.

Quasi-ane fun tions are often also alled null Lagrangians.

438

Ë

6 Relaxation in Variational Cal ulus: Ve torial Case

An important property of any quasi-ane fun tion is that the orresponding Nemytski mapping is sequentially weakly ontinuous if restri ted on gradients, namely8 w-lim

k Ù

y k # y in W 1 p ( ; Rm ) á ;

v(›x y k ) # v(›x y) in L p

/

w-lim

k Ù



(

)

(6.4)

p ¡  and v is quasi-ane ontaining subdeterminants up to the order  ¢ min(m ; n). From this and from the weak lower semi ontinuity of onvex fun tions,

provided

it easily follows that poly onvex fun tions reates weakly lower semi ontinuous fun tionals on

W 1 p ( ; Rm ), namely9 ;

w-lim

k Ù

y k # y in W 1 p ( ; Rm ) ;

âá for any

v ò C p (Rm,n )

liminf X

k Ù



v(›x y k (x)) dx £ X v(›x y(x)) dx

(6.5)



poly onvex. A tually, (6.5) holds even for quasi onvex fun -

tions.10 Having an arbitrary

v :

Rm,n Ù R, we dene its poly- (respe tively quasi-, or

rank-one) onvex envelope, denoted by

vPC (respe tively vQC , or vRC ), as

vPC (F) :# sup{ v (F); v ¢ v & v poly onvex};

(6.6)

vQC (F) :# sup{ v (F); v ¢ v & v quasi onvex};

(6.7)

vRC (F) :# sup{ v (F); v ¢ v & v rank-one onvex}:

(6.8)



















The quasi onvex envelope an be also equivalently dened by the formula11

vQC (F) #

inf

y ò W 01; ( ;Rm ) 

1



X v(F

% ›x y(x)) dx :

(6.9)

8

This result is by Reshetnyak [636℄ and it is based on the observation that any subdeterminant of

›x

y

an be expressed as a divergen e of some ve tor eld omposed from two fa tors, one onverg-

ing strongly and the other weakly. (For example, in ase

u

# (y y /x ; "y y /x 1

2

2

1

2

m

# n # 2 we have det(›x y) #

div(

u) with

1 ).) In fa t, it is a spe ial ase of a general phenomenon alled  om-

pensated ompa tness; f. Ball [59℄, Ball and Zhang [68℄, Ciarlet [216, Chap. 7℄, Da orogna [241, 242℄, Evans [287℄, Murat [564℄, Tartar [747℄, et . Let us point out that, in fa t, the weak ontinuity (6.4) har-

p # , the onvergen e in (6.4) holds in the sense L1 ( ) provided one onsiders only sequen es {y k } su h that adj › y k £ 0 for all omponents involved in the quasi-ane fun tion v . x

a terizes the quasi-ane fun tions. Moreover, for of distributions. By Müller [558℄ this holds even in

9 10

This observation is due to J.M. Ball [59, 60℄. This was rst observed by Morrey [552, 553℄. For the growth onsidered here, the lower semi onti-

nuity was established by Ball and Zhang [68℄ and, for

p

# 1, also by Fonse a and Müller [322℄. In fa t,

the weak lower semi ontinuity (6.5) is equivalent with the quasi onvexity of

11

v.

See, e.g., Da orogna [242, Se t. 5.1.1.2℄. The independen e of the right-hand side of (6.9) on a parti -

ular hoi e of

follows by the Vitali- overing argument.

6.1 Prerequisites around quasi onvexity

X of ontinuous fun tions Rm,n Ù X an X - onvexi ation of v , denoted by v , as Having a set

v X (F) :#

inf

X

òr a%1 (Rm,n )

R, we an also dene generally v( F ) (d F ) : 

Rm,n

:vòX: PRm,n v(F ) (dF )£v(F)

Ë 439

(6.10)



X - onvexi ation depends monotoni ally on X in the sense that X1 ’ X2 implies v X1 ¢ v X2 . Some parti ular ase are of spe ial importan e:

