Optimal Control of Distributed Systems: Theory and Applications [1 ed.] 082181382X, 9780821813829

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Translations of

MATHEMATICAL MONOGRAPHS Volume 187

Optimal Control of Distributed Systems. Theory and Applications A. V. Fursikov

EDITORIAL COMMITTEE AMS Subcommittee Robert D. MacPherson Grigorii A. J'vfargulis

James D. Stasheff (Chair) ASL Subcommittee Steffen Lempp (Chair) IMS Subcommittee Mark I. Freidlin (Chair) A. 8. c'[lypcHEOB

OITTHJ\IAJTLHOE YITPABJTEHHE PACITPEl!EJTEHHLIMH CHCTEMAMH. TEOPH.Fl H ITPHJTOJKEHH.fl HOBOCv!Bli!PCH. HAYLIHAH 1-(Hli!rA. 1999 Translated from the Russian by Tamara Rozhkovskaya with the assistance of Scientific Books (IDl\.U), Novosibirsk, Russia. 1991 !1.1athematics Subject Classification. Primary 93C20; Secondary 35Q30, i6D05. i\BSTI-lACT. The book is devoted to the analysis of optimal control problems for systems described by partial differential equations. The methods proposed by the author cover the ca..-;cs when the

controlled system corresponds to a well-posed or an ill-posed boundary value problem, which, in addition, can be linear or nonlinear. The uniqueness problem for solutions of nonlinear optimal control problems is analyzed in various settings. In the last two chapters the author applies the general methods to the study of two problems connected with optimal control of fluid flows described by the Navier-Stokcs equations. The book cau be used by specialists and graduate studentes worldng in optimal control, partial differential equat.ions, and applied mathematics.

Library of Congress Cataloging-in-Publication Data Fursikov, A. V. [Optimal'noe upravlenie raspredelcnnymi sistcmami. English] Optimal control of distributed systems : theory and applications / A. V. Fursikov. p. ern. - (Translations of mathematical monographs, ISSN 0065-9282 ; 187) Includes bibliographical references and index. ISBN 0-8218-1382-X (add-free paper) 1. Control theory. 2. IVIathematical optimization. :~. Distributed parameter systems. I. Title. II. Series. QA402.:3 .F8713 1980 G29.S':n2-dc21 89-0-!8.377

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Contents Preface

ix

Chapter 1. 1.

2. 3. 4. 5. 6. 7. 8. 9.

The Existence of Solutions to Optimal Control Problems Function spaces and boundary value problems Abstract extremal problems Linear stationary e..xtremal problems Optimal control problems for linear parabolic equations Rigid control Nonlinear stationary extremal problems Nonlinear parabolic extremal problems Hyperbolic extremal problems Comments

1 1 12 19 27 36 40 46 54 63

Chapter 2. Optimality System for Optimal Control Problems 65 1. The Lagrange principle for an abstract problem 65 2. Linear regular stationary problems 76 3. Optimization for the Cauchy problem for the Laplace operator 83 4. Linear regular evolution problems 89 5. Problem for the backward heat equation. Distributed control 94 6. Problem for the backward heat equation. Boundary control and initial control 104 7. Nonlinear stationary problems 112 8. Optimality system in the case of control of singular semilinear parabolic equation 118 9. Comments 128 Chapter 3. 1. 2. 3. 4. 5.

6.

The Solvability of Boundary Value Problems for a Dense Set of Data The solvability of the Cauchy problem for an elliptic equation in the case of a dense set of data The approximate controllability of parabolic equations The solvability of the backward parabolic equation with the right-hand side belonging to a set with empty interior The Navier-Stokes equations The unique solvability· ~'in the large' 1 of the Navier-Stokes equations in a three-dimensional domain with right-band sides in a dense set Comments vii

129 129 134

1-:14 148 158 163

Chapter 4. 1. 2. 3. 4. 5.

The Problem of Work Minimization in Accelerating Still Fluid to a Prescribed Velocity Statement of the problem and the existence theorem Necessary and sufficient conditions for a minimum The uniqueness of a solution Asymptotics of solutions to extremal problems Comments

170 174 184 192

Chapter 5.

1. 2. 3.

4. 5. 6.

Optimal Boundary Control for Nonstationary Problems of Fluid Flow and Nonhomogeneous Boundary Value Problems for the Navicr-Stokes Equations Formalization of the problem Exact statement of the extremal problem A nonhomogeneous boundary value problem for the evolution Navier-Stokes equations The existence of optimal solutions Optimality system Comments

165 165

201 214 217 232

Chapter G. 1. 2. 3. 4. 5. 6. 7. 8.

The Cauchy Problem for Elliptic Equations in a Conditionally Well-Posed Formulation Statement of the problem A model problem. Formulation of the theorem on the main estimate The change of variables The main estimate The definition of a quasisolution The construction of a quasisolution Properties of a quasi-inverse operator Comments

193 193 198

233 233 237 241 248 253 257 260 2M

Chapter 7. 1. 2. 3. 4. 5. 6. 7. 8.

The Local Exact Controllability of the Flmv of Incompressible Viscous Fluid 267 Statement of the problem and formulation of the main result 268 Reduction to a linear controllability problem 270 An alL--..::iliary extremal problem and the solvability of the corresponding optimality system 274 Properties of the function w 276 The solvability of the linear controllability problem and estimates for solutions 281 The proof of the main results 285 The Carleman estimates 287 Comments 298

Bibliography

299

Index

305

Preface The book is devoted to the study of optimal control problems for distributed parameter systems, i.e., for systems described by a boundary value problem for partial differential equations. A typical optimal control problem can be \Vritteu in the form

J(y, u) ...., inf,

(0.1)

(0.2)

F(y, u)

(0.3)

= 0,

u E Ua,

where.!(-,·): Y xU-; !R: is a functional, F(·,·): Y xU-; Vis an operator, Y, U, F are Banach spaces, and Ua C U. The spaces Y and U usually referred to as the space of states and the space of controls. The variables y and u are the state and the control respectively. The functional J is called the cost functional. The set Ua is a set of constraints. The operator F(y, u) is given by some boundary value problem for partial differential equations. The simplest. example of the problem (0.1 )-(0.3) is the following optimal control problem for a distributed parameter s:ystem in a bounded domain 0 C JR:d with boundary iJ!l of class c=: (0.4) (0.5) (0.6)

J(y, u) =

r(lu(:r)- w(J:)I' + Nlu.(J:)- .f(J·)I') dx-; inf,

.In

~y(x) = u.(x),

Ylan= 0,

u(.1:) E Ua,

where w E £ 2 (!1) and .f E £ 2 (!1) are given functions, N > 0 is a parameter (the so-called cost parameter), and Ua is a convex closed subset in the space L2(S1). The state y(x) and the control u(x) a.re unknown. Thus, the abstract equation (0.2) is given by the relations (0.5) which for a fixed u E £,(!1) can be regarded as the Dirichlet problem for the Laplace operator. We can begin to study the problem (0.4)-(0.6) by solving the boundary value problem (0.5), i.e., by expressing y in terms of u. Then "\Ve substitute the result in (0.4), and reduce the problem under consideration to the minimization problem for a functional depending only on u in Ua. Such an approach is used for many optimal control problems for distributed parameter systems described by well-posed boundary value problems. However, this method does not always work. In more complicated situations, another approach is possible, which is based on some properties of the functional .] . Let the functional J(y, u.) be bounded from below and tend to +oo as (y, u) -; oo. For such a functional we can considerably ea.se the requirements on the boundary value problem (0.2), thus including in the framework of the theory those extremal problems for which the boundary value problem (0.2) is not necessarily well ix

PnEFACE

posed in the sense of Hadamard. For example, in this book we study the following optimal control problem for a distributed parameter system in a domain n c IPJ.d ·whose boundary consists of bvo disjoint manifolds fo and r 1:

an

J(y,

UJ,

Ho) =

(0.7)

r

lro -f-

(0.8)

(0.9)

tly(.-r)

/y(,T)- w(;r)/ 0 do-

r (!u.r(x)- .ft(.T)/ 0 -f- /u.o(.T)- j,(:r)/ 0)d 1/2. \Ve will use the following known Poincar8-Steklov inequality (the proof can be found, for example, in [23]). :!That is, from any bounded sequence {ulo(:r)} in 1Vf~ 11 (f2) we can choose a subsequence that converges in \-\-"1~~ (rl). \Ve note t:lmt the bouudcdness of the domain n is essential for the v 1/p and ru = hh11 u, 0 ,:; j < s- 1/p}, m·e the operators (1A). Then the trace opemlor r defined on cx(fl)

THEOREM L3, Let 1

where

"1/m

can be e:-dended to a contin'UO'US opemto'f'

r ·• TY'(!1) p

(LlO)

Jo ~II B·'-)·-ll"(an) p ' j=O

whe1·e Jn is the lm:qest integer such that Jn < s- 1/p. The operator (L10) 1.s on epimorphism, a.nd

r

has the linear continuous righl inverse ope-rator.

We note that Theorem L:J remains valid if W;;(!1) is replaced with Bj~(fl). By (LS) and (L9), for p = 2 we can formulate Theorem 1.3 as follows,

1. FU!\"CT!ON SPACES AND UOUNLMHY VALt:E PHOlJLEi'dS

· THEOREA1 1.4. Ld s > 1/2 and f'u. = {'T~nu, 0 :s;; j :s;; m L where '"'fhn are the operators (1.4) and m < s- 1/2. Then the tmce opemtor· r defined on cx(Q) can

be c:rtended to a. continuous opendot (1.11)

r: W(11) ~II w-HI'(an), j=U

and this operator is a sw:jection.

1.2. Elliptic boundary value problems in Sobolev spaces. The general theory of elliptic bounda.ry value problems is well presented in the literature. The book by J.-1. Lions and Magenes [114] is best adapted to our purposes. This book presents the elliptic theor:y in the spaces Hs. The case Hl1~ for p /::. 2 is treated in [113] and [119]. These works also deal with interpolation spaces and the solvability of elliptic boundary value problems in these spaces. The proofs of all the theorems in 1.2 can be found in the above reference:_~s. In the present book, we mainly use only results concerning the Dirichlet problem and the Neumann problem for the Laplace operator. Therefore, we consider only these problems and do not study general elliptic boundary value problems. "\Ve begin with the Dirichlet problem for the Laplace operator in a bounded domain n c ]p~d:

(1.12) (1.13)

'\vhere d

.,

(

-La-y I).·.· ,.-"'

)

6.· y-

j=l

. j

is the Laplace operator. THEOREM 1.5. Let s ;:, 2, 1

< p < oo. Then for any f(:r) E W;;-'(fl) aod

g E B~-l/p(IJll) the·re e:cists o uniqne solution y(:r) E W;;(n) to the problem (1.12), (1.13) .su.ch that

(1.14) where the constant C is -independent off and g.

"\Ve replace the Dirichlet condition (1.13) \vith the Neumann condition

(1.15) where 'Tc\n is the operator (1.4). (1.15).

Consider the boundary value problem (1.12),

THEOREM 1.6. Let·' ) 2. Let f(:r) E W;;-'(ll) and y(:c) E Bi~-l-i/J•(IJll) be snch that (1.16)

l,

f(:c) d:c-

.~n g(.1:') rh:' =

0.

Then the·re e:eists a .solution .ijCr) E lF,;(Q) to the problem (1.12), (1.15). The set of solutions of this problem consists of functions y(:r) + r~ where r is an arbiJra.ry

I. THE 1 0, 1 < p < oo. The operators (1.29) are e:r:tended from ex (Q) f,o coniJnuous operators 'lr:

(1.30)

w;:·"'(Q) _,

B;'-"I"(D),

s

> 1/p,

7::.: H'J~,:!s(Q)-+ B;:~tf2p.28~!/P(E), 7~8":

Wj;·''(Q) _,

2s

> 1/p,

Bj;- 1/ ' - 1/IJp),J,-l-l/p(E),

PROOF. We refer to [14] and [132].

2s > 1 + 1/p.

0

L TilE EXISTENCE OF SOLl 1 riONS

10

\Ve note that the operator ( /'n /'~-> {'~/:)~~) :

(U1)

. lF,;·:!s(Q)......;.

.,

,

B~-~-p(D.) X B;~~lj(:2p),:2s-l/Jl(~) X Bj~-1/:.!-1/(:!p),:!s-1-1/p(I:)

is not a surjection since functions from the range of this operator satisfY the compatibility condition for (1;,.1:) E {r} x rJ!1:

•·''.. rJ"(·,-1\i ("'-rJ"u) t r~ .Jn u) = •y,-iJi ~ 11 I· 1 for some k and j bounded from above by constants depending on s and p. In the case p = 2~ the anisotropic Sobolcv space coincides with the anisotropic Besov space and is denoted by H""·:!.s:

TF;'·'' (IP: !+d) = B~·'' (IP: l+d) = }]''·'·' (IP: !+d), W;'·''(Q)

= Jl"''(Q),

B.~·''(E)

= W·''(E).

\Ve consider the following Dirichlet boundary value problem for t.he hca.t equation \vith p > 0: ( 1.32)

!i(l,:c) -fL!:J.y(l,:r) = f(t,:r),

(1.33)

boY)(.r) = vo(>:),

(l.:H) where .Y = iJyjiJI. We also consider the Neumann problem, i.e., the problem (1.32), (1.:33) with the Neumann boundary condition on 2.::: (1.35) To solve the bounda.ry value problems in some class of smooth functions, it is necessary that the compatibility conditions for the data (f, Vo, 9) or (f, vo, ·1./1) be saJisiled. If we assume that the solution y, as well as the data (f, uo, 9) of the problem (1.32)~(1.3~1), are infinitely differentiable, then it is easy to 'ivrite recurrence formulas for DfuL=n in terms of Vo and f: k=1,2, ....

By the c:amJw.libilily conditions of order rn for the problem mean the equalities ( 1.36)

k

""/i)~(/'(1 i) I

,k

.IJ = "'/OUt 9,

k=0,1, ...

(1.32)~(1.34)

we

,'17l.

Similarly, using the same recurrence formulas, we obtain the following compatibility conditions of order m for the problem (1.32), (1.33), (1.35):

"=

(1.37)

0,1, ... ,111.

\Ve formulate the solvability theorems for mi.>;:ed boundary value problems for the heat equation. THEOREM 1.14. Let 1 < p < oo and s ;;, 1. Assume that. s - 3/2p > 0 is not an integer. Then for f E H',\·'~!), 2 (-•~I)(Q), un E Bf, 1 ·'~l/p)(il), and 'P E " sn.h.sjyrng . . . . . . . condJ.U.ons · · Bp_,~ l/2p.,(.,~lf'p) (""-') f.hc co·mpatdnhiy of orda 1s -3· /''-fJ1 there

c:rists a unique solution

u E H/-1~·~s(Q)

to the problem (1.32)~(1.3:1). This sol'IJ.t.ion

II

l. FU~C'I'ION SJ-',\CE.':i AND BOITNDAHY VALL'E 1-'llOL\LEl\IS

satL5fies the estima{e

(1.38)

llullll,;·''IQI,;; C(llfll 11 ·,;··-'>.'=l), where the constmd C

> 0 is indepen.dent

off~

vu_. andy.

PROOF. We refer to [132].

0

THEOREM 1.15. Lei 1 < J1 < oo and s ? 1. Assume that s- 3/(2p)- 1/2 > 0 is not an integer. Then for f E n~:·•-I).O(., .. l)(Q), v11 E B,;(.•-l/p\!1), and 'lj.1 E B;:-l/:l-l/:lp.:l(s-l/:1-l/'2p)('E) snUsJ!;ing the cmnpaiibifity confhlions of onler [s-3/(2p)-l/2] there e:rists a ·unique solnt.ion !IE H';:· 0 ·'(Q) to the problem. (1.32), (1.33), (L:l5). This solu.tion sal.isjies the estimate ( 1.39)

llullwr''IQI ,;; C(llfllw,r,"-''·"'-''IQI

+ llvoiiLI~·"-:.!h'(n) + 111/'lln;~ -1/~-1/1~1').'2(.'-t/~-

where t.he constant C

> 0 is independent

1/l~plJ(~lL

of j', Vo, a.ndl/1 •

PROOF. We refer to [132].