Obviously, the

Proposition 6.1.12 Let v £ v for some poly onvex v . Then: p m , n ); v onvex}, then v X is just the onvex If {&[adj ℄ i ; 1 ¢ i ¢ mn } ’ X ’ { v ò C (R X envelope of v , i.e. v # v , f. (5.36). min m n m , n p m , n ); v poly onvex}, (ii) If { v ò C (R ); & v is a subdeterminant} ’ X ’ { v ò C (R X X PC then v is the poly onvex envelope of v , i.e. v # v . p m , n ) has a p -growth (i.e. v ( F ) £ "  F  p " for some ò R% (iii) If p ¡ 1, v ò C (R m , n ); v quasi onvex}, then v X is the quasi onvex large enough), and X # { v ò C p (R X QC envelope of v , i.e. v # v . (iv) If v # v on {F £ %} for % ò R% large enough and if X # { v ò C(Rm,n ); v rank-one onvex}, then v X is the rank-one onvex envelope of v, i.e. v X # vRC . 0

(i)

0

1





**

(

;

)









1





**





Considering still the onvex envelope

v

**

, one gets as a dire t onsequen e of (6.2)

and of Proposition 6.1(ii-iii) the following hain of estimates:

v

**

¢ vPC ¢ v X ¢ vQC ¢ vRC ¢ v

(6.11)

v ò Cmin m n (Rm,n ); & v is a subdeterminant} ’ X ’ { v ò C p (Rm,n ); v quasi onvex} p m , n ) has a p -growth. and if p ¡ 1 and v ò C (R if {

(

;

)





A probability measure if there is

k ò N su h that



ò r a% (Rm,n ) will be alled pairwise rank-one onne ted k is 2 -atomi and takes the form 1

#

12

The

points

(i)

and

Rm,n ; &v

Cmin(m;n) (

)

(ii)



with

X

#

2

k

H al ÆF l l #1

(6.12)

&[adj ℄i ; 1 ¢ i ¢ mn}

{

1

and

with

X

#

v

{

ò

is a subdeterminant}, respe tively, have been basi ally proved by Da orogna [242,

1%mn)-

Se t. 5.1.1.2℄ if one realizes that the inmum in (6.10) is attained on ( atomi measures, respe tively. The enlargement of

X

mn 1% min  )#

and (

(

;

)

1

as admitted in the points (i) and (ii) then does

not hange the respe tive set of probability measures involved in (6.10). As to the point (iii), (6.10) then

Rm,n

F ( f. Corolv; f. the proof of (6.62) (expressed in ) whi h exploits the oer ivity of v. Eventually, for the

involves pre isely all gradient probability measures of

whose rst momentum is just

lary 6.13) so that (6.10) yields just the quasi onvex envelope of terms of a Young-measure representation of

point (iv) we refer to Pedregal [596, Thm. 4.1 with Lemma 4.3℄.

440 with

Ë

6 Relaxation in Variational Cal ulus: Ve torial Case

a l and F l satisfying the following re ursive onditions13 a l # Akj#1 l"1 2j"k %1 j ; F l # F l k ; l # 1; :::; 2k

2i j F2i j % 2i"1 j F2i"1 j # F i j"1 ;

2i j % 2i"1 j # 1; 2i j ; 2i"1 j £ 0; Rank(F2i j " F2i"1 j ) ¢ 1; i # 1; :::; 2j"1 ; j # 1; :::; k ; F1 0 ò Rm,n arbitrary; [(

;

)

;

℄

;

;

;

/ 7 7 7

;

;

;

;

;

;

;

(6.13)

? 7 7 7 G

;

;

where [-℄ denotes the integer part. The rank-one envelope an alternatively be expressed in the limit by the following modi ation of the formula (6.10):

Proposition 6.2 (Da orogna [242℄). vRC (F) #

Let

v £ v0 for some poly onvex v0 . Then:

inf

òr a%1 (Rm,n ); PRm,n F (dF )#F

v( F ) (d F ) :

X

(6.14)





is pairwise rank-one onne ted

Rm,n

Remark 6.3 (Firoozye's envelope). denote it by

For m # n another type of an envelope of v (let us vF ), whi h satises vPC ¢ vF ¢ vQC , has been proposed by Firoozye [312℄

who dened it by the formula

vF (F) #

One has always ( envelope

v

F

inf

v( F ) (d F ) :

X





òr a%1 (Rm,n ); PRm,n F (dF )#F; PRm,n det F (dF )#det F Rm,n  ‹F  (d F  )# F ‹ F % # i a i ‹ b i ‹ a i ‹ b i ; # i £0 ; a i ; b i òRn PRm,n F

v " q)