0

To conclude the section, we note that for y E lF1}· 0 (Q), o E H',i· 0 (Q), 1/p + 1/q,;; 1, the following Green formula holds (c:L t\,r example, [115]):

r(iJ,y-

(lAO)

}q

-

/1

/Lf:!.y)o rf:r dt =

!' (

.~

0, !J · 0

-

.Inrhr!J(:r)')'ro(;r) -/n!J(:t'J-illo(:r)) rf.T

J!On 0 )d:c' rJ/ -

r y( 01Z -1-

}q

/Lf:!.O) d:r rff.

Vv"e formulate one result concerning the solvability of parabolic boundary value problems which is beyond the scope of the theory presented in 1.4) but will be used later in the book. \Ve consider t.he follmving mixed boundary value problem for the heat equation with zero Dirichlet data: (1.41)

.i;(t,:c)- pb.y(t,:e) = f(l,:r),

We assume that the space

f

'loY= !Jo,

'i>:.Y = 0.

E L,(Q), .linE Lo(ll), f1 > 0. For the space of solutions we take

(1.42) THEOREM 1.16. (a) For any

T

E (0, T) the mapping

-is conLinv,ous and su"T:jective. (b) The m.apping

-is an

-isomorphism.

In both cases (a) onrl (b). Y is the space (1A2).

PROOF. We refer to [114].

0

I. THE EXISTENCE OF SOLUTIONS

1:!

2. Abstract extremal problems In this section, we prove existence theorems for some abstract extremal problems. \Ve will use these theorems in the proof of various existence theorems. \Ve begin '.Vith linear problems, i.e., extremal problems with linear constraints. For linear problems we also establish the uniqueness of a solution. 2.1. Linear extremal problem. Let ..Y be a linear normecl space, ~Yo a convexr) closed subset. of the space X, and .J(:e) a functional defined and bounded from below on )(u: (2.1)

oo) J(x) ) C

> -:xo Vx

E Xa.

\Ve assume that J(:c) is lower semicontinuous on ..Ya with respect to the weak convergence in _y, i.e., for an element X E X a and a sequence of elements :rn E .Xa converging weakly to X we have J(x)

~

lim J(.r,). 11-•X

Let X 1 be a reflexive Banach space continuously embedded in _y, and let \1 be a linear normed space. Consider a linear continuous operator

L: X 1

(2.2)

__,

V

and the extremal problem

(2.3)

JCr) __, inf,

(2.4)

L:z:+ F!J = 0, :c E Xa,

(2.5)

where Fo E 17 is a given vector. By an admissible element of the problem (2.3)-(2.5) we mean a vector :c E ..-Y1 satisfying (2.4), (2.5) and the condition J(:r) < cxo. The set of admissible elements of the problem (2.3)-(2 ..5) is denoted by 2l. \Ve assume that the following conditions hold. CONDlTION 2.1 (nontriviality). The set 2l of admissible elements of the problem (2.3)-(2.5) is nonempty. CONDlTION 2.2 (coercivity). For each R > 0 the set {:r E 2l bounded in the space X 1 •

J(.r) < R} is

By a solu.tion. to the problem (2.3)-(2.5) we mean an element X E 21 at which the functional J attains the minimal value: (2.6)

J(x) = inf J(J:). .rE2!

THEOREM 2.1. The problem (2.3)-(2.5) has a solution PROOF.

Since 2l -::J. 0, there exists a minimizing sequence r!!-·X

~\Vc recall that a set .Xo C

+ (1 ~ a:)r:1

.Tm

E 2t:

lim J(:z:m) = inf J(.T);

(2.7)

a:r,

x E X1.

:rE~t

X is said to he convex if E Xo for any o: E [0, 1].

.T'J

E Xn and :r:1 E Xo imply that

2. A13STRACT EXTRE!'v!AL F'flOBLEi\IS

moreover, J(:rm) have

~

C

< oo, where Cis independent of m. By Condition 2.2, \Ve

(2.8) Passing , if necessary, to a

~ubsequence,

from (2.8) we find that

(2.9) Since the embedding is also continuous on

~Y t ~\1

(2.10)

C ).; is continuous, any linear continuous funct.ional on .X and, consequently, from (2.9) we conclude that ;r 11

--:--

X weakly in

~Y.

Since the set Xu is convex and closed in .X, it is sequentially weakly closed by the lVlazur theorem0 in X (d., for example, [89, Chapter 3, §3.1]). Therefore, (2.10) implies that E Xo. Let L' : ]!'' __,X[ be the adjoint operator to the operator (2.2) 7 Then (2.9) implies that for a.nJ. w E v~-

x

= (:r,,L'w)s, __, (x,L'w)s, = (Lx, w),·. have L:rm + F(J = 0. This equality and (2.1.1.)

(2.11)

(L:rn,w) 1•

Since :r 111 E Q{, we imply that LX+ F(J = 0. Since .J is lower semicontinuous with respect to the >.veak convergence in X, from (2.10) ·we have

J(x):;; lim J(:z:,).

(2.12)

m--x

x

Thus, E 2t. By (2.7) and (2.12), we conclude that xis a solution to the problem D (2.3H2.5J. R.EI\lARK 2.1. The proof of Theorem 2.1 remains unchanged if the continuity condition on the operator (2.2) is weakened as follows: there exists a dense subset 5 C F·" such that for any s E 8 the functional :r --:-- (L:r, s)F is extended bJ-' continuity from the domain of the operator (2.2) to a linear continuous functional on Xt.

\Ve recall that the functional .J(:r) given on a convex subset ~Yo of the space .X is said to be conve:z: if for any .TJ 1 :c'2 E Xn and n E [0 1 1] the following Jensen inequality holds: (2.13) If for any ;r 1 f=. .1:2 and o- E (0, 1) we have strict inequality in (2.13), then the functional J is said to be st·rictly corwe:c. REfvlARK 2.2. A lower semicontinuons convex functional is lmver semicontinuous with respect to the weak convergence (cf., for example 1 [32, Chapter 1, §2.2]). Therefore, in Theorem 2.1 instead or the assumption that the fundional .J(.T) is lower semicontinuous in the sense of the weak convergence in the space X, we can require the following conditions on .J: (a) J is lower continuous on )(o with respect to the strong convergence in the linear normed space .X, (b) J is a convex functional. !iThat is, if the sequence :r, weakly converges in X t.o x, and :r, E Xn for every n, then :r E Xo. 7 An usual, z~ denotes the dual space to the linear norrncd space Z and(·, between Z and Z"".

·Jz

!.he dualit:;r

l. TilE

].]

A solution to the problem

EXISTl~NCL·~

(2.:~)-(2.5)

OF SOL!_TTJONS

is unique.

THEOREl'vi 2.2. If J is a strictly conue:r functionaL lhen a solnUon to lhe problem (2.3)-(2.5) is unique.

PROOP. Let J.·, and:eo be two solutious to the problem (2.3)-(2.5). The set 2l of admissible elements is convex. Therefore: from :r1 E 2l and :e:; E 2l it follows that. {.1:1 +:r:;)/2 E 2t. Since the functional J i~ strictly convex, for :l't f. :r:2 \VC have 1 J((:r 1 + :r,)/2) < (.r(:r!} + .J(a:o)),

2

which contradiet.s the ass11mpt.ion t.hat :c 1 and :r'2 are solutions to the problem (2.3)-(2.5). D The follmving convexity condition is sufficient (cf. [125]). PROPOSITION 2.1. Let .J(.T) be a. functional defined and twice diff'crcntiable in the Gateau:r sense on a conve:z; subset. Xu of a lineor normed space--~\. If for any X E ~~n we have J" (X)(h, h) ?: 0 for ull h E X, then J is n. conve:r: functional on Xn. .J"(.r)(h, h)~ ctlll1ll' for all hE X and et > 0 is independent of /1, then .Tis n. strictly convc:r functional.

rf

COROLLARY 2.1. Let X be 11 Hilbert space equipped with the rw'rm "'"EX. Then the functional .J(:r) = ll:r- .Toll' is strictly convc:r:.

11·11: and let

x

PnooP. Since for any EX we have .J"(x)(h,h) = 2llhll', the required aHsertion follows from Proposition 2.1. 0

2.2. Linear control problem. Cont5icler an important special case of the problem (2.3)~(2.5) \vhich will be referred to as an abstract linear control pmblem. Let Y and V be linear normed spaces. Consider reflexive Banach spaces Y1 and U such that (2.14)

}~'i

is continuously embedded in Y.

Consider a convex closed subset. Ui) of the space U and a convex functional

J(y, u) defined on Y x Uu that is lower semicont:.inuous and bounded from below. 1vioreover, let a linear (2.15)

continuou~

operator L:Y1 xU-;J7

be given. Consider the following extremal problem: (2.16)

.J(y, u) -; inf,

(2.17)

L(y, u) -1- Fn = 0,

(2.18)

n E Un,

where Fu E \f is a given vector. This problem is called a control problem,, u is called a control, y is called a state, and the functional J(y, n) is called the cost Jnnctional. To give a pn.~ci:::;e statement of the problem) we introduce (d. 2.1) the set 2t of admissible pairs (y, u) for the problem (2.1G)-(2.18), i.e .. the set of pairs (y, u) E Yr x U satisfying (2.17) aud (2.18) such that .J(y, u) < oo. "\Ve assume that the follmving nontrivialit.y condition and coercivity condition hold lor the problem (2.16)-(2.18). CONDITION 2.3 (nontriviality). The set Ql of admissible elements is nonempty.

~-

AFIST!l:\CT EX"!TlEC\lAL PBOULE?\·!S

CONDITION 2.4 (coercivity). For every R > 0 the set {(y,u) E 21: J(y, u.) is bounded in Y 1 >: U.

As in 2.1, by a solution to the problem (2.16)~(2.18) we at which tlw functional .] att.ains the minimum:

J(!f,'li) =

lllf~au

~

R}

a pair (ljJi) E Qt

.l(!J, u.).

inf

(!/.11 .I '~'21

TI-IEORET\[ 2.:3. Lel the assumption. stated in 2.2 be satisfied. Then the pmblcrn (2.16)-(2.18) has a solalioa (if, 'li) E Y1 " U. If the .faor:tirmal.! is strir:lly conve:r, then lhis solul'ion is uniqu.c. PROOF. \Ve show that. Theorem 2.3 is reduced to Theorems 2.1 a.nd 2.2. YVe set X = Y xU, X 1 = Y1 xU, and Xa =X x Un. Then (2.14), Conditions 2.:3 and 2.'!, as well as the otlwr assumptions formulated in this subsection le~ U. Hence

(2.32)

J(fj,U) ~ lim J(yn 11 um)· IT/~·=·

Indeed, IIYm lh· ,;; C, in view of (2.27), (2.14) andy, -; TJ weakly in Yin view of (2.28). From (2.31), (2.32) and the inclusion 1i E Uu it follows that. (y, u) E 2l, whereas (y, u) is a solution to the problem (2.23)-(2.25) by (2.26) and (2.:32). D REl\.lARK 2.3. In Theorems 2.3 and 2.4, the condition that the operator (2.15) is continuous can be replaced with the following weaker condition (cf. Remark 2.1): there exists a dense subset. S C y-+· such that for every 8 E S the functional (y, u.) -+ (L(y, u), s)F can be extended to a linear continuous functional on the space Y] x U. We prove that the set. !2l of admissible pairs of the problem (2.16), (2.23), (2.18) is sequentially weakly closed. LEi\.·IMA 2.1. Let the assumptions of Theorem, 2.4 hold and let the functional J(y, u) be bounded an each bounded subset oJY X Uo. Then the set 2l of admissible pa.i1·s of the problem (2.23)-·(2.25) is sequentially weakly closed in Y xU.

PROOF. Let (Y 111 , Um) E 2l for every m E PT, and let (!Jm, 'II m) -+ (Til U) \Veakly in Y xU. Since sup,(llvmlh· + lln,llu) < oo, we have sup, J(y,,u,) < oo. By Condition 2.L1 (coercivity), we have (2.28). As was shown iu the proof of Theorem 2.4, from (2.28) we obtain the relations (2.31), (2.32) and the inclusion 1i E Uo, which imply (TJ, u) E 2l. 0

:..!. ADSTR.r\CT E.XTH.E?\IAL PrtODLElvlS

IT

2.4. Nonlinear extremal problem. To conclude the section, \Ve consider a nonlinear analog of the problem (2.3)-(2.5). Assume that spaces ~Y, .X 1 : V: a set ..-Yu, and a functional J(:r) satisfy the conditions formulated in 2.1 before formula (2.2). Consider a. linear normed space X _1 such that

(2.33)

the embedding X C X -1 is continuous:

(2.:34)

the embedding

xl

(S:

x_l

is compact.

Let F:x,~v

(2.35)

be a nonlinear continuous operator satisfying the following condition. CONDITION 2.6. There exists an evcr:y""\vhere dense subset S C V"' such that for every wE S the functional :r: """"""7 (F(:c), w)1· is extended by continuity from X 1 to

x_,,

Consider the relation

(2.36)

F(x) = 0

and the extremal problem (2.3), (2.36). (2.5). As before, 21 denotes the set of admissible elements for this problem: i.e., the set of :c E Xa satisfying (2.36) and the condition J(:r) < oo. V\.Te assume that Conditions 2.1 and 2.2 are satisfied where '2! is the set of admissible elements for the problem (2.3), (2.36). (2.5). Under the conditions formulated in this section, the follmving assertion holds. 1

THEOREM

2.5. The1·e e.~ists a. solutia.n'

xE

X 1 to the problem (2.3), (2.36).

(2.5). PROOF. Arguing as in the proof of Theorem 2.1, we establish the existence of a sequence .1:m E Qt satisfying the relations (2.7), (2.9), (2.12) and prove the inclusion E Xa. From (2.9) and (2.34) it follows that

x

:L'm """"""7

X strongly in )(_ 1 •

B:y Condition 2.6 and the inclusion :rm E 2l., we have 0 = (F(a:,), w)l·

~

(F(x), w) I'

=0

as m

~

oo

\:fw E 11'.

Thus, x E Ua. F(x) = 0. By (2.12), x E 21. By (2.7) and (2.12), xis a solution to the problem (2.3), (2.36), (2.5). 0 2.5. Counterexamples. The follmving simplest examples demonstrate that the main assrnnptions (coercivity and compactness) of the existence theorems .for extremal problems are very essential. The nontriviality condition is also important. However, it is not necessary to confirm this fact by a special example because the extremal problem loses the meaning if the nontriviality condition fails. \Ve begin with an example \vhere the coercivity condition fails. tiA.s before, by a solution to the problem (2.3), (2.;{{)), (2.fi) we mean a function T E 21 such that .!(X)= infJ"E'~l.J(:r).

ltl

I. TilE EXISTENCE OF SOLL:TIONS

EXMIPLE 2.1. Consider the problem (2.3)-(2.5). We set X = X 1 = V = iF:, Let .J(:r) be a lower bounded continuous function defined on IR, let J~J = 0, and let L be the operator sending each function to 7,ero. ln this case, the problem (2.3)-(2.5) can be written in the form )(a= IP~.

J(.r)

(2.37)

~

inf,

.T

E iF:,

aud the set 2l of admissible elements coincides with iR. It is obvious that all the assumptions of Theorem 2.1, except possibly· the coercivity condition, are satisfied. The coercivity condition for the problem (2.:37) is equivalent to the condition ~

J(:r)

(2.:l8)

~

oo as :r

±oo.