**

% q ¢ vQC provided q is quasi onvex. The philosophy of the

is lear from the formula14

vF (F) #

sup

(

quasi-ane q 2 quadrati rank-one onvex q#q1 %q2 ; q1

Remark 6.4 (Pedregal's envelope).

v" q)

**

(

F) % q(F) :

;

In [600℄, P. Pedregal proposed another mode of

onvexity between the quasi- and the rank-one- onvexity, say P- onvexity, dened by the requirement

:n ; n ò Rn ; n 1

2

v(F) ¢

2

2

1

-

2

HH H i #1 j #1 k #1

n2 # 0; :a1 ; a2 ; a3 ò Rm :F ò Rm,n :

2 " ("1)i%j%k 1 v„F % (("1)i a % ("1)k a ) ‹ n 16 2 1 % (("1)j a % ("1)k a ) ‹ n 2 1

3

1

2

3

2

One an introdu e the orresponding envelope, denoted by

Always,

13



:

(6.15)

vP :

vP (F) :# sup  v (F); v ¢ v & v P- onvex : 

…



(6.16)

vQC ¢ vP ¢ vRC . Sometimes it is even known15 that vP   vRC .

This ondition was invented by Da orogna [242℄, alled (

H k )- ondition there; f. also Pedregal [596℄

for a detailed investigation.

vF # v X for a spe ial hoi e of X . ¡ 3 and n ¡ 2, Pedregal [600℄ demonstrated this fa t on the ’verák's example [741℄

14

See Firoozye [312, Thm. 4.1℄. It is not lear whether

15

Indeed, for

m

of rank-one- but non-quasi- onvex fun tion

v.

Ë 441

6.2 Gradient generalized Young fun tionals

6.2

Gradient generalized Young fun tionals

H ’ Carp ( ; Rm,n ), we shall investigate in p m , n ), dened this se tion the set of the gradient generalized Young fun tionals G ( ; R H Considering a separable linear subspa e

like in (5.8) by:

GH ( ; Rm,n ) :# p

† òH

*

; Rm ) : i H (›x y k ) #  ‡ :

; ; {y k }kòN bounded in W *

w -lim

k Ù

The main dieren e from Se tion 5.2 is that

1;

GH ( ; Rm,n ) p

p

(

(6.17)

is not a onvex subset of

min(m ; n) ¡ 1 unless H is very small.16 Supposing p ¡ 1, H separable mn and (1 ‹ v ) ò H with v quasi onvex, as a dire t and C ( )-invariant, id # adj ò H H

*

provided

1



onsequen e of (6.5) we get the (generalized) Jensen inequality:17

: ò GHp ( ; Rm,n ) :

1 ‹ v) DZ  £ v(1 ‹ id) DZ  :

(6.18)

(

We will onne ourselves to homogeneous generalized Young fun tionals be ause the obtained results an be straightforwardly extended to pie ewise homogeneous ones, whi h will be su ient for our purposes.18 Let us rst investigate hattering generalized Young fun tionals:

Lemma 6.5 (2-atomi gradient Young fun tionals). Let  # a i H (u ) % a i H (u ) with p u l (x) # F l ò Rm,n , a ; a £ 0, and a % a # 1. Then  ò GH ( ; Rm,n ) provided19 F and F are rank-one onne ted with ea h other, i.e. Rank(F "F ) ¢ 1. 1

1

2

1

2

2

1

2

1

2

’ W p ( ; Rm ) su h that i H (›x y k ) # . As Rank(F "F ) ¢ 1, there are some a ò Rm and n ò Rn su h

Proof. We have to onstru t expli itly some sequen e { y k } k òN w*-lim k Ù

1

2

1

1;

2

Indeed, let m # n # 2, let H ontain the integrand 1 ‹ det, and let  l # i H ( u l ) with u l ( x ) # F l , # 1; 2. Then ea h  l belongs to GHp ( ; Rm,n ) be ause obviously u l # ›x y l for y l (x) # F l x. Yet  % p  òÖ GH ( ; Rm,n ) provided Rank(F "F ) # 2. Indeed, then det F % det F #Ö det( F % F ), whi h shows that the fun tional  ( y ) # P det(›x y ) d x is either not onvex or not on ave. On the

p ( ; Rm ), f. (6.4). As the fun tional  ( y;  ) # <  ; 1 ‹ det> other hand, it is weakly ontinuous on W

16

l

1

2

2

1

1

2

1

2

1;

is ertainly ane, by Proposition 6.22(iii) below, we an see that

p

GH ( ;

1

2

Rm,n

1

2

2

1

1 1 2 1 2 2



) annot be onvex.

P of su h that any open subset of ontains some E ò P and p H and a suitable extension of  ò GH ( ; m,n ))  E ‹ v ò H for any E ò P. Sin e H an be still onsidered as separable, there is a sequen e { y k }kòN bounded in W 1;p ( ; m ) su h that i H (›x y k ) Ù  weakly* in H * . Then ›x y k Ù (1 ‹ id) DZ  weakly in L p ( ; m,n ) and (6.5) implies P (1 ‹ v ) DZ  d x # limkÙ P v (›x y k ) d x £ P v ((1 ‹ id) DZ  ) d x for any E ò P, from E E E

17

We an take a ountable overing

R

suppose (after a possible enlargement of

R

R

whi h (6.18) already follows.

18

A general nonhomogeneous ase requires rather ne te hniques; we refer the reader to Kinder-

lehrer and Pedregal [426℄.

19

A tually, if

a1 ; a2

¡ 0 and H is ri h enough, the rank-one onne tedness of F and F is not only H # L ( ; C (Rm,n )) f. Ball and James [63, Proposi-

the su ient but also ne essary ondition; for

1

1

2

0

tion 2(iii)℄. This is related with the so- alled Hadamard ondition: gradients of two ane deformations must be rank-one onne ted if the deformation is globally ontinuous.

Ë

442

6 Relaxation in Variational Cal ulus: Ve torial Case

n # 1 and F1 " F2 # a ‹ n . For a urrent xed k ò N, let us divide onto layers with a1 /k and a2 /k su h that n is the ommon normal to the interfa es between

that  

the width

the adja ent layers as outlined in a two-dimensional ase in Figure 6.1.

PSfrag repla ements

a2 /k

a1 /k

F1

F2

F1

F2

F1

F2

F1

F2 n

Fig. 6.1:

Layered stru ture in

;

a so- alled simple laminate.

yk

Then it is possible to dene

y k (x) #



F1 x " (a1 %a2 l)k"1 a F2 x % a1 lk"1 a

l ò N. Let us note k, one has

where

lim

y k ( x) xÙ x l % a 1 ¡ k 

by20

yk

that

if if

l ¢ k   l % a1 ; l % a1 ¢ k   l % 1;

is ontinuous be ause, for

xò

su h that

l % a1 #

" lim y k ( x ) # „F x " (a %a l)k" a … " „F x % a lk" a … 1

1



xÙ x l % a 1   k

1

1

2

2

1

# (F "F )x " (a % (a %a )l)k" a 1

2

1

1

2

1

# (a ‹ n )x " (a %l)k" a # x ; n a " (a %l)k" a # 0 1

1

1

while, for

1

x ò su h that l # k, one an al ulate

lim y k ( x ) " lim y k ( x ) # „F x " (a %a l)k" a … " „F x % a (l"1)k" a … 

xÙ x l ¡ k



xÙ x l   k

1

1

2

1

2

1

1

# (F "F )x " (a %a )lk" a 1

1

2

1

2

# (a ‹ n )x " lk" a # x ; n a " lk" a # (l " l)k" a # 0: 1

1

1

F1 were not rank-one onne ted with F2 , su h onstru tion a1 or a2 vanishes). It is now evident that ›x y k os illates between F 1 and F 2 , and i H ( y k ) weakly* onverges for k Ù  just to  . Å Let us emphasize that, if

would not have been possible (ex ept a trivial ase when either

k

Corollary 6.6 (2k -atomi gradient Young fun tionals). Let  # l# a l i H (u l ), p u l (x) # F l ò Rm,n and a l and F l satisfy (6.13).21 Then  ò GH ( ; Rm,n ). 2

1

20 21

where

We refer to Ball and James [63, Proposition 1℄ for more details on erning this formula.

 possesses a Young-measure representation # { x }xò with x inò and pairwise rank-one onne ted. Contrary to k # 1, for k £ 2 the ondition