If J(.r) = 1/(1 + ,r'), then the condition (2.38) is not satisfied and the problem (2.37) has no solution. Regarding the compactness conditions, 0 we have

~

{(y, u) E 2l: J(y, 11.) ~ N} =

{c4,uJ,

c {(y, u):

.[ciJ' + (1- .~i'J'Jdt ~

R, u.=

1-

ll.u11L,~ 1 , 1 ~ (1 +Vii)',/·' u'dt = 'lJ

.li'} /.r (1- .i/) dt

. ()

~ R}.

Since the set on the right-hand side of the last inclusion is bounded in TT--.1 ((), 1) X L,(O, l), the coercivity of the problem (2.:39), (2.40) is established. We show that the problem (2.39). (2.40) has no solution. By Theorem 2.4. the problem (2.39), (2.40) does not satisfy the cmnpactness condition. Using (2.40). we express n in terms of iJ and substitute the obtained expression in (2.30). Then the problem (2.39), (2.40) is written in the form (2.42)

I(y) =

~·I (y'(l) + (t/(1)- 1)') rl! ~ inf, ·"

y(O) = y(l) = 0.

\Ve show that the infimum of the functional (2 ..:12) defined on {-l---.~(0, 1), vanishes. For these purposes, we consider the follmving sequence of functions: ·I

Un(/) =

!

sgnsin(27rnT)dT.

·"

It is obvious that y, (0) = y, (1) = 0, Iii, (I) I' = 1 for all t except a finite set of points and y,(t) ~ 0 as n ~ oo uuifonnly with respect to IE [0.1]. Tlwrefore, I(y,) ~ 0 as n ~ cxJ. On the other hand, if y(t) 0, then !(y) =Land I(y)? y'dt > 0 for y(·) iE 0.

J:

=

3. Linear stationary extren1al proble1ns In the rest of the chapter, we construct the existence theory for optimal control problems for distributed parameter systems. \Ve clarify specific features of this thcor:y and emphasize differences from and connections with the theory of bouudary value problems for partial differential equations. As a rule, the existence theorems will be proved using the abstract scheme in §2. In this section, we consider stationary problems. Hereinafter, fl is a bounded domain in IR~rl whose boundary DO. is a closed (d- I)-dimensional manifold of class c·=. 3.1. The simplest controllable system. Distributed control. Let functions w(:r), f(:r) E L:2(0) be given, and let Ua be a nonempty convex closed subsr~t of L,(rl). WP consider the following extremal problem. Find a pair (y(:r), u(:r)) such that (3.1) (3.2) (3.3)

J(y, u)

=

l,

(ly(:r)- w(:rJI'

+ Nlu(:r)- J(:r)I'J d:r ~ inf,

b.y(:r) = u.(:r),

uliln= 0,

u.(-) E Ua,

·where :r = (:r:1, ... , :rr~), d:r = d:r1 · · · d:cr1, L1 = L~=l ()'2 /D:r.] is the Laplace opcra.tor, and N > 0 is the cost JHlil'amct.cr. The function u(:r) is referred to as a contml

1. TilE EXISTENCE OF SOLUTIONS

:.!!1

and y( .T) is called a slate of the controllable system. The minimization problem for the cost functional (3.1) provides a compromise solution of t\vo mutuall}r contradicting problems. Namely) we must choose a state function y as close to a given function tv as possible and the cost of control must be minimal. (The cost of control is zero if u =.f.) The cost functional (3.1) with the state function JJ and control u is called a compromise functiorw.l. \Ve indicate typical examples of Uu: Ua = (u(x) E Lo(D):

lluliL(n)

~ R'},

Ua = {u(~·) E L,(D) : u(~·) ? 0 for almost all J: E D},

Uu

= {u(x)

E L,(D): a(x)

~

u(J:)

~

f3(x) for almost all xED},

where a(x),;3(x) E Lx(D) and a(x) ~ [3(.1:) for almost all :rED. The convexity of the first set is proved on the basis of the Jensen inequality for the functional .J(u) = lluiiL(n)· This inequality is established in Corollary 2.1. Indeed, let ·u 1 ,u 2 E Uu and a E (0,1). Then

llau1 + (1- o)uJL(n)

~

ailu1ll' + (1- n)llu,ll'

~R

2

Using the definition of a convex set, it is easy to prove that the remaining sets are also convex. It:. is obvious that the first set is closed. To prove this fact for the remaining sets, it suffices to use the known assertion that for every sequence converging in L~(O) there is a subsequence converging for almost all .T E 0:. To formulate the problem (3.1)-(3.3), it is necessary to choose function spaces indicated in the formulation of the abstract problem in 2.2. The form of the functional (3.1) dictates the choice of Y and U. We set Y = U = L 2 (D). By (3.2), it is natural to set L(y,u) = !::.y-u, Y1 = H 2 (D) nH1\(D), F[1 = 0. For the chosen Y1 , U, and L the operator (2.15) is continuous if we set \f = Lo(D). THEOREM 3.1. There exists a unique solution (y, u) to the problem (3.1)-(.3.3) in (H 2 (D) n HA{D)) x L 2 (D).

PROOF. It suffices to shmv that the problem satisfies all the assumptions formulated in 2.2 before Theorem 2.3. \Ve have already chosen the function spaces and the operator L. For.] we take the functional in (3.1). Then Condition (2.14) holds and.] defines the squared norm in the Hilbert spaceY xU= (L,(D)) 2 . Hence it is continuous and bounded from below. By Corollary 2.1 1 it is strictly convex on Y X Lf. By Theorem 1.5, for any u(:r) E L 2 (D) there exists a unique solution y(J:) E H'(D) n H,\ (D) to the problem (3.2) such that

llullu'l"l

~

q,.,IIL,( 0 such that B1

={v

B,

E JI"+l(ru):

={v

E I-I"(ru):

llv- u.~llu• ,, < .J} C u,, [[u- u.~llu•· < .5} c U,.

These relations prove (:3.28o) and (3.29 1 ). 4. Optimal control problerns for linear parabolic equations

In this section: we study the solvability and uniqueness of solutions to extremal problems for an evolution equation of parabolic type. \\leU-posed, as "\Vcll as illposed boundary value problems describing a controllable system are considered.

4.1. Distributed control and observation. ~ 1 Let rl be a bounded domain in IR" with boundary of class T > O. Q = (0, T) x 11, ).; = (0, T) x We assume that w(t,.1.·) E Lo(Q), f(t,.T) E L,(Q), and Yu(:r) E L2(1l) are given functions and JV > 0. In Q, we consider the following optimal control problem:

an

J(y(4.2)

w,"- f)

c=,

1

=

an.

')

211.'1- wllrdQJ +

_N 2 lin-

:lj(l,:r) -ul>.y(t,:r) = H(i.,J>), (! . .1:) E Q;

')

..

fllr.,(Q) __, ml,

Yl>== 0, vl,~u= !/n,

u(l,:r) E Uo,

(4.3)

where wE L,(Q), f E L,(Q). !JoE L2(1l), Un is a noncmpty convex closed subset of L 0 (Q), .IJ(l,:r) = C!y(t,:c)jat, IJ i" 0. For fixed 11. and u > 0 the problem ('1.2) is a mixed boundary value problem for the heat equation, which is known to be well posed. If 11 < 0, then (L1.2) is a boundary value problem for the backward heat equation. This problem is ill posed. Typical examples of t.be set Un from the problem (4.1)-('1.:3) arc similar to examples of the set Un in the case of stationary problems:

(-!A) (4.5) (4.6)

Uu

Un = {11 E L,(Q): [[u- Hn[[ ~ R}, au E L,(q), L,(Q): u(I,.T) ~ 0 for almost. all (l,.r) E Q},

= {u E

Ua

=

{u E L,(Q): a,(t,:r)

~

u(t,:r) ~ n2(i,:r)},

where o;(l,:c) E Lx(Q) and O:J(I,:r) < n,(t,:r) for almost all (i.,:r) E Q. Introduce thE.~ space (4.7)

equipped with the norm

11=11~

=

ll=lli.c,QI + 11:- uL'>=llLun·

This space will be used in t.he proof of the solvability of the problem ('1.1)(:1.3). \Ve study thf~ trace of u. E Z on the lateral surface and on the base of the uTI. is well known that an observation is a function Cy placed in a cost funct.iounl, where y is the state and Cis au operator dcpewlin_r,o; on the optimal control problem. In (~1.1), Cis the identity operator, but for instance, in (::1.2--!), is the operator -il of restriction to r.

c

l. THE EXISTENCE OF SOLLTIONS

cylinder Q. Introduce the operator "h of restriction of u(t 1 :r) to the set t = the operator/'~ of restriction of u(t,:c) to the lateral surface E:

T

and

(4.8) LEI\Uv!A

4.1. The following tmce opera.tong are defined and are continuous:

,, : Z ~ H~ 1 (fl), (4.9)

TeO~: z~Ir'f'~;(Ofl;H~ 1 (0,T)),

j =0,1,

a

where 111 is the dijj"e'renliaiion along the outward normal toE and a;~ is the identity opera.lor. PROOF. By (4.7) and the continuity of the operator i'. : L;(fl) ~ H~ 2 (fl), from the relation z E Z C L;(O, T; £ 2 (fl)) we have i'.z E Lo(O, T; H~ 2 (fl)). Consequently, 0 1 z E L 2 (0, T; H~ 2 (fl)). Therefore, by the trace theorem (cf. [114]), we conclude that the first operator in (4.9) is continuous. With each function cp(t) E H,j(O,T) we associate the mapping

(4.10)

r ~:

Z

~ H~(fl), (r ,z)(:r)

T

=

z(t,x)cp(t) dt .

{

.fo

To prove the continuity of the operator (4.10L we use the relation

f'.f 9 z = -(f ¢Z + f 9 (:0- vf'.z))jv in order to deduce the estimate

l!r ~zii~J;i()

=

l!r ,zliL() + lli'.f ~z11Ln) 2

q

')

2

')

,;; l!r A£.()+ --ollf ~(i: -I;f'.z)ll£.(1 + --ollf l/u- ~zii:L.(n) ,;; (11z11LQ) + +

,;;

,~,liz- ''i'.ziiL(CJ)) II'PIIL(o '/)

2 v'llz II''r.,(Q) II.'f I" li,(o.T)

Gllzll~lll"lln•(n,T)·

This estimate proves the continuity of the operator (4.10) and 1 in addition, the continuity of the embedding

Z

c

H~(fl;Ir'(O,T)).

HX.,

The last relation, Theorem 1. 7 about traces of functions from and the relations D (1.8) and (1.9) imply that. the second operator in (4.9) is continuous.

By Lemma 4.1, the follmving space is well defined: (4.11)

Z1 = {z E Z: -r"z = 0},

llzllz,

=

llzllz,

where Z is the space (4.7). It is natural to look for a solution y to the problem (4.2) in the space (4.11) if u E L,(Q). We require that the set. Un in (4.3) sat.isf'y the following condition. CONDITION 4.1. There exists a pair (y, u) E

z,

:
'> ~ I'Vy(t,:r:)l-d:cdt •> -1 /' (y-(T,x)-y 0(x))d:c+v 2. n n ,n

(4.29)

;·T

=

.o

r g(f, X)!J(f, X) d:z: dt + .;·Tu lan r u(f, .In

.T

1

)y(f, :r') dx' df .

PROOF. Let Ej (:r) and 0 < J.IJ .s; I'·' .s; · · · .s; J.lk .s; · · · respectively be the eigenfunctions and eigenvalues of the spectral problem -6.Ej

For f-LJ =/=

+ Ej

= f-LJEj,

a.,ej(:z:') = o,

:r E fl,

:z:' E an.

integrating by parts, -..ve find

f-1-k:

d

(cj, clc} H'( 0. By the theorem on the solvability of parabolic boundary value problems (cf. [115, p. 86, 87]), any subset Ua # 0 of the space L,('E) satisfies Condition 4.3. If v < 0, then Uu satisfying Condition 4.:3 docs not exist for all y 0 E L 2 (rl) and g E Lc(CJ). If, for cxarnple, g = 0 and Yo = 0, then such a set Ua exists. Indeed, a set Uu C D2 (L,) containing 0 satisfles Condition ·:1.3 because, in this case, the pair (y, u) = (0, OJ satisfies ('1.25). In Lemma 2.6.1 (below), we will present more general examples of the set Un such that the nontriviality condition is satisfied for the problem (4.24)-( 4.26) with g = 0 and !In = 0 for IJ < 0. The set of pairs

;j.J

I. THE EXISTENCE Of SOL!_iTJONS

(y, u) E L 2 (0, T; H 1 ( rl)) x U11 satisfying (,J.25) in the weak sc•nse is called the set of admissible pairs lor the problem (4.2•1)~(4.26) and is denoted by 2l. By Lemma 4A, lor each pair (y, u) E 21 the functional .1 in (4.24) is defined. In accordance with the general definition, by a solution to lhe problem (4.24)~(4.26) we mean a pair (fj, U) E 2t such that .J(y- w, ii- y.,J

(4.31)

=

inf

(y,u) E'~l

J(y- w, u- y,J = s

(the number s is introduced for convenience). THEOREM 0 from (4.:32) we get

lluiii,(o.TJPI)) ,:; C(II!!IILQ)

(4.34)

+ II 11 IIL1~1 + lluoiiL(o)l·

Let (!/n: u 11 ) E 2l be a sequence minimizing the functional .J, i.e., lim J(yn - ·w:

(4.35)

lin -

y~) = 8,

!l-•X

where sis the number defined in (4.:31). By (4.35), (4.24), (4.33), and (4.34), we have

+ lluniiL,(>:),:; C,

IIYnllr.,(li."l x~ n.n.d let Condition 6.1 hold. Then there exists a solution (fj, li) E (W,2 (11) n L 0 (11)) x £ 2 (11) to the problem (6.2)-(6.4), where

(6.6)

r = min{2,a/x}.

PROOF.

\\.Te use Theorem 2.4. Let us verify that all the assumptions of this

ll

theorem hold. We set Y = £,,(11), U = £,(11), Y1 = W,'(l1) n Lu(D) n 1 )(11), V = £,.(11), Lr = Lx(l1), L(y, u) = 6.y- u, F(y) = b(y), J is the functional (6.2). By (6.6), the operators (2.15) and (2.19) arc continnous. If (y, u) is a pair from Condition 6.1, then b(y) - u.. E £,.(11). By Theorem 1.8 on the solvability

=

0

of the problem 6.y = f(x) u- b(y), 'IY = 0, we have y E W,'(l1) n w;.(l1). Therefore, Condition 2.3 (nontrivialit:y) holds. Using the estimate for solutions to the Dirichlet problem for the Laplace operator with the right-hand side J = u.-b(y), for (y, u) E Yr x U satisfying (6.3) we have

IIYih, + llullu

=

llulln? + llvlk, + llu[[L,

~

C 1 J(y- w, u- f)+ C,,

~

C(l[b(y)[[L, + IIYik, + llui[L,)

where J is the functional (6.2). Consequently, Condition 2.4 (coercivity) is satisfied. To verify Condition 2.5 (compactness), we prove the following assertion. D LEM:rvlA 6.1. Let G be a bmmded domain in JRd~ a.nd let n nonned space .:\(G) of fu.nctions y(:r), .T E G, be compactly em.bedded in. L 1 (G). Then for (3 > '!;;, 1 the embedrliug X(G) n L 1;(G) 1 ford= 2,

hold. where d is the dimension of the domn:in Sl. Then the-re cx·ists a sobdi,nn (Tf, u) E (H 0 (ll) n L,,(ll)) :< L2(ll) to the problem (6.2)-(6.4). PROOF. We set Y = Lr = Lx(ll), U = L,(ll), Y, = H 0 (ll) n H,j(ll), V = L2(fl), and define operators and the functional in the same way as in Theorem 6.1. From (6.Q) we obtain the inequality d(1/2 -l/(2x)) < 2, which, together with Theorem 1.2 (the R.ellich-Kondrashov theorem on compact embedding) (6.10)

H'(ll)

L,x(ll)

1, p > 1,

since for d(1-1/p) < 7, Theorem 1.1 (Sobolev embedding theorem) implies that the dual embedding lF1~1, C L-x(Sl) is continuous; here l/p 1 = 1- 1/p. Assmning that ~~ C (0, 1) and (y, u) satisfies (6.3), from Theorem 1.8 (solvability of the Dirichlet problem for the Laplace equation D.y = b(y)- u) and the inclusion (6.11) we find that y E H~;--, (ll). By Theorem 1.1,

(Ci.12)

r;. 1\"0NLINEAH ST:\TrONAHY EXTHE!\IAL PBOBLEl\lS

For'd = 2 we can take p 1 = 2x in (6.12). Hence b(y) + u. E L,(rl). Therelore. by (G.3), we have y E !{2 (0). From the above arguments: ·we obtain the estimate

(6.13)

llullrF "'C( lluiiL," + llu.llr.cl"' C\(llvll 11,;-• + llullr.cl "'C,(llb(y) + ullw,,' + IIHI!L,) "'c,(llvllr." + IIHIIL.,) "'c,.J(v- w, n + c,., (l-

which implies Condition 2.4 for r1 = 2. In the case r1 ):- 3, we choose p and "l in ((ill) so that. for f.lt in (6.12) we have Pt > x. This can be clone in view of (6.9). Since y E Lp, we have b(y) E Lp,j%· If p,jx < 2, then from (6.3) we have u E TFI~t/.v. C Lp.2 , '.vhere l/p'.2 = x/JJI- 2/d. By this equality, the ine:_~quality Pt > x, and the relation (6.9), \Ve have

Pl ;-

(6.H)

P2

=X-

2 < -jPl c.

Y.

(·1- -j2) < l. c.

Repeating the above arguments, from the condition y E Lp~ we conclude that y E L 1,, for p,fx < 2, where, as for (6.14), p,fp;J < x(1- 2/d) < 1. Repeating these arguments several times and using inequalities similar to (G.14): we lind that P1J x ):- 2 for some k. Hence y E TF]. By the above arguments, we can deduce estimates that are similar to (6.13). These estimates imply Condition 2.~1. By the above arguments, Condition G.l implies Condition 2.3. Thus, the existence of a solution to the problem (6.2)-(6.4) for a = x under lhe condition (6.9) follows from Theorem 2.4. D REMARK 6.1. If the functional (6.2) can be replaced with the functional

J(y- w, u- f)= ;· (ly(:1:)- w(:c)l''

+ iu(:r)- .f(:c)i 11 ) d:r,

'!l

where 1 < ;3 < oo, then Theorems 6.1. and 6.2 (with natural changes of formulations) remain valid. No essential changes must be made in the proofs. REl'dARK 6.2. The assertion of Theorem 6.2 remains valid if the condition (G.9) is replaced with the assumption x = rl/(d- 2), r1 ):- 3. In this case, we impose the following additional conditions on the nonlinear term b:

b(,\)A:;::, CJ!AI%+ 1 - C0 , where

F(z) =

{'

Jo

b(O

d~:;::, ab(z)z- c,

a> (d- 2)/(2d), Co> 0, Cr > 0.

The existence of a solution to the problem

(6.2)~(6.4)

is proved in the space

H 1\(rl) x L,(rl); see [33]. 6.2. Distributed control and observation. II. \Ve consider the case where the functional J is defined b:y using not the Lp-norms but the norms of the Sobolev spaces }/" ( rl):

1V . llu-} IIL.., 1n 1 --' mf, 2 \vhere the functions u1 E J-J"(fJ) and .f E L:2(0) arc given, k ):- 0 is an integer, and JV > 0. Instead of the semiliucar equation (6.3), we consider the boundary value problem for the more general quasilinear stationary equation (6.15)

(6.16)

J(y- w, u- .f)=

Ay(:r)

+ B(y(:r))

1

•)

2llu- wll;1 , i 0 is independent; assume that

of~

and .1.:. Regarding the nonlinear operator B, we

B(y) = b(y(:r)),

(6.22)

where bE c=(!ft 1 ) is a function satisfying the estimates (6.1) and

(6.23)

b(y)y) 'IIYI%+l -

-y

'(l;

> 0,

'(l

E 1ft.

In this case, the existence theorem for the problem (6.15), (6.16}, (6.19) holds for k = 0, 1 and x > 1. We note that for an operator B(y) of the form (6.22}, i.e., if B(y) is independent of 8y/8>:j, Theorem 6.4 holds fork> d(1/2-1/x) provided that x) 2. Therefore, ford = 2, 3, 4 1 Theorem 6.4 holds for all 1.: ~ 2 and r. > 1. Consequently, it remains to consider the case 1.: = 0, 1 and r. > 1. THEOREM 6.5. Let A and B be the opendors (6.17), (6.22) satisfying the conditions (6.21), (6.1), (6.23). Suppose that k = 0, 1, x > 1, Uo io 0. Then the·re exists a solution (fj, u) E (W!'x+l)/x(O) nLHl(O) nHJ(S1)) x L,(S1) to the problem (6.15), (6.16), (6.19). PROOF. We set Y = H'(S1) n Lx(S1), U = L,(S1), Y_t = Lx(S1), Y} = W('x+ll/x(S1)nLx+t(S1)nHMS1), V = L(x+ll/x(S1), L = Ay-u, F(y) = B(y). By Lemma 6.L the embedding Y1 = W?x+ll/x(S1)nLx+t(S1)nH,\(S1) (y(t, :r)) =

u( I, :r ),

/u.'/ = Yo(:r ),

-y-,;y = 0,

(7.8)

in Q, where the nonlinear term b(y) is given by a continuous function b(,\), ,\ E lR: satisfying the following; condition. CoNDITION 7.2. (a) Wehavceithersup~1 " 1 lb(A)/AI < worlim.1~~ lb(A)/AI = oo. In addition, if 1>(,\)j"\ __,-co as,\__, QO, tl1en there exist E > 0 and I\> 0 such that for any .\ > 0 we have

(7.9) (b) We have either sup-1-:;-tlb(.\)j,\l 0 and I,-> 0 such that for any,\< 0 the inequality (7.9) holds. \Vt.~ note that Condition 7.2 is satisfied for a sufficiently large class of functions that includes all polynomials and functions &(,\) = ±1.\l" .\, where cr ;;, 0, b("\) = ±e\ b(.\) =±cosh"\, etc. For n ;;;:: 0 and !3 ;;;:: 0 we put

X,. a= {y(t,.T) E Lo(Q): llullt ,, 0 and l\._- > 0 such that for any /\ ~ 0 we h;:we (7.29) Taking the inner product (in L,(ll)) of (7.7t) with Y+(t,:c), after simple transformations "\Ve find that

1 ,, :;11!1+(1, -)lli.,(r!i .:...

+

[,'

,, IIV'Y+IIi.,(l'idT

' ()

1 q ,;: :;II.Y+(O, -)11;., 1 ~>;

...

+ h + C1)

!'' ,()

,

q IIY+(,-, -)11i-, 1n;t1' +:;1 /,'' llullr,, 1,nt1T +c.

-',()

By the Gronwall lemma, (7.30)

llv+IIL(Qi ,;: C(IIY+(O, -)lli.,("i

+

llulli_,(QJ

+ 1).

By (7.29), we have I,

is the space (7.10).

THEOREM 7.3. There e:cists a solution (Y, u) E X,,l x L,(Q) to the problem. (7.3.5)-(7.37), and the integral (7.38) is .finite for fj.

PnooF. The set 2t of admissible elements is the set of pairs (y, u) E ~Y2 , 1 x Un satisfying (7.36) and (7.38). By Condition 7.3, we have 2l # 0. Therefore, there exists a sequence (y 11 , un) E Qt minimizing the functional J in (7.35). By (7.35) and (7.14), we have

(7.39)

llYn llx, ' +llYn llr,,(QI +

./qr(T- t)lb(y,)y, I d:r dl + llunllr,,(Q) ~ C,

where C is independent of n. Passing, if necessary: to a subsequence that

\Ve

can assume

(7.40) The following lemma will be proved after the proof of Theorem 7.3. LEMMA 7.1. Far a. function 'P E Cr\'0 (Q)

(7.41)

lim

r b(y, )'P cb: rlt ./qr b(fi)'P ci.?: dt. =

n-x}Q

Since a convex closed set is sequentially weakly closed. from (7.L10:2) we have TiE Un. Substituting (!Jn, '11 11 ) in (7.36L passing to the limit as n---+ CX), and using (7.401) and Lemma 7.1, we find that (fj, li) satisfies (7.36). Therefore, (fj, li) E '2l. Since CIJn, U 11 ) minimizes the functional J in (7.35): we conclude that Cif, U) is a solution to the problem (7.35)-(7.37) in view of (7.40). 0

T. NONLINEAFl PA!lr\BOLIC' EXTHEl\IAL PHODLE:\·!S

'PROOF OF LEMMA 7,L Since II(T- t)ullir• 1q 1 ~ C(llull:\-,,, + IIYIIL 1ml, from (7.40 1 ) we conclude that (T- t)y, ~ (T- t)fj weakly in H 1 (Q) and, consequently, strongly in L'2(Q). Passing to a subsequence if necessary, ·we can assume that Yn (t, "') ~ fj( t, .T) for almost all (t, :r) E Q. Consequently, by the Fatou lemma and (7.39), we have (7.42)

.~ (T- t)lb(!J)!JI rLz: rlt ~ }il_r'c .~ (T- t)lb(y, )Ynl d.r dt ~ C


::Y = 0. Then the pair (y, u) "'(y, .ii- L>y+ b(y)) satisfies (8.2)-(8.5). In the case Uo "I L,(Q), we can guarantee the validity of Condition 8.1 by imposing some additional conditions on Ua. For example, if Yu = y 1 0, then the inclusion b(O) E Un guarantees the validity of Condition 8.1 since the pair (y, u) = (0, b(O)) satisfies (8.2)-(8.4). If we do not impose any additional conditions on Ua f:. 0, then we can guarantee the validity of Conclition 8.1 by imposing additional conclitions on b(y) and applying the known theorem on the solvability of nonlinear hyperbolic boundary value problems (cf., for example, [108]). To est.ablish the solvability of the problem (8.1)-(8.5), we need the following knmvn lemma about solutions to the boundary value problem for the \vave equation (cf. [110, Chapter 2, §3])

=

(8.14)

jj(t,:r)- L>y(t,x) = F(l,:r),

'I"Y = 0,

'loY= Yo,

-roil= Yr·

LEMrdA 8.1. Let Yo E 7-i'-'(!1), Yt E 7-{-'(ll), FE L,.(Q). and let y(t,.1:) E Y, be a. solution to 1./tc pmblcm (8.14). where y, is the space (8.13). Then u(t,.T) satisfies the estimate

whe1·c 1( ...\) is a coniimwus positive monotone increasing function on Itt+={.-\;? 0}. PROOF. Since s = d(l/r- 1/2) = d(1/2- 1/r'), where 1/r' = 1 - 1/r, !rom the embedding 7-i"' C L,.,(D) we have L,.(D) C "}-{-·'. Therefore, FE L,.(O, T; "}-{-'). Taking the inner product (in 7-{ 0 = L0 (D)) of the first equation in (8.14) with A-'y and integrating with respect to t, we find

(8.16)

('(jj(T, ·) - L>y(r, · ), [J )H-' dT =

h

('( F(T, · ),

h

fi)·H-' rlr.

Let P;, : 7-1 11 _, [e,, ... , ek] be the orthogonal projection (in 7-1°) on the subspace [e 1 , . . . , e,,.J generated by the first. k eigenvectors of the operator .4. Since P,,y is determined from the system of ordinary differential eqnat.ions obtained after

b. H'{i"'EIUJOLIC' EXTHE:'v!i\L 1-'IIOIJLEJ\IS

applying the operator P~.- to (8.H), we have P,,.y E lY,'(O, T; ,, /, ,()

/,'' 1

;;D,(IIn.ull~-· .o-

(:ij- t;.y, P,!i)H.-· dT = =

ieJ, ... , c,,]).

Therefore,

+ llhYII~·-··)riT

~(llhW. ·lll~ + IIP,,u(t, ·lll~·-·· - llhvdl~- - IIP,,yoll~· -··l·

Since the left-hand side of the first equality and the two last terms on the righthand side of the second equality have limits as k . . . . ., oo, thE.~ first two terms of the second equality have limits in vie\v of the Beppo Levi theorem. Passing to the limit ask_, oo and using (8.16), we conclude that for every t E [0, T], 1

.

')

')

2(11v(t, ·lii'H-· + llu(t, lii'H• ) 1 ., ., ,::; ;;Uiu,ll?-i-· +llvoii'H•-·)+ ~

;,·' . 0

(.ij-/:;.y,,i;)H.-dT

,::; ~ (IIYtll~- · + II Ynll~ •- ••) + [' II F(T, ·)II H.-·· II !ill'1' II .till' I•' -

.h)

since 1/r -1- 1/r' = 1. In view of the continuity of the embedding £,.(0.) C '}-{-', this inequality implies that

ll!i(l., ·)II~-·+ IIY(I, (8.17) Setting o(t) =

,:;

llvi!I~-

llli(t., ·)ll'it-.., ±(I.)=

+

)II~·-

IIYoll~·-·· +

l

UIIF(T, ·)III,,I"I + ~) lllill dT.

from (8.17) we Bnd that

(7.11F(t, ·)11/.,lnl + ~ + P(t)) yG(t)

for some j3(t) ,::; 0. We solve this equation by eliminating j3(t) from the obtained formula. Using (8.17), we obtain (8.15). D Now we prove the solvability of the problem (8.1)-(8.5). THEOREM 8.1. Let the following condition hold, together with (8.6): '2dx

(8.18)

(l

> d + 2'

Then the problem. (8.1)-(8.5) has a solution (Y, 'ii) E Y, space (8.13).

X

L,(Q), whe·re Y, is the

PROOF. We assume that Y = L2(Q), Y_ 1 = L%(Q), Y 1 is the space (8.13), U = L 2(Q), V = L,.(Q) x W,;- 2(0.) x lY,;- 2(0.), where r is the number (8.7), L(y, u) = (.tj-/:;.y-u, 'lolJ, "io!i), F(y) = (b(y), -y0 , -yt). If (y, u) E Y 1 xU satisfies (8.2)-(8.4), then Lemma 8.1 implies that

llulh·, + llullu = IIYIIL,,/f/1 +liD- t;.yiiL,IQI + ll.tiiiL,. 11 ,_,(n.T;'H-'I + IIYik.;,,._,,(II.T;'H'-•) + llu.llldQ) ,::; IIYIIL,(Q) + Clluiiidq) + "t(IIYnllu,:lnl + IIYtiiL,(nl + llb(y)III.,(QI + llu.IIL,ItJi) ,::; IIYIIL,(QI + ClluiiL,(QI + "t(C, + C2(11uiiL,,(QI -1- lluiiL,(QJ)).

L TI-lE EX!STEi\'C'E 01' SOLUTIONS

tiH

Hence Condition 2.4 holds. B)· (8.13), the embedding

Y1

(8.19)

c Z"

={!IE L, 11 ,.~ 1 J(O,T;H 1 ~,,): [J E L,. 11 ,.~, 1 (0,T;H~')}

is continuous, where " = r/(1/r- 1/2). By (8.18) and (8.7), we have s E [0,1). By Lemma 1.2, is compactly embedded in Lc(Q). In view of (8.13) and (8.8), the embedding Y1 c Z'2 n Ln(Q) is continuous. By (8.6) aud Lemma G.l, the embedding Y1 C L:v.(Q) is compact, which proves (2.21). The verification of the rcmainiug assumptions of Theorem 2A is obvious. D

z,

We note that ford= 1 Theorem 8.1 remains valid even if o = "'(cf. [110, Chapter 2, §3]). If d = 2, then the inequality (8.18) is equivalent to the first inequality in (8.6). If d ~ 3, then the inequality (8.18) is more restrictive than (8.6). 8.2. A priori estimates. The condition on a in Theorem 8.1 can be considerably weakened if we require that the nonlinear term b(/\) from equation (8.1) satisfy a certain condition that is not. too restricted. Using this condition, we can deduce a priori estimr~tcs for y satisfying (8.2)-(SA). \Ve will derive these estimates iu this subsection. Thus, we assume that the nonlinear term b(,\) in the problem (8.1)~(8.4) satisfies the follmving condition. CONDITION 8.2. b(,\) E C(JR), lim.\-±x (b(A)(\( = oo, and there exist c > 0 and [\- > 0 such that for every /\ E IR:,

(,\) =

(8.20)

/"'

./o

b(p.) dp.;;, -(1/2- c)[b(,\)AI- I\.