This means pre isely that

dependent of

x

6.2 Gradient generalized Young fun tionals

Ë 443

Sket h of the proof.22 Let us demonstrate the ase k # 2, the generalization for k ¡ 2 being straightforward. Let us noti e that the matri es appearing in (6.13) form the following graph (= a binary tree): ;

F1 1 ❅ ✠ ❅ ❘ F1 2 F2 2 ;

j#1

F2 1 ❅ ✠ ❅ ❘ F3 2 F4 2

;

;

;

;

(THE ROOT)

j#0

F1 0 ✟❍ ❍❍ ✟✟ ✙ ✟ ❥ ❍

(THE LEAVES)

j#2#k

;

The ondition (6.13) just says that ea h matrix is a onvex ombination of two respe tive matri es below it, whi h are mutually rank-one onne ted. The root, formed by the matrix

F1 0 , is arbitrary. The resulted  is then omposed from the leaves23 of this ;

tree. We an use the onstru tion from the proof of Lemma 6.5 for ea h pair of the rankone onne ted matri es in the above binary tree. We begin at the root level and on-

y k1 as in Figure 6.1 by means of the pair of matri es F1 1 and F2 1 , the parti ular layers having the width 1 1 /k1 and 2 1 /k1 , respe tively. Then we take ea h parti ular layer orresponding with the matrix F i 1 ( i # 1 ; 2) and, for k 2 ò N mu h larger than k 1 , onstru t a ner layered stru ture inside it by means of the pair of matri es F 2 i "1 2 and F 2 i 2 with the width of the parti ular layers 2 i "1 1 / k 2 and 2 i 1 / k 2 , respe tively. The ner layered stru ture must be suitably modied in a stru t the layered deformation ;

;

;

;

;

;

;

;

;

(small) neighbourhood of the interfa es of the oarser layers in order to satisfy the rank-one onne tivity of the adja ent regions;24 f. Figure 6.2. Thus we get a deformation

y k1

;

k2 .

(6.13) is not ne essary for

 to be a gradient Young fun tional be ause, as shown by Bhatta harya at

al. [120, Thm. 3.1℄, there exists a 4-atomi gradient Young measure supported on mutually not rankone onne ted matri es. This 4-matrix te hnique has been proposed by Casadio (see [242, p.116℄) and later independently by Tartar [749℄.

22

Alternatively, a non onstru tive proof follows from Corollary 6.13 below, using also the fa t that

1 ‹ v) DZ  £ v((1 ‹ id) DZ ) for any v ò C p (Rm,n ) rank-one onvex ( f. Da orogna [242, Se t. 4.1.1.3℄),

(

hen e for any

23

v quasi onvex, too.

A leaf is, by the usual graph-theoreti al denition, the end of the bran hes. Here the leaves are just

all matri es in the bottom row.

24

We refer to Ball and James [63, 64℄ or Kohn and Müller [431℄ for details.

444

Ë

6 Relaxation in Variational Cal ulus: Ve torial Case

PSfrag repla ements

1;1 / k 1

2;1 / k 1

F 1;2 F 2;2

F 3;2

Fig. 6.2:

Two-level layered stru ture of

F 4;2

; a so- alled 2nd-order laminate.

Then a areful limit passage drives

i H (›x y k1

;

k 2 ) to

.

More spe i ally, it holds

i y k2 tending to  su iently fast in omparison with k1 .

 #

w*-lim k 1 Ù w*-lim k 2 Ù H (›x k 1 ; k 2 ) so that one an sele t a suitable diagonal subsequen e with

Å

Let us emphasize that, in general, not every homogeneous gradient Young fun tional has the pairwise rank-one onne ted Young-measure representation and even need not be attained by su h fun tionals.25 Therefore it is worth turning our attention to homogeneous generalized Young fun tionals whi h need not be hattering. Then the ee tive riterion to de ide whether su h a fun tional is a gradient one or not is absent in general. Nevertheless, some riteria an be found in spe ial ases when

H

is small enough.