Condition 8.2 is satisfied for any polynomial of degree at least 2 and for functions of the form b(,\) = ±[A["A, where o > 0, b(,\) =±cosh,\, b(A) =±sinh,\, etc. We set

(8.21)

£.,,, (Q)

= {u(t, .1:) :

llu[[i,."

=

/Q [T -I['' [u(t, J:)['dx dt < CXJ}.

Assume that a pair (y,n) E Xn,n x L 1 ,, (Q) satisfies the relations over. let

(8.2)~(8.4);

more-

lrr-

ti" lb(y )yl d.1: dt < DO. ·C! \Ve recall that Xo,u is the space (7.10). \\.Te specify n· in the formulation of the theorem below. (8.22)

THEOREM

8.2. Let b(,\) 80tisfy Condition 8.2, and let

Suppose that

l,

(8.23)

!/II

E JJ,\(r!),

!/1 E'

L,(r!).

[(uo)l d:r < oo.

(a) If a pair (y, a) E Xn. 11 x L,.n(Q) satisfies (8.2)~(8.4) and (8.22), n = 0, and

llu(T, ·)lli., 1111 < oo, then T

(8.24)

llull\,, +

r llb(y)y[[c,(!l)dt 0 for some .TJ E K 0 • Consequently: the vectors :ro E /( 2 and :q E J\.2 are situated on difi'erent sides of j-L 1 which contradicts the assumption. Therefore: (1.1) holds with :~;J· = 0 and .1:~ = f. Thus: \Ve can assume that.

If J\2 is situated on one side of the hyperplane

(1.2)

M

=fJ. n K

2

#

0,

M

#

0,

3e E K 2 \M,

(f, e) < 0.

\Ve prove that Af n J\"1 = 0. Indeed, otherwise for :L'(j E 1H n I\ I there exists E > 0 such that :~: 0 + Ee E K 2 n K 1 (the set I\ 1 is open). But (.f, :r 11 - ce.) = c(f, e.) < 0, which contradicts the inclusion f E (/\. 1 n/\·2 By the above relation and the first separation theorem (cf., for example, [6, §2.1.4]): there exists a functional g EX\ separating AI and l\.1:

r.

( 1.3)

(g, :r) > 0 ;:, (g, y)

V:r E K,

'ly E M.

Note that the possibility to put zero in the inequality (1.3) is based on the fact that /\"1 and AI are cones and zero is their limit point. Let N = fJ. n gJ., X/N the quotient space, and rr: X--; X/N the natural projection (i.e., 1r:r = :r + JV E X/1V for ;rEX). Since codimJV = 2, we conclude that X/N is a two-dimensional plane. Let c; EX, i = 1,2, (g,c,) = 0, (f,E·,) = 0, (.f, EJ) = 1, (g, c'2) = 1. Set Ct = 7iE 1 and e~ = JTE:::. Define the functionals and g on X/N by the formulas (.{, rr:c)s.;N = (.f, :c)s and (g, rrJ:)s.;N = (g, :r)s for all :r E _,.y. This definition is obviously unambiguous. The vectors e1 and e2 form a basis for )(jN; moreover: l.J = !JJEt +JJ2C::! for ally E ...-YJN, where .111 = (y,el)s;N and 'Y'2 = (y,e'!.)_\)N· It is obvious that 7r/( 1 and 7il\"2 are cones in X/N = .!Pi.2 = {Y = (Yt,"Yo)}; moreover, by {1.3), we have rrK, C {y = (y,,y,): y, > 0}. Let rays l 1 , 1'2 C ..Y.jJV starting at the origin be elements of the cone 11!('2· Since rrK, n rrK0 # 0, from (1.3) we find 11 c rrK 1 C {y = (!1~:!1 2 ): y 2 > 0}. By (1.3), we have 7rJ\{ = {y = (-y 1 , y2 ) : y 1 = 0, llJ. < 0} C Jrf(2 and the cone 7il\"'2 is convex. Hence we can assume that

f

Let 'Pi be the angle between cot 9'2 > 0. Take o: such that

c1

and /i. Since

1rf('2 i~

cot

0

h on )\jJV by the formula

= f(y)- c@y) = y, -

ay,,

y

= (!It, Y2),

!. TllE LAGBr\NGE PfUl'JClPLE FOH AN r\LISTHACT P[(OL\LE-1\'1

\vhere the second equality follows from the definition of

J and fl.

G7

Let us show that

( 1.5) For y E {y = (y,, y,): !fl ? 0, y, 0. If y E rrK, n {!I= (y,, y,): y, < 0, y, < 0}, then

h(y) =

!fl-

oy, = /y,/(a- y,jy,)? luci(o- cot>p,)? 0.

Similarly, for y E rrh·,n{y= (u 1 ,y0 ): y 1

O,y,

:;:,

> 0} we lind

h(y) = U1- ctyo = y,(.IJJ/.'1,- n)? y,(col: 'Pl- a)? 0. Thus, if his defined on X by t:he lonuula h = rr'h, then from (1.5) we lind (h, :c) :;:, 0 for all :r E 1\·'2· By (1.4) and the definition of the functionals and g) we have f = ng +h. Thus, we have obtained (1.1) with 1:] = ctg and 1:2 =h. 0

J

THEOREi\1 1.2 (Dubovitskii and fviilyutin). Let l\" 1 and ]('2 be conve:z; cones in .X 1 and let f(t be open. Then [\"t n ]('2 = 0 1j and only if ther-e exist nonzem fund.-ionals .TT E K{ and :r~ E J\"2 such that :r]' + ;r2 = 0.

PROOF. LeL 1( 1 n I\ 2 = 0. By the first separation theorem, there exists a nonzero functional .1/ E _'\:" -t· such that inf (y', :c) :;:, sup (y', .:) '"'"'inf,

(1.8)

L:r

+ F11

= 0,

:t E ~Ya.

(1.9)

Let :r 0 E X satist'y (1.8). Recall that the set A= {:c E Xn n {:c 0 + Ker L} J(:r) < oo} is called the set. of admissible elements and a vector X E A satisf~ying the equality J(J:) = inf"c..-tJ(J:) is called a solu.l.i.on t.o the problem (1.7)-(1.9). We will assume that a solution X to the problem (1.7)-(l.!J) exists. An optimality system for the problem (1.7)-(1.9) can be deduced using the Lagrange principle. To apply this principle) it is necessary to construct the LagTange function

L.(x, ~\,p) = A./(.1:)

(1.10)

+ (p, L.T + F0 ),

v-t.

where,\ E IR+, p E By the Lagrange principle) there exists (/\JJ) f- 0 such that the solution J: to the problem (1.7)-(1.9) is also a solution to the problem (1.11)

L.(:c, ~\,p) _, inf,

:r E Xn.

Gb

2. OPTI!\IALITY SYSTEJ\1 FOB OPTil\lAL CONTROL PROBLEMS

If the functional J : X ~ !R is Gateam-difl'erentiable at the point x, then the necessary condition for a minimum in the problem (1.11) has the form (L~(x,A,p),,r-x)s?

(1.12)

0 'h: E Xa,

·where .C~1 : is the GateatLX derivative of the function (1.10) ·with respect to x. If we prove that A fo 0 in (1.11), then we can set A = 1. The relations (1.8), (1.9), and (1.11), where L is the function (1.10) with ~\ = 1, are referred to as an opiima1ity system of the problem (1.7)-(1.9). Let us justify the Lagrange principle for X a with nonempt_y interior, i.e., Iut ~Yay'::. 0.

(1.13)

Using the linearity of the operator L and the convexity of the functional J, we prove that the optimality system provides sufficient conditions for a minimum in the problem (1.7)-(1.9).

x

THEOREM 1.3. Let be a solution io the problem (1.7)-(1.9), and let J be GateouJ:-dif/erentiable at x. We assame that the condition (1.13) is satisfied and the set ImL o= LX is closed in V. Then there e:cists a pair (~\,p) E (llt+ x v~)\{0} such !.hat the function (1.10) satisfies (1.12). f/ IntXu n {x + Ker L} fo 0, then we ca.nJYU.t .\ = 1 in (1.1.0) and (1.12). Con·ueTsely, ·1j J i8 a conve:r functiona.l and a. function E Xu sa/.isjies (1.8), (1.12) joT some p E V' and ,\ = 1, then is a solution t.o the problem (1.7)-(1.9).

x

x

PROOF. Recall that by the cone generated by a. set l\I ·we mean the set Cone 111 = {m:: :rEM, o > 0}. By the definition of a solution t.o the problem (1.7)-(1.9), foro E (0, 1) we have

x

O~J(x+nv)-J(x) ~ 0:

(J'(:r),v)

\lvE(Xa-x)nKerL,

n-0

where J'(:r) is t.liC GateatL\: derivative of the functional J. Therefore, (1.14)

.J'(x) E (Cone(Xo- x) n Ker L)'.

If IntXa n {X+ Ker L} fo 0, then Cone(Int(Xa- x)) n Eer L fo 0. By Theorem 1.1 and the inclusion (1.14), there exist :r; E (Ker L)j_ and :r; E (Cone(Xa- x)t such that (1.15) f'(x)- "'; = "''· IfintXon{x+KerL} = 121, then Cone(Int(Xa-x))nKerL = 0 and, by Theorem 1.2, there exist nonzero :rj E (Kcr L)j_ and J·; E (Cone( Xi!- x))' such that

(l.lG) Since ImL is closed in V, we have (EerL)j_ = ImL' (cf. [150, Chapter 7, §5]). Therefore, (1.17)

for some J1 E V'. Substituting (1.17) in (1.16), (1.15) and recalling the definition of the conjugate cone, we find

( 1.18)

(>-f'(x)

+ L'p,y)? 0

\lu E Xa-

x,

where~\= 1 in the case (1.15) and~\= 0 in the case (l.Hi). By the definition (1.10) of the Lagrange function C.., the inequality (1.18) is equivalent to the iuequalit:r (1.12), where J: = U + x.

1. THE LACHANCE PH.INCIPLE FOH AN ADSTl1ACT PHOBLE0.-1

(i~)

We prove that the relations (1.8), (1.9) and the inequality (1.12) for A = 1 and some p E V' are sufficient for the vector X to be a solution to the problem (1.7)-(1.9) in the case of a convex functional J. From (1.10) and (1.12) we have 0 ~ (J'(x),x- x)

+ (L'p,1:- x)

= (J'(x),:r- x)

V.r E Xa n {x

+ Ker L}

since :t·- X E Ker L and L'1']J E (Ker L )J... This inequality and the Jensen inequality imply that for any .r E Xa n {X+ Ker £}, G E (0, 1), we have J (X·)

-

,J(") ,t ):

.J(x + o(J:- x))- .J(x) ~. ( .J '("). .. X , ,z;

") ): Q. - J.

u-0

0:

Consequently, xis a solution to the problem (1.7)-(1.9).

D

We consider a special case of the problem (1.7)-(1.9), where Xa = X. From Theorem 1.3 we obtain the following assertion.

COROLLARY 1.1. Let; x be n solation to the problem. (1.7), (1.8), let J he a. fnncf'ional, and let Im L be closed in F. Then ther·e e:risls p E If"' such that. the function (1.10) satisfies the equality Ga.tea.v:.r-d~fj'eren.tiable

(1.19)

(.C,(x, l,p), :r)s

=

0 '11: EX.

Conversely, if J is a conve:e functional and x sat.isfies ( 1.8), (1.19) .for some p E V', then xis a solution to the problem (1.7), (1.8). PROOF. It suffices to show that for Xa =X from (1.18) we obtain (1.19). For this, it suffices to set y = (x ±h)- x in (1.18), where x ± h EXa= X. D 1.3, Xa with empty interior. Condition (1.13), under which Theorem 1.3 has been proved, is too restrictive in many applications to ill-posed controllable systems. Our next goal is to (:.~liminate the condition (1.13) or 1 at least, to weaken it. Let X and _Yo be Banach spaces, _yiJ a convex set in X, and X E )(a. Instead of the condition (1.13), we impose the follmving condition: (1.20)

Intx,((Xa- x) n Xu)

i

0,

\Vhere Int_-'lo J.\f denotes the interior of a set. AI in the topology of the space ..tYo. In applications, the embedding _\0 C ...:\ is usually continuous. For any convex set -Ya containing at least one point different from X, it is easy to construct ..Y0 such that (1.20) holds. Indeed, if a: E X a and x #- X, then for X 0 we can take the line passing through the origin and the point :1:- X. Hmvever, to deduce exact necessary and sui-licient conditions for a minimum in the problem (1.7)--(1.9), it is useful to choose _Yn in such a way that the set (X·a- X) n _.\0 is as large as possible. LEI\IIVIA 1.1. Let ..Yo be a conve:D closed separable (i.e., _Ya contn.·ins a countable dense subset) subset of fl. Banach space X 1 and let, X E _Yn. Then there exists a Banach space -Yn that is con.timwnsly ernbedded in X and dense in Lin(-Yo- X)

su.ch that the condition (1.20) holds. Hereinafter, we use the following notation: Lin(-Ya- X) is the linear span of the oet (Xn- x) and Lin(Xn- x) is the closure of the set Lin(Xu- x) in the topology of the space );_.

70

::!. OPTTi'd:\LIT'l. SYSTEM FOB OPTI!\'1:\L CONTD.OL PBORLEldS

x

PROOF OF LEMMA 1.1. Let Y = Xu and let {C.i} be a countable subset everywhere dense in Y. \Ve put e, = e,/IICIIIs and construct C:J,C:J, •.• by induction: Cm = C~;m-1/IIC~c,_l,h·, where e,(} = e1 and e~...m-1 is the first vector among ek,-~+1, ek"'_ 2 +:J, ... , that lies outside the linear subspace Em-1 spanned by the vectors e1, ... , em-J. The follmving equality is obvious: oc

U Em = Lin(Xa- x).

(1.21)

111=1

For each j there exists O:j yf 0 such that O:jCj E Y. \Ve can assume that supjlo.:JI < cxJ. Since X E ~Ya, we have 0 E Xa -X. Since Y is convex, for an:y j we have aei E Y if sgna = sgnaj and fa[ ~ faJf· Let Pi > 0 be a sequence such that 2::~ 1 Pi = 1. We set

(1.22)

{:r = f.TJCJ: Vj sgnx

Q=

1

= sgnaj, [J:Jf < faJfPJ }·

)=1

\Ve prove that Q C Y. Indeed, let :z: E Q, i.e., x = the conditions in (1.22). \Ve set :r"

= 2::~:= 1 :r.ieJ.

2::.:;: 1 :rJCj,

Let QJ ~ PJ, j

'\Vhere .1;J satisfies

= 1, ... , k,

and let

k

I>j = 1.

(1.23)

J=l

It is obvious that [x;i[/q.i ~ [aJfPJfqJ ~ fa.if· Therefore, (xJfqJ)e;i E Y. In view of the convexity of Y and (1.23), we have k

.Tk =

'\'

~

T

qj'.....l.Cj

j~l

E

Y.

q.i

Since lf.T - .Tk [[ x --; 0 as k --; oo and Y is a closed set, we have x E Y. \Ve set

(1.24)

Xo =

{x = f

xie.i: [[.T[[x., =sup ([.Tii/PJfaJIJ < oo}.

j=l

.I

The space _Yo is a Banach space. Since for :z: =

[[:r[[x

=

= I I;xje 1

II .}=1

•

.\

L: :rJe.i

we have

= [xj[ = ~ LPJ[a;[pfa[ ~ [[x·[[x., LP.ifaJf, ]=l

J

J

)=1

!:he space Xu is continuously embedded in X. From (1.21) it follows that Xo is ever:vwhere dense in Lin(Xa- X). \Ve set

(1.25)

V = { J: E Xu: [[:1: -yuf[x., < 1/2,

The set Vis open in X 0 . Moreover, if :r = I;.TJej E V, then from (1.24), (1.25) for any j we have

p·ct·l < pfa[ -'-'-. I.:r·] 2 2 ....1..__1_

From the last inequality it follows that f:rJf < PJfo:Jf and sgnxj = sgnCY;. Consequcntly, :r E Q in view of (1.22). Since Q C Y, we have V C Y. 0

71

l. THE LAGRANGE PH!NCIPLE FOH AN ABSTHACT PB.ODLEI\I

\Ve continue the study of necessary and sufficient conditions for a minimum in the problem (1.7)-(1.9). We assume that Xu is a Banach space, Xu eX,

(1.26)

and, instead of the condition (1.13), the condition (1.20) holds. Let 1/(1 be a Banach space, and let the follmving conditions hold:

(1.27) £)(0

C

~~h

L: _.{Yo---:.

l~1

is continuous,

L~Yo

is closed in the topology of

\~ 1 .

Consider a functional J that is GateatLx-differentiable along _.{Yn at a point X, i.e., there exists a linear functional .J'(x) : Xu _, IP: such that .J(x + Ah)- J(x)U'(x)h =a(,\), lla(A)II/IAI _, 0 as A_, 0 for any hE Xu and,\ E R Note that x does not necessarily belong to Xu.

x

THEOREM 1.4. Let. be a salu.t.'ion to the problem (1.7)-(1.9). let J be Ga.t.eau.xd'ifferentiable along X 0 at. !.he point and let !.he conditions (1.26), (1.27), and (1.20) hold. Thea there exists a. pair (A,p) E (IP:+ x l'.n\{0} .such that (1.28)

x,

(C:.. (x,A,p),x)s,;,o

\lxE(Xa-x)nXu,

where£. is the La.gra.nge fu.nction (1.10). If

(1.29)

Intx,((Xo- x) n Xu) n Ker L

#

0,

x

then (1.28) holds for ,\ = 1. Conve1·.sely, if a. fur~ction E X .sa.t.isfies (1.28) for some p E 1',) and,\= 1, and al.so (1.8) and (1.9), a convex functional J is Ga.teau..~­ differentiable a.lang Xo at. and

x,

(1.30)

Xu n (Kerx L n (X a - x)) is dense in Kerx L n (Xn - x)

with respect to the topology in which the functional J 'is continuous, then X is a. salut.ion t.o the problem (1.7)-(1.9). In (1.30) we use the not.a.t.ion Kerx L {x EX: L:r = 0}.

PROOF. For y E X 0 we set g(y) = .J(x + y) and consider the problem (1.31)

g(:r) _, inf,

Lx = 0,

.T

E Xo n (Xa- x).

where x IS a solution to the problem (1.7)-(1.9). It is obvious that the vector :c = 0 is a solution to the problem (l.:ll). Moreover, the problem (1.31) satisfies the assmnptions of Theorem 1.3, \vhere for the spaces ~Y and F \Ve take _.{Yo and V(J respectively and for _.{Ya we take the set (Xa- X) n X 0 • Applying Theorem 1.3, we obtain the necessary condition (1.28). By this theorem, from (1.28) for ,\ = 1 and (1.31 2 ). (1.31:1) (i.e., the second and third relations in (1.31)) it follows that :c = 0 is a solution to the problem (1.31). Since g(x) = (x +:c). this means that

(1.32)

.J(x),;; .J(x + J:)

\l:r E Xo n (Kerx L n (Xa-x)).

In view of (1.30) and the continuity of .J, (1.32) implies that xis a solution to the problem (1.7)-(1.9). D RErviARK 1.1. Condition (1.26) is not essential. Theorem 1.4 remains valid if (1.26) is replaced with the assumption that .11 = 0 is a solution to the problem (1.31). Sometimes, in the study of sufficient conditions for a minimum it is useful to replace (1.30) with another condition.

~-

OP'JT:'dALITY SYSTEi\1 FOH OPT!i\IAL CONTHOL PHODLEi\IS

x

LT"IMA 1.2. Let EX satisfy (1.8), (1.9), and the inequality (1.28) for some p E l~i and~\= 1. Assume that. the solu.tion (x,p) /.o the sys/.em. (1.8), (1.9), (1.28) for..-\ = 1 'is nnique in the space X X 1~t. rYe also USS'/l.TTIC that a soln!Jon f,o the problem (1.7)-(1.9) exists and satisfies (1.28) for som.e )Jt E I~) and,\ = 1. Then xis a solution /.o the problem (1.7)-·(1.9). PROOF. Let .r 1 be a solution to the problem (1.7)-(1.9). By assumption, there exists PI E ~~ such that (a: 1,p 1 ) is a solution to the system (1.8), (1.9), (1.28) for ..-\ = 1. Since a solution to this system is unique, we have .r 1 = X and p 1 = p. Consequently, xis a solution to the problem (1.7)-(1.9). D

1.4. Well-posed controllable system. A special feature of Theorems 1.3 and 1.4 is a rat.her \Veak condition on the operator L, which is eompensated by conditions on _)\.rh namely, the condition (1.13) in Theorem 1.3 and the conditions (1.20) and (1.27) in Theorem 1.4. Theorems 1.3 and 1.4 are convenient in the study of optimal control problems for ill-posed distributed parameter systems. If the controllable system is described b,y a well-posed bounclar:r value problem, it is more convenient to use other theorems justifying the Lagrange principle. In these theorems,~\= Y xU, where Y is the space of states and U is the space of controls. The set of constraints Un is a subset of U, and, in fact, no conditions are imposed on Ua. This is compensated by more rigid conditions on the equations describing the controllable system. Nevertheless, they can be nonlinear with respect to y. Let Y, U, and If be Banach spaces, J : Y x U ~ JRi. =ill: U { oo} a fundiona,!, J..' : Y x U ---+ V a mapping, and Uu a convex subset of the space U containing more than one point. 'Ve consider the extremal problem

J(y, u)

(1.33) (1.34)

F(y, u.)

~

= 0,

inf, u E Uo.

As above, the set 2l of admissible pairs is defined by the relation 2l = {(y,u.) E Y x Ua: F(y, u) = O,J(y, u) < oo}, and a solution (fj, ii) E Ql to the problem (1.33), (1.34) is determined by the relation

J(y, ii) =

inf

.J(y, u).

(y,u)E'2l

The Lagrange function of the problem (1.33), (1.34) is defined by the equality L(y, u, ~\.p) = U(y, u)

( 1.35) where /\ E JR.+, p E

+ (F(y, u),p),

F'~.

THEOREM 1.5 (Iofle and Tikhomirov [89]). Let (y, ii) E Y x U be a. solution to the problem (1.33), (1.34) for any u E Uo, let the mappings y ~ J(y, u) and y ~ F(y, u) be continuously differentiable for y E O(fj), -whe·re O(y) is some n.eighborhood of the point fj, and let Im P~Cfi, U) be closed and !za.'l!e finite codimension in \f. In addition, for y E O(!i), let the function u ~ J(y, ·u) be conve:z:, the Junctional J Ga.tea'U:J;-dijJerentiable with resped to u at the point. Cfj, U), and the mapping u---+ F(y, u) continuous from U lo V and affine. i.e.,

(1.36) F(y, ou 1

+ (l- o)u 0 )

=

aF(y, u 1 )

+ (1- n)F(y, u 0 )

'fu 1 , ·u.o E U, o E llt

1. THE Lt\GHANCE PHINCIPLE FOB AN AUSTIL\CT F'flOBLEi\1

Then there e:rists a pair (,\,p) E (!8:+ x V')\{0} such that (L~(fj, ii,

(1.37)

A,p), h)= 0

for each h E Y and

(L;,(!i, u, A,p), a)~ 0

(1.38) JflmF~(!j,

VuE Uu-

u.

u) = V, then we can assu.rne that,\= 1 'in (1.37), (L:JS).

RE:rvlARK 1.2. By (1.35), the Gateaux derivative of c~l exists (which is assumed provided that the operator u __, F(y, u.) is differentiable at the point u.

in (1.38)) However, for any y F;,(y, u)h

tram (1.36) it follows that the operator u __, (F(y, u.)- F(y, 0)) is linear E O(fj). By continuity, the mapping u --;. F(y, u) is differentiable and = F(y, h)- F(y, 0).

RErviARK 1.3. Let the functional J in (1.33) be convex ·with respect toy and n 1 and let the operator Fin (1.34) be jointly affine, i.e., F(y, u) = L(y, 11.) + F 0 , where L : Y x U ----+ F is a linear continuous operator. Then the necessary conditions for a minimum in the problem (1.33), (1.34) are also sufficient. lvlore precisely, let a pair (y,fi) E Y x U satisj)· (1.34), (1.37), and (1.38) for ,\ = 1. Then (fj,ii) is a solution to the problem (1.33), (1.34). To prove this assertion, it suffices to set (y,u) = .T, J(y,u) = J(:r), L(y,u) = L:c and rcpmt the corresponding part of the proof of Theorem 1.3 (cf. the end of the proof).

\Ve \Vill shmv that we can apply Theorem L5 to deduce optimality systems for a large class of optimal control problems for distributed parameter systems. However, examples of singular distributed parameter systems are known for which it is more convenient to use not Theorem 1.5 itself~ but a certain generalization. \Ve formulate the corresponding result. Let (fj, U) be a solution to the problem (1.33), (1.3,1). In addition to the spaces Y, U, and 1f in which the problem is stated and the solution is defined, we introduce Banach spaces 1'(11 U0 , and 1'tJ. \\.Te set

g(y, u.) = J(fj + y, u + u)- J(fj, u),

(1.39)

G(y, u.) = F(!i + y, u + u)- F(y, u).

\Ve assume that the follmving condition holds. 1.1. (a) The mappings g(y, u) : Y(1 x U0

__,

Yo introduced in (1.39) are defined for (y, u) E

1~1

CONDITION

l'i1 x U0

__,

iR: and G(y, u) : x Uo, and the set

(Ua - U) n U0 contains more than one point. (b) For each u. E (Uo- u) n U0 the mappings y __, g(y, a) andy__, G(y, u) are continuously differentiable for y E On, where Oo is a neighborhood of zero in the space Y(J. \Ve consider the extremal problem (1.40)

g(y, u) __, inf,

G(y, u)

= 0,

a E (Uu- u)

n Uo

and formulate a theorem on an optimality system for the solution (fj, U) to the problem (1.3:J), (1.34). THEOREM 1.6. Let. (y, a) = (0, 0) E Yi1 x U11 be n solution to the p-roblem (1.40), and let Condition 1.1 hold. Suppose that Im 1 (0, 0) is closed and has finite codimens·ian in 1~J· Afm·eover1 assnmc that u --;. g(y, u) is conve:r a.nd GateauxdijJerentiable on U11 at the point (y, u) = (0, 0). whereas the mapping u __, G(y, u) 'is contin'lto'IJ.S as a mapping from. Uo to V(; and is affine for a.ny y E 0 0 _, where On

c:

2. OPTII\IAL!T'{ S'.{STr..-::1\-1 FOI!. OPTlA!AL CONT!lOL PHODLEMS

7-1

is a neighborhood of zero in 1(1• Then there e:cists a po.i:r (A,p) E (R+ x V[;)\{0} 8Uch that the relation (1.37) holds for h E Y(, and the relation (1.38) hold8 for u E (Ua- u) nUn, where L is the Lagrange fa.nclion (1.35). rf ImG;,(o, 0) = Vo, then we ca.n 08S"Ume that.,\= 1 in (1.37), (1.38).

PROOF. Br the assumptions of the theorem, the solution (y, a) = (0, 0) to the problem (1.40) satisfies the assumptions of Theorem 1.5, where for Y, U, 17, Un, /, F, and (!i, u) we talw 1·(, Un, \"'' (Un- u) n Uo, g, G, and (0, 0) respectively. 0 The Lagrange function L of the problem (1.40) has the form

(1.41)

l.(y, u, ~\,p) = ,\g(y, u)

+ (G(y, u),p)l;,

,\:;, 0, p E V(1'.

Br Theorem 1.5, there exists a pair (A,p) E (R+ x l~/)\{0} such that

(1A2) for u E (Ua-u)nU0 . From (1.35), (1.41), and (1.39) it follows that (1.42) coincides with (1.37), where h E 1"[, and with (1.:38), where -u = u E (Ua- u) nUn. REMARK 1.4. Let the functional J in (1.33) be convex and let the operator F in (1.34) have the form F(y, o) = L(y, u) + F 0 , where L : Y x U ~ V is a linear continuous operator. If 1'{"! is dense in Y, V{1 is dense in F, and the operator 1(, ~ \"' is an epimorphism, then the relations (1.34), (1.37), and (1.38) for .\ = 1 are sufficient for the pair (y, u) to be a solution to the problem (1.33), (1.34). As in Remark 1.3, it suffices to repeat the end of the proof of Theorem 1.3.

L;, :

1.5. Extremal problem without inclusion-type constraints. To conclude the section, \Ve give simple necessary conditions for an extremum in a nonlinear extremal problem \vit.hout constraints of type :r E ..-Ya. Let ~Y and 1/ be Banach spaces, and let a functional J : )( . . . . ;. 1Ft and a mapping F : "'\ -o- V be continuously cliiierentiable. Consider the extremal problem

J(o:)

(1.43) (1.44)

~in[,

F(:r) = 0.

Let X be a solution to this problem. As before, we introduce the Lagrange function of this problem by the formula

( 1.45)

£(.1:, A,p) = M(:r)

+ (F(.1:),p).

x

THEOREM 1.7. Let be a solution to the problem (1.43), (1.44), and let the range of the opemJor F' (X) : "'Y. . . . . ;. V be closed. Then there c:.rists a. nonzero pair (A,p) E iPi x V" such that (1.46)

(£~,('1', ~\,

p), :r)

If, in addition, ImF'(x) = V, then,\

= 0

V:r EX.

io 0 in (1.4fi)

(and we can as.mme that>\= 1).

Proof of Theorem 1.7 can be found in [89] and [6]. \Ve. present a generalization of Theorem 1.7. Let X EX be a solution to the

problem (1.43), (1.44), and let X 0 • \1(1 be Banach spaces. As in the case (1.39), we .set

(1.47)

y(x) = J(x+:r)- J(x),

G(:r) = F(x+:r) -F(x).

Assume that the following condition holds.

1. TilE LAGHANGE PlllNCIPLE FOB AN ADSTilACT PJ108LE!\'1

{.')

"coNDITION 1.2. (a) The mappings g : X 0 -;. JR. and G : .X0 -;. 1-~ 1 introduced in ( 1.47) are defined for all .T E ..Yo and are continuously difFerentiable in a neighborhood of :r = 0. (b) The range of the mapping G'(O): ..Yo-;. 1~J is closed.

x

THEOREM 1.8. Let be a .solution to the problem. (1.43). (1.'14), and let the nwppings (1.47) satisfy Condition 1.2. Then there e:r'ists a nonzero pair ( ~\, p) E lP!. x 1't/' such that the La.gmnge function (1.45) satisfies the equality

{L;,.(x,A,p),x) = 0

(1.48)

\hE Xu.

If, in addition, the range of the operator G'(O) : Xn ~ l'u coincides with 1'r,, then in (1.48) is nonzero (hence we can assume that~\= 1).