P , we shall suppose that there p m , n are some linear subspa es V ’ C (R ) and G ’ L ( ) su h that To have at our disposal a homogenizing operator

*



G ‹ V ’ H ’ l(G ‹ V) ; H where  l refers to the natural topology of

is

G-invariant; G “ C( );

(6.19)



Carp ( ; Rm,n ). Furthermore, we shall sup-

pose26

P : h ÜÙ ”(x ; F) ÜÙ

1 



X h ( x ; F ) d x • is a ontinuous proje tor on

Let us note that the operator



P



(6.20)

makes the integrands spatially onstant, and (6.20)

requires, in fa t, a ertain ompatibility of of Proposition 3.86, the adjoint proje tor

25

H:

P

*

H

as well as of its norm with

P.

In view

sends the generalized Young fun tionals

For the former fa t see Bhatta harya at al. [120, Thm. 3.1℄. The latter one follows from the exis-

v : Rm,n Ù R whi h is not quasi onvex; see ’verák [741℄. ÜÙ on { ò GHp ( ; Rm,n ); (1 ‹ id) DZ  # F}, being equal QC ( F ), diers (for a suitable F ò Rm,n ) from the inmum of this fun tion on the set {  ò to   v p YH ( ; Rm,n );  fulls the assumptions of Corollary 6.6 with some k ò N and F # F}, being equal RC ( F ) #   v ( F ) ¡   v QC ( F ); f. Proposition 6.2. to   v

ten e of a rank-one onvex fun tion Indeed, the minimum of



1;0

26

Let us again remark that, as previously, the ontinuity of

suitable norm on

H , e.g. the norm from Example 3.76.

P

an be always ensured by taking a

6.2 Gradient generalized Young fun tionals

Ë 445

YH ( ; Rm,n ) onto the homogeneous ones and from its proof, using also the onp m , n ) ’ G p ( ; R m , n ) so stru tion (6.23) below, one an also see that even P GH ( ; R H p p p m , n m , n m , n ). that obviously P G ( ; R ) # G ( ; R )  P Y ( ; R H H H from

p

*

*

*

Let us begin with the following essential observation:

Lemma 6.7 (D. Kinderlehrer and P. Pedregal [424, 426℄, here modied.).27 m,n , and (6.19)(6.20) be valid. Then, for any F ò R

Let

p £ 1,

M F # { ò P GH ( ; Rm,n ); (1 ‹ id) DZ  # F a.e. in } p

*

is a onvex subset of

H

*

(6.21)

.

 # N , {nite subsets of H}, dire ted by the ordering ¢ , ’. Let  ;  ò M F . Then there are bounded nets28 {y kl }kòN ’ W 1 p ( ; Rm ) su h l that w*-lim k Ù i H (›x y k ) #  l , l # 1; 2. We an suppose29 y l (x) # Fx for all x ò l l 1 m . Moreover, we an suppose y ( ; R ), although the whole net { y k ò W k } k òN 1 m naturally is not supposed bounded in W ( ; R ). Let us now take a xed ountable overing (up to zero measure) P0 of by pairwise disjoint subsets of the form x j % " j , " j ¡ 0, j ò N; the existen e of su h a overing

an be shown by Vitali argument; for a two-dimensional domain an example of su h Proof. Let us put (1)

( )

(2)

( )

;

( )



( )

( )

( )

;

;



self-similar overing is outlined on Figure 6.3.

Ω Fig. 6.3:

A two-dimensional domain

"

Then  j òN ( j )

jòN2 (" j

n )

#

and its ountable self-similar (a.e.) overing.

# 1 and there are N ’

n

1

1 2 . Then we put

y k (x) #

x"x j "j ) % (2) x " x j " j y k ( "j ) % > 6 6 Fx F . 6 6

"j yk

(1)

(

N and N # N 2

\

N1

"

su h that  j ò N ( j ) 1

Fx j

for

x ò x j % " j ; j ò N1 ;

Fx j

for

x ò x j % " j ; j ò N2 ;

n

#



elsewhere



(6.22)

:

Now we an take the sequen e of su

essively rened overings P k , k ò N, of the form Pk # {x kj % " kj ; x kj ò ; " kj ¡ 0; j ò N} as in the proof of Proposition 3.86. Then, for

27

Here the assertion is expressed in terms of generalized Young fun tionals instead of Young mea-

sures used in [424, 426℄. Also the proof is slightly adapted.

 an be onsidered independent of l be ause hosen ri h enough.