~\

PROOF.

Consider the extremal problem g(a:)

(1A9)

~

inf,

G(:r) = 0,

where g and G are defined in (1.47). Since xis a solution to the problem (1.43), (l.H), 0 is a solution to the problem (1.49). It is obvious that the problem (1.49) satisfies the assumptions of Theorem 1.7 with X = 0, X = ~Yo, V = ViJ, J = gl F = G. By this theorem, there exists a nonzero pair (~\p) E JR. x \~j' such that the Lagrange function M(:~:, ,\, p) = Ag(:~:) + (G(:r ), p) satisfies the condition (1\f~(O,A,p),.T) = 0 for all :c E X 0 . Since AI;(o,A,p) = [/,(x,,\,p), where£ is D defined in (1.45). we have (1.48). \Ve consider the following abstract extremal problem. Let ~Y 1 and ..Y·2 be Banach spaces, f : _yl -;. lP!. and .9.i : X1 -;. 1P~ fuuctionalsl and F : ~Yt -;. ~Y2 a mapping. \Ve look for w E )\. 1 such that

f(w) =

( 1.50)

inf

f(u),

ttE1Vurl

\vhere VVad = {u E ~Yt: F(u) = 0, gj(u) ~ 0, j = 1, ... ,m}. The Lagrange functional of the extremal problem (1.50) is defined by the formula £(w, .\, q) = ,\of(w)

"'

+ (F(w), q) + L

,\JgJ(w)

-i=l

for all 'WE X1, A= (~\n,~\J, ... ,,\ 111 ) E :iftm+l and q EX~', where X~ is the dual space to Xo. The following Lagrange principle holds (d. [6]). 1.9. Let w be a. solution to the problern (1.50), let f, 9), and F be let the mn.ge of the operator P'(w) : X1 -;. ..-\2 be closed. Then theTe exist q E x;· a.n.d A = (~\o, A1, ... , Am) E JR.rn+J such that (q, .\) io (0, 0); mo'reover, the pair (q, .\) satisfies the relations THEOREM

conLirl'lW'lLsly diJferentia.ble, and

(£;,.(w, .\, q), h)= 0

(1.51) (1.52)

-\;;,0,

j=O,l, ... ,m,

If hE X 1 ,

AJ9J(w)=O,

j=1, ... ,m,

wheTe c:L'(·, ·, ·) denotes the FrechCt deTivaiive of the Junctional [ with Te5pect to the .first argument. 1\Ioreovcr, if F'(w) : .X1 -;. X:.! is an epimorph-ism and the problem (1.50) does not con.ta·in. the cons{ral.n.ts gi(w) ~ 0, f,hen ~\n f=. 0 and we can set ,\ 0 = 1.

7G

2. OPTll\lALTTY SYST8i\l FOB OPTD.IAL CONTROL PHODLEAIS

2. Linear regular stationary problems This section deals with necessary and sufficient conditions for solutions to extremal problems ·whose stat.e function is defined by a linear stationary boundary value problem (such a collection o[ necessary and sufficient conditions is called an optimality system.). In many examples: this bounda.r_y value problem is "\veil posed. However: we also consider the case of underdetennined ill-posed boundary value problems. In all the examples: the range of the operator determining the boundary value problem has finite codimension. Therefore 1 in §2 we use Theorem 1.5 to derive an optimality system.

2.1. Problems with distributed controls. \:Ve begin with an optimal control problem for s;ystems described by well-posed problems. Consider the problem ( 1.3.1 )-( 1.3.3 ): 1 211 Y- w II''i.,(.:' = 0

\lz E H 2 (fl).

By Theorem 1.1.4, the mapping (-y, -(8,) : H'(fl) __, H"I"(Bfl) x H'I"(Bfl) is an epimorphism. Hence (2.21) implies the equalities (2.22)

-yp(x')

= 0,

'ffJ,p(:c')

= 0,

x' E fJfl.

From (2.10) and (2.22) we obtain (2.19). Moreover, from (2.10) and (2.22) it follows that p E JJ'2(f!.). The assertion about sufficient conditions for a minimum is valid D in vimv of Remark 1.3.

:2. LINEA\l U.EGCLAH. STATIONAn.Y PHOIJLEi\lS

7U

2.2. Regular problems with boundary controls. Consider the problem

1

2 1 Y- w II''T.2(rn + N2 I u-.I'll''L~(nn)

(2.23) (2.24)

!;.y(.T)

= g(.T),

(2.25)

;z;

E fl,

'(Un!J

.

--+ m f' ,

=

u,

u E Ua,

where wE L,(fl), f E L,(fJfl), g E L,(fl), N > 0, Ua is a convex closed subset of L~(O); moreover, Uo n L 9 =!=- 0, ·where L 9 is the set (1.3.13'). By Theorem 1.3.5, 2 the problem (2.23)-(2.25) has a unique solution (fj, u) E H},! (!1) x L 2 (fJfl). 2

2.4. A pair (fj, u) E Hjf (fl) x L,(fJfl) is a solution to the pmblem (2.23)-(2.25) if and only if there exists p E H 2 (!1) such that the triple (y, p) E Hjf2 (!1) X L,(fJfl) x H2 (!1) satisfies, in addition to (2.24) and (2.25), the ·relations THEOREM

(2.26) (2.27)

u,

!;.p(x)

+ (17(.1:) + w(x)) = 0,

:c E !1,

'lnP

=

( (JV(li(.T')- f(:z:')) + 'fp(:r'))(u(x')- u(:r')) ch'

ion

PROOF.

0,

~0

lc!u E Ua.

We set

(2.28)

U)i = Uu n {u E L,(fJfl): ;· uds = ;· . un

. rm

ud,,}

and deduce necessary conditions for an extremum in the problem (2.23L (2.24) \vith inclusion

u E u)i.

(2.29) We set Y = H[,·~r- (!1), U 1.1.9, the problem

= L,(i3!1), and F(y, u) = (!;.y- g, '(U,y- u). By Theorem

(2.30) has a solution o E H[,'l/''-(!1) if and only if h 1 E £ 2 (!1), h 2 E L 2 (i3!1), and the following relation holds: ( h 1d:c -

./r1

(

./on

hods = 0 .

We set (2.31)

V

=

{(h 1 , h 2 ) E L,(fl) x L 2 (i3!1): ( h 1 dx- (

.In

Jon

h 2 ds

=

o}.

It is obvious that F~Y = V and, by (2.28), we have F(Y, U)i) C V. Consequently, F(Y, U8) = Y. Thus, the assumptions of Theorem 1.5 hold. To use this theorem, we need to describe the space V'. Let us identify the space L, (!1) x £ 2 (i3!1) with its duaL Then the dual space V' to (2.31) consists of pairs ('rh ,Tj2 ) = { (~ 1 + ,\, ''!' + ,\), ,\ E IR}, where T/r E Lo(fl), T)o E Lo(ufl), and the value of the functional (i/r, if,) at

::!. OPTE\TALITY S'Y.STEi\f FOB. OPTil\IAL CONTHOL PRODLEl\.·IS

(hr, ho) E Vis as follows:

((rii, Tfo), (hr, ho))l=

(2.32)

= =

r(qr(x) + ,\)hrdx- .!a~1.(ryo(:r') +,\)hods

./n

~ h ds+,\( ( h 1 d,;- (.h 2 ds) . rm 2 2 ln. luu

(qrh,cLr-1

ln.

r 'll!h,rh- .fan ( 'l)oho ds.

ln

The Lagrange function of the problem (2.23), (2.24), (2.29) has the Jorrn - ij) = L(y, u,p,

(2.33)

1

;zlly- w II"'"i,(r>J + Nil 2 u- j "II"'Lc(i!n) + ((p, ij), (t;.y- g, '(fJ,y- u))F,

where (jj, ij) = {(p + ,\, q +A), A E lB.} E V', p E Lo(D), q E Lo(8!1). From (2.33), (2.32), and (1.37) it follows that (2.34)

CiJ- w, z)L,(i11 + (p,6.z)L.,(!lJ

= (q,'(fJ,z)rdiill· vd.T = JnrHL'>ud:r =

ln

0

\fu E HR(Il)

in view of (2.11). Consequently, H C (L'>Hij(ll))l.. On the other hand, if v E (L'>H1}(0))l., then L'>u = 0 in view of (3.8), i.e., v E H. Let Q: L 0 (0) ~ L'>Hrl be the orthogonal projection on L'>Hri, and let (3.9)

" E

ui)

=QUu.

It is obvious that the solution (ij, u) to the problem (3.1)-(3.3) is also a solution to the problem (3.1), (3.2), (3.9). \Ve begin by deducing necessary conditions for an extremum in the problem (3.1), (3.2), (3.9). We set Y = Hri(ll), U = V = L'>Hif, and F(y, u) = L'>y -11. It is obvious that t.he operator }~ = 6 : Y ___,. V is an isomorphism. Consequently, all the assumptions of Theorem 1.5 hold. The Lagrange function C has the form

(3.10)

1

,, L.(y,u,p)=211.1J-wlli~c(n)+

Nl lu-.f II''l.,(!l)+(L'>y-u,p):;rr,;, 2

where p E (L'>H,jt, (·, ·) Mic is the duality between L'>H,j and the dual space. By (3.7), we can identify (L'>HrlJ" with Lo(fl)/H. Thenp=pn+H, wherepn E L'>H;}, and for any v E 6f!cf we have

(3.11) From (1.37), (3.10), and (3.11) we obtain (3.4) with p = Pu E L'>Hif. By (1.38), (3.10), and (3.11), we have

(3.12)

(N(u- .f)- Po, u- u)L,(U) :;, 0

\111 E

ug.

Let us deduce necessary conditions for an extremum in the problem (3.1)(3.3). Take p = flu + }JJ, where p 11 coincides with the function ]Jo in (3.12) and Pt E .His such that 1V.f + p 1 E 611a. Such a choice of p 1 is possible in view of the decomposition (3.7). Since L'>pu = w- y and L'>p1 = 0, for p =Po+ p, the equality (3.4) holds. By (3.7) and (3.9), lor u E Uu there exist un E ug and 111 E H such that u = Uo + 'Ut. Taking into account the inclusions U,-uo,Pn,iVf + Pl E 6Hff, UJ,]h E Hand the relation 6H(~ .l H, we find that for u E Uo we have

(N(u- f)- p, I I (3.13)

u)L,(I1)

= (N(u- .f)-pu, Uo-1iJL,(n)- (p,, uu- il)L,(n)

+ (N(u- f)- Po- p,,ui)L,(n) =

(N(u- f)- pu, uu- fi)L,(''i ?

o,

where the last inequality holds in view of (3.12). By (3.13), we have (3.5). The sufficiency of the conditions for a minimum follows from Remark 1.3. 0

3.2. Optimization of the Cauchy data. Statement of the problem. Let 0 C IRd be a connected bounded domain with boundary 80 E consisting of two closed disjoint. manifolds fo and ft such that fo u rl = ao, fo n rl = 0, dimrJ = d- 1, J = o, 1.

c·x

:3. Of'Til\"l!ZATION FOH Tl!E CAUCHY PROBLE!\1 FOil THE LAPLACE OPJ::Il:\TO!l

SS

In the domain 0, we consider the optimal control problem for a srstem described b:y the Cauchy problem for the Laplace operator:

J(y, u, v)

(3,14)

1 = ;zii'IIU-

wl I''fdC,J

N1

·)

No + 2lluf 'II''L,(C,.) .

+ 2llv- 9llc,(r,.) __, mf, (3.15)

6y(:r) = 0,

"loV =H.,

')'oDnY = v,

u E Ua,

(3,16)

where x = (:c 1, ... ,:z.·d) E 0 and '"'/j 1 j = 0, 1, is the operator of restriction of functions defined on 0 to r j' all is the derivative along the outward normal to the boundary of 0, 1Vo > 0, 1V1 > 0 are given numbers, w E L:2(f!), f E L::!(fn), g E L::!(f 0 ) are given functions, Un C £2(fo), Vu C L::!(fo) are convex closed sets. Assume that there exists a triple (y, u, v) E H}j"(rl) x L"(fn) x L"(f 0 ), satisfying (3.15) and (3.16). By Theorem L3.8, there exists a unique solution CiJ, u, v) E lj'l He, -(rl) x H(f 11 ) x L"(fo) to the problem (3.H)-(3.1G). We clecluce an optimality system for the problem (3. 14)-(3. 1G) only for two examples of the sets Ua and 11/J. ExAMPLE

3.L We set

(3,17) where

Lj(fo) = {u E L"(fn): u(:r);;, 0 a.e. on fn}, L 0 (fu) = {u E L"(fo) : u(:r) ~ 0 a.e. on fo}. EXAMPLE 3.2. Let Oj E C(f 11 ), {3J E C(r 11 ), j = 1, 2. Assume that there exists 17 > 0 such that Oj(:z:) < flJ(:r)- ry, :r E fo, j = 1, 2. We set Ua = {u E Lx(f 11 ): crt(:r) ~ u(:r) ~ ;J 1(.T) for almost all :r E r 0}, (3.18) \fc1 = {u. E Lx(ru): a"(:r) ~ u(:r) ~ (32 (:r) for almost all :z: E r 11 }.

To deduce an optimality system, in addition Lo the Sobolev spaces H'J(O), the Bcsov spaces B(3(f 0 ), and the spaces of the form

(3.19)

Wjc,(rl) = {y E W!i(rl): !:>y E La(rl)}

defined in Chapter 1, 31, \Ve need the following spaces:

Y

(3.20) (3.21)

=

{y E H}j"(rl): 'loiJnu E L,(ro)},

Pa = {p E \·V~.c,(rl): loPE (C(ro))', 'loiJ,.p E (C(ro)n.

The norm in the space PJ is defined by the equality

IIPIIv, = IIPIIL,,(n)

+ IIC>PIIL,,(n) + lhoPII(c(r,))· + lhoiJnPII(qr,))'>

\vhere ;3 > 1 and

llqll(c(r,l)' =

sup

l(q,rp)(r,,JI·

II·:Piic(ro)=l

Denote by (q_. r..p)(.'1) the inner product in £:.2(8) and the corresponding duality as well. It is obvious that Y is a Hilbert space and P,.3 is a. Banach space.

:::!. OPTJl\·L\LITY S'lSTEl\! FOB. OPTD.!AL C'ONTBOL PHODLE?dS

3.3. Optimality system. Example 3.1. We deduce an optimality system of the problem (3.H)-(3.1G) in the case of sets Uu and v,, of the form (3.17). THEOREM :l.2. A t1·iple C!f.u,v) is a solulion to the problem. (3.14)-(3.17) if and only if there e.r,istspE ?,3 ,;3< dj(d-1), such that (y,v,v,p) E Y xH 1 (f 0 ) x L,(fo) x Pa satisfies (:3.15). (3.16). and the following conditions: (3.22) (3.23) (3.24)

D.p(:r)

= 0, :r E 0.,

'!1Y -,,EJ,p

= w,

'Ill'= 0,

(Nn(li- f)- 'foG,p, un)(r") ;;-, 0 \iva E H'1.6)

Jj(/,:r)+b.p(t,:r)=y(t,:r)-w(t,.r), (t,J;)EQ;

JQ{ (N(U:(t, J;) -

'f~p=O,

·up(:r)=O,

f(t, :r)) - p(l, :r) )( u(t, :r) - u(t, .r)) d:r dl. ;, 0

Vu E UiJ.

PROOF. For Y we take the space (4.4). We use Theorem 1.5. Let. U = L2(Q), V = L,(O, T; IJ- 1 (S1)) x L,(S1), and F(y, n) = ([I- b.y- u, "lo!l- y 0 ). By Theorem

1.1.16, the operator F~ = L~1 : Y-;. Vis an isomorphism bet\veen the spaces Y and 11. Hence the assumptions of Theorem 1.5 hold. The 'Lagrange function of the problem (4.1), (:1.2) has the form

(4.7)

L(y, u, p, q) =

1 ., Nil u. - .fllrdQJ ., ;ziiYw[];,.,(QI + 2

+ (!i- b.y- u,p)L,(CJI +hoY- vu,q)L,(nJ, where (p,q) E V' = L,(O,T;HMS1)) x L,(S1).