28

Note that the index set

29

Cf. Kinderlehrer and Pedregal [424, 426℄ for details.

Ë

446

6 Relaxation in Variational Cal ulus: Ve torial Case

 # (k ; {h l }) ò , we put y k (x)

" kj y k (

.

#

> F

Fx

x " x kj ) " kj

% Fx kj

for

x ò x kj % " kj ; j ò N ;

elsewhere

(6.23)

:

W 1 ( ; Rm ).  Then we denote  k # lim  ò  i H (›x y ) and  # limk Ù  k in H . As the set k  p m , n {›x y } k òN  ò  is bounded in L ( ; R ), the existen e of these limits (at least if k we would sele t appropriate ner nets30) is guaranteed by the  - ompa tness of p YH ( ; Rm,n ). Obviously, both

yk

and

y k

are in

;

*



;

We want to prove

k #

1 P i (› y 2 H x k *

(1)

)

1 P i (› y 2 H x k

%

*

(2)

:

)

(6.24)

l # 1; 2, let us put l # U jòN l (x j %" j ). Clearly,  1  #  2  # 21  . Similarly as in (3.166), for any v ò V and any g k ò L ( ) pie ewise onstant on the partition P k , we For



an al ulate31

 i H (›x y k ) ; g k

1

#



#

‹ v #

X v (›x y k ( x )) d x





H Xv (›x y k ( x )) d x X g k ( x ) d x

;

 l#1 2

l

1 H 2 l#

(l) ¼i H (›x y k ) ; P ( g k

‹ v)½ #

1;2

whenever with

1 

#

X g k (x) dx

1 H 2  l#

1;2

1 H 2 l#

¼P

*

X v (›x y k

l

( )

x dxX g k (x) dx

( ))



i H (›x y kl ); g k ‹ v½ ( )

1;2

 ò 

 # (k ; {h l }) with k £ k. Passing l  l #1 2 ¼ P i H (›x y k ) ; g k ‹ v ½. The identity 

is su iently large, namely

 to the limit, we get #

1 2



( )

*

;

(6.24) an then be proved by the density and the non- on entration arguments analo-

y

gous with the proof of Proposition 3.86; re all that ›x k

L



(

; R

m , n ).

k ò N , we get  # 12 P  1 % 12 P  2 . In parti ular, # limkÙ limò i H (›x y k ), we an sele t  # () *

Passing to the limit in (6.24) with it shows

òP

*

p YH ( ;

R

m , n ). Sin e 

l was supposed to belong to

( )



(

)

*

(

)

() p so that  # lim k Ù i H (›x y k ), therefore we have proved also  ò GH ( ; p (l) * m , n * (l) (l)  ò P YH ( ; ), we have P  #  (it follows simply from P Ž

R

therefore

30

 # 12  1 % 12  2 (

)

(

)

Rm,n . Sin e

Let us note that we do not need to hange (rene) the index set

(

A tually, if possibly

gk

)

 be ause it has been taken ri h

enough; f. also Example 1.4.

31

)

P # P), and 1 1 . Furthermore, we have obviously (1‹id ) DZ  # (1‹id) DZ  % 2

‹ v òÖ H , we an extend all involved fun tionals by ontinuity.

6.2 Gradient generalized Young fun tionals

Ë 447

1 (2) 2 (1 ‹ id) DZ 

# 12 F % 12 F # F . Hen e, in view of (6.21), we an see that M F has thus been proved onvex. Å For further onsiderations, it is ne essary to impose additional requirements on the subspa e

V . Supposing we have suitable sets V0 and V1 given, we require: V0 € V1 ’ V ’ C p (Rm,n );

(6.25a)

adj ò V ; V ’ v ò C p (Rm,n ); v is quasiane; V ’ v ò C p (Rm,n ); v is quasi onvex; mn

1

(6.25b)

0

0

(6.25 )

1

(6.25d)

and, for some onvex subset

Y # ’ YH ( ; Rm,n ) spe ied for ea h parti ular ase later, p

:v ò V : vQC #Ö "

âá

;{ v l }

: ò Y #

’ V; £ v; : lim

0

0

1; : : : ;

1; : : : ;



1

1

1

1 



/ 7 7 7 7 7 7 7 7 ? 7 7 7 7 7 7 7 7 G

(6.26)

! l : Rk1 Ù R non-de reasing means ! l (v1 ; :::; v k1 ) £ ! l (v1 ; :::; v k1 ) whenever v k £ v k for all 1 ¢ k ¢ k1 . Let us point out that, if vQC is equal to " at some point of Rm,n , then it is " identi ally.32 Some examples of V satisfying (6.25) and (6.26) will 

where





be mentioned below.33 The following theorem shows that, for

:v ò V : :v ò V : 0

1

V0 and V1 given, (6.26) makes the relations

1 ‹ v) DZ  # v(›x y) ; (1 ‹ v ) DZ  £ v (›x y )

(6.27a)

(

(6.27b)

su ient for attainability of a homogeneous (generalized) Young fun tional by gradients.

Theorem 6.8 (Gradient generalized Young fun tionals).

p £ 1, V0 and V1 satisfying (6.25b-d) be given, let a separable H full (6.19)(6.20) with V satisfying (6.25a), p m , n )  Y # . If the relations (6.27) are satised for some and (6.26), and let  ò P YH ( ; R p m , n m , n ). y # y(x) # Fx with F ò R , then  ò GH ( ; R Let

*

32

This result is by Kinderlehrer and Pedregal [424, Proposition 8.1℄.

33

Let us remark that often it is possible to take

vl

# v and then, of ourse, Y # # YHp ( ; Rm,n ). It simpli-

es onsiderably (6.26). Also, it is then possible to see that every

V , whi h satises (6.25a) and (6.26)

simplied as mentioned, is ontained in another subspa e with these properties whi h is maximal;

f. [669, Proposition 3.1℄.

448

Ë

6 Relaxation in Variational Cal ulus: Ve torial Case

P YH ( ; Rm,n ) ’ YH ( ; Rm,n ), there is a sequen e {u k }kòN bounded p m , n in L ( ; R ) su h that i H ( u k ) Ù  in H . First, we want to prove that, for all su h sequen es { u k } k òN , it holds Proof.34 Sin e

p

*

p

*

: v ò V :  ; 1 ‹ v £  vQC ((1 ‹ id) DZ ) : For

vòV

! l , {v l }lòN , and, for ea h l ò N, {v lk }k#1 k0 and { v lk }k#1 k1 ò V0 ’ V and v lk ò V1 ’ V , we have in the sense of the weak

given, let us take

l from (6.26). For every v k 1 L - onvergen e35

(6.28)

;:::;

v lk (u k ) # (1 ‹ v lk ) DZ i H (u k ) Ù (1 ‹ v lk ) DZ  # v lk (F)

# (1 ‹

v lk (u k ) 

Ù (1 ‹

v lk ) DZ i H (u k ) 

v lk ) DZ  

Then (6.29) and the onvexity and monotoni ity of [

;:::;





!l

£

and

(6.29a)

v lk (F) :

(6.29b)



implies

v l ℄QC ((1 ‹ id) DZ ) # [v l ℄QC (F) # ! l (v1l (F); :::; v lk0 (F); v 1l (F); :::; v lk1 (F)) 

l

v lk0 ) DZ  ; (1



l

v lk1 ) DZ …

¢ ! l „(1 ‹ v ) DZ  ; :::; (1 ‹ ‹ v ) DZ  ; :::; (1 ‹ 1 ¢ lim inf X ! l „v l (u k (x)); :::; v lk0 (u k (x)); v l (u k (x)); :::; v lk1 (u k (x))…dx 1

k Ù

# lim inf k Ù

# lim

k Ù

Realizing that











1

1







1



X [v

1

l QC ℄

i H ( u k ) ; 1

(

1 

u k (x)) dx ¢ lim inf

‹ vl #

1 



k Ù

 ; 1

1 





X v

l

(

u k (x)) dx

‹ vl :

v ¢ v l , whi h implies vQC ¢ [v l ℄QC , and passing to the limit Ù

 ; 1 ‹ v> as supposed in (6.26), we get eventually (6.28). For any v ò V , let us put v F # inf < M F ; 1 ‹ v > with M F from (6.21). As the ase v F # " is trivial, we will onsider only v F ¡ ". For any y ò W 1 p ( ; Rm,n ) su h that y (x) # Fx for any x ò , we have P i H (›x y ) ò M F , and therefore v F ¢