:2. 01-'TL\!ALlTY SYSTE?\f FOil OPTII\IAL CONTHOL l-'HOBLE?\IS

!Jfl

By (4.7) and (1.37), we have (4.8)

(y- w, =h,(Q)

+ (i- tl.o,p)c,(Q) + bu=, qJL,(n)

= 0

't/o E Y.

Hence \Ve obtain the equality (4.9)

]i

+ D.p =

Y-

W1

which is understood in the sense of distributions. Ffom (4.9) it follows that p E

L,(O, T; H- 1 (1?,)). Integrating by parts in (4.8) and taking into account (4.9), we obtain the equalities "rrP = 0 and (4.10)

/nP = q.

H,: (

Since {c:;p = 0 for p E L, (0, T; ll)), the relations (4 ..5) hold. From (4. 7) and (1.38) we obtain (4.G). Let (!j,u,p) E Y x L 2 (Q) x Y satisfy (4.2), (4 ..5), and (4.6). We define q by formula (4.10). Then (4.10) and (4.5) imply (4.8), whereas Ii·mn (4.6)-(4.8) and Remark 1.3 it follows that CiJ, u) is a solution to the problem (4.1), (4.2). D

4.2. Initial control. Final observation. Consider the problem (4.11) (4.12)

1

•")

JV

·1

;ziiTrY- wll£,( on(m·), then,\> 0 rrrul we can assume that A= l. If,\= 1 and (Y, u,p) E Y x Lo(Q) x Lo(Q) satisfy (4.30), (4.31), (•1.33), then (y, u) is a solution to the problem (•1.29)-(4.31).

PROOF. We use Theorem 1.3. Let X= Y xU, where Y is the space (4.32), L(y, u) = (y- D.y- u, "IT !f), Fi1 = (0, -ur),

U = L,(Q), ]I= L2(Q) x H 1\(rl), LT (4.34)

Uo =

{o E

L,(Q):

=

llniiL,(q) ,:;; a},

Xa

= Y x Ua.

It is obvious that Tntx ~Yo i=- 0. To prove that LX = Y: we show that for any g E L 2 (Q), hE H 1j(rl) there exists a pair (y, u) E YxL 2 (Q) such that [J-D.y-u = g and "trY = h. It suffices to take y E Y such that ''lT!J = h (such functions \Vere constructed in the proof or Proposition 1.5.2) and put 'U = iJ - D.u -g. Thus: \Ve have verified the assumptions of Theorem 1.3. The Lagrange function of the problem (4.29)-(4.31) has the form (4.35)

AII ull r.,(q) ., L(y, u, A, P) = 2

+ ([I -

L1y- u, JJ)(QI

+ (/rY -

where,\ ;;:, 0, p E L 2 ( Q), q E H- 1 (rl). By (1.12), Lu = 0 end (£." (y, 0 1 u E Un. The first equality and (4.35) lead to the relation (4.36)

A(Y, z)(q) -1- (i- L1z,p)(Q) -1- (Trz, q)(n) = 0

y.,-, q)(n),

u, p, A), u-u) ;;:,

If: E Y,

which implies (4.33 1 ). By Lemmas 1.4.1 and 4.1, this equality and the inclusion p E L:2(Q) guarantee the existence of the restrictions ''/oP: TrP: 1-::.P: -r:::.BnP of p and the validity of the Green formula (4.2fi). From (4.36), ('1.33 1 ), and (4.26) we obtain the equalities (4.332) and '~''Jl = -q. From (•1.35) and (1.12) we obtain the inequality (4.37)

-(p,u-u)(QI;;:, 0

VuE

Uo,

where Uu is the set from (4.34). Since Un is a ball, (4.37) implies either (4.33") for Ar ;;:, 0 or p 0. But the last equality is also realized by ('1.33:J) for At = 0. If _,.\ 1 = 0, then p := 0 in view of (4.33:.1) and, consequently, .\ -f=. 0. Hence ]J 0 and (4.33 1) imply y(t,.r) 0. Therefore, YT = ·rr!l = 0. Let a > n 11 (y·r), let Yo E Y be a solution to the problem (1.5.16), and let un = iJo- D.yo. It is obvious that IIHuiiL~(Q) = a·o· Therefore: llo E Int Uu, where U0 is the set in (4.34). It is clear that (y0 , u 0 ) E (Y x Jut U) n (Ker L + (Y, 11)). By Theorem 1.3 1 we have ,\ > 0. The sufficiency of the conditions follows from Theorem L'l. D

=

=

Consider the problem (4.38)

=

:!. OPT[l\lALlTY SYSTE~I FOR OPTll\L-\L CON'l'HOL PR08LEJ\:IS

\J·l

with the condition (4.30). By Theorem 1.5.4, there exists a solution (fj, u) E H 1 (Q) x L 2 (0,T;H~ 1 (ll)) to the problem (4.38), (4.30). •1.5. A problem (4.38), (4.30) THEOREM

pai1·

(!i,u) E H 1 (Q) x L 2 (0,T;H~ 1 (ll)) is a. solv.tion to the

zf and only if -jj- t!.y + y = 0, ul~= 0,

(4.39) (4.40)

'faY= 0,

/TY

= Yr,

u=[J-t!.y.

We use Theorem 1.3. Set X= Y xU, where Y = {y E H 1 (Q) : 'fo;Y = o,,11 y = 0}, U = L 2 (0,T;H~ 1 (ll)), V = L 2 (0,T;H~ 1 (ll)) x Lc(ll), X 0 =X, Lx L(y,11) = (il- t!.y- u,11'Y), F11 = (0,-yT)· Since L : X ___, Vis an epimorphism, the assumptions of Theorem 1.3 are satisfied. The Lagrange function has the form PROOF.

=

£(y, u, p) =

~ IIYII~J' (QJ + (if- t!.y- u, P)(q) + (i,y -

YT, q)(n),

where p E L 2 (0, T; HM>l)) and q E L2(ll). Since X a = X, the inequality (1.12) turns into equalities which 1 in the case under consideration 1 have the form

(v, p)(q) (y,z)H'(QJ

0 \lv E U,

=

+ (£- t!.z,p)(q) + ('!Tz,q)(!l)

= 0

\lz E Y.

These relations and (4.30) imply (4.39). The sufficiency of the conditions ('1.39) D and (4.40) follows f\'Oln Theorem 1.3.

5. Problem for the backward heat equation. Distributed control In this sedion 1 we deduce an optimality system of an optimal control problem in which a controllable system is described b:y an ill-posed boundary value problem for the baclnvard heat equations and the control is distributed. \Ve use the Lagrange principle (Theorem 1.4) \vhich leads to some integral relations. The main efforts are spent to deduce the adjoint boundary value problem from these relations. 5.1. Case IntL-~ Uo f:. 0. \Ve consider the following optimal control problem which looks like the problem (4.1), (4.2) but. contains the backward heat equation:

1

(5.2)

'l

2IIY- wlli,(QJ +

(5.1)

y(t, .1:) + t!.y(t, :r)

JV ') . llu- Jllr,,(QI ___, mf, 2

= u(t, :r);

/o;Y

= 0, 111Y = !/o, n E Uu,

where w E L 2 (Q). f E L 2 (Q). N > 0, y 0 E L 2(ll), and Uu is a nonempt.y closed convex subset of the space L-2 ( Q). Let (5.3)

Z,

=

{y(t, .1:) E L-,(Q) :

if+ t!.y E L,(Q), />::!/ = 0}.

As \vas proved in Theorem IA.l 1 if the nontriviality condition is satisfied, then there exists a unique solution (fi, x) E Z 1 x L,( Q) to the problem (5.1), (5.2). where is the space (5.3). Deduce an optimality system ofthe problem (5.1), (5.2). The complexity of this problem depends on the properties of the set of constraints Ua. \Ve begin with a rather simple case where

z,

(·5A)

lntL,(QJ Ua j 0.

."i. BACKWAHD HEAT EQCATJOI\". DISTB.IEIUTED CONTHOL

()5

THEOREM 5.1. Let. /.he condition (5.4) hold and y0 E H,\(!1). A pa.i·r (y,u) E x L2(Q) is a. solution to /.he problem (5.1), (5.2) if and only if there e1:is/.s p c L2(Q) s·uch that the triple (TJ,u,p) satisfies the condition (5.2) and the relations

z,

(5.5)

p(t,x)- /',p(t,.T) = y(t,:c)- w(t,.1;),

(5.6)

(N(u- f) - p, u-

'YIP= 0,'f>:P = 0,

u)IQI ) 0, Set X = z, X

\fu E Ua.

PROOF. We use Theorem 1.3. L"(Q), where zl is the space (5.3), V = L2(Q) :< H,i(fl), Xu= Z, x Ua, L.c- L(y, u) = (.i; + !:;.y- u, /o.Y). We show that the operator L : .X __,. V is continuous. It suffices to prove the continuity of the trace operator 'Yu : zl ~ HI\ (!1). Let y E zl· Then

=

(5.7)

tu

Since y E L 2 (Q), there exists 1 (5.8)

E (T/2,T) such that ?

llu(tu, ·liiLI"I ~ -r~

iT llv(t., ·lll'dt

, T/'.!.

?

~ -r~ llulli,IQI

(the first inequality in (5.8) is easily proved by contradiction). The energy estimate for solutions to parabolic bouncla.r_y value problems gives (5.9) As in the case (5.8), we establish the existence of t 1 E (T/4, T/2) such that

(5.10)

llv(t,, ·)ll~''i"l ~ 0

d

-T·

iT/' llv(T, )117;• n1dT

, Tj-1

u

1

4 ~-1,

~ -T

• 0

IIY(T, ·)lli,, 1" 1dT. °

Using the estimate (1.1.38) for solutions to parabolic problems 1 we have (5.11)

Using Theorem 1.1.13, and the estimates (5.11), (5.10), (5.9), (5.8), we get

(5.12)

lhnYIIi;,\1"1 ~ Cllvlli,,.,IIO.r,JxnJ ~ · · · ~ CI(IIgiiLu!l + IIYIILu;nl = C'! llulll, · For any (g, Uo) E V we choose a function y E JJ1.'2(Q) such that "loY = !Jn, = 0 (this is possible in view of Theorem 1.1.13), and set u = [J- /',y- g. Then L(y, u) = (g, .Yn). Therefore, LX= Y. We show that

T>:Y

(5.13)

(Z 1 x lnt Uo) n {(y, u)

+ Ker L}

c:;f 0.

Indeed, by Lemma l.L1.2, there exists a pair (z, v) E Zt x Int(Ua- U) satisfying the relations::+ ~z = v, 7~.::: = 0, and "(nZ = 0. It is obvious that (.:::, u) E Ker L. By the construction of v, we have Cif, U) + (.:::, v) E zl X Int Ua. This proves (5.13). Thus, the assumptions of Theorem 1.3 hold. The Lagrange function for the problem (5.1), (5.2) has the form (5.lcl)

C(y, u., p, q) =

1

2II.Y -

') wllr"o(Q)

+

JV llu2

')

fiiL,(Q)

+ (il + !:;.y- u, P)(QI + bo!l- Yo, q)(n): 1 In further arguments, using a strong lifting, \Ve choose n. function clcfincJ for all ( /,, ·) E Q from the class of functions y E L:!.(Q) coinciding for almost all (t, ·).

:!. OPTII\·1:\LITY SYSTEI\.·1 FOH. OPTII\Jt\L C'ONTHOL PHOBLEI\IS

where p E Lo(Q), q E H- 1 (0). By (1.12), we have Applying the first relation to (5.14), we find

(y- w, z)(Q)

(5.15)

£;1

+ (i + f',.z,p)(Q) + boz, q)(ct)

= 0 and

= 0

(£;, u- u)

:;, 0.

\lz E z,.

Hence \Ve have the equality

p- b.]J =

(.5.16)

u-

w,

which is understood in the sense of distributions. By (5.16), we have

pE Z= {p E L 0 (Q) :p-/',.p E L,(Q)}.

(5.17)

By Lemmas 1A.1 and 4.1, the restrictions "(o]J, ""(rp, "("f:..]J, and ''l'~~ElnP are defined and the Green formula (4.26) holds for u = -1. From (5.15) and (4.26) for u = -1 \Ve obtain (5.5) in the same way as in Theorems 4.1 and 4.4. From (1.12) we obtain (5.6). 0 5.2. Case IntL, Ua = 0. The use of the Lagrange principle. First of all, we impose the following conditions on the initial function Yo(:r) of the problem (5.1), (5.2): y 0 (:r) E C"(l'l) and-yuuYu(:r) "'0. In the ease under consideration, the form of the set Uu plays a significant role. \Ve study only the case where Ua = {u(t.,:r) E Lo(Q): a(t,x) ~ u(l.,:r) ~ ,13(1,:~:), a.e. in Q},

(5.18)

where n(t,:r),,d(t,:r) E C(Q) and ;3(t,x) -a(t,:~·):;, 17 > 0 for any (t,.r) E Q, where ' 7 > 0 is a number. lvloreover, we assume that Uo and Yn(.r) are consistent in the following sense: ~mna(O, :c)

< "Yun/',.Yo(J:)
1/J.

From (5.25), (5.35), and (5.36) it follows that (5.37)

(Ck'Pj,Pl)(Q)

= (ek'{i,PI) =

1

for every j. Let Zj(t) be defined by the relations

':J(t)- ,\kz1 = 'PJ(t), where

,\~cis

Zj(O) = 0,

the eigenvalue of the problem (5.33) corresponding to the eigenfunction

e,,(l:). It is obvious that {)

(5.38)

Dt (zJek)

+ Ll.(z1ek)

= cp1

e,

7,;(z1e,,) = 0,

-m(zje,..) = 0,

(5.39) as j __, oo. Substituting (5.38) and (5.23) in (5.19) and taking into account (5.37), we find

(y-

tv, ZjCf,)( rl/2: m. is an integer: e~; are the eigenfunctions, and .-\h are the eigenvalues of the problem (5.3:-3). Introduce the functions (5.42)

z,(t,x)

=

~

zk(t) L.. ( -'- _, )"'ek(:c),

k=l

where

1

'

z~o(t) = (z(t,J:),e~o)(n)·

(5.43)

Z;; E 1'(-h

/;./\/;

It is obvious that

z~V E Yu,

Z(';

+ ilzs

= '1/'s,

z~V

+ ilz~V

= 'lj1~V.

By (5.41,) and (5.34), we have (0;'',p,)(CJ1 = 0. Substituting the last equality from (5.43) in (5.19) and taking into account (5.23) and (5.34), we find (5.44)

(y- w, z~v)(Q)

+ uJ;v ,p)(Q) =

0.

It is obvious that '1/J~v-;. '~/'(';weakly in £.J(Q) as JV-;. oo and (5.4.5)

z;v-+ z, weakly in L,(Q) as N-+ oo.

Since rn > d/2, using the Sobolev embedding theorem and the inequalit:y llekiiH"'(n) :s;; c/\~; 112 , where c is independent of k, we conclude that for almost all t E [0, T],

(5.46)

X·

oS;

E~'

I: k=N+l

:X:·

~~'k(t)l'

I:

.\;,-:"'-+ 0

k=N+l

as N -+ oo. Moreover, the right-hand side of the inequality (5.46) is finite since ~\~.- ~ k'fd as k-+ oo form> d/2. From (5.46) it follows that ·1/J;v-+ 1jJ, as N-+ oo in L.x(Q). By (5.45), we can pass to the limit in (5.44) as N-+ oo. We find (5.47) We set (5.48) where k=0,1, .... By (5.41), we have 1/•,(t,.r) = 4J'"(t.,:r) for almost all t E [O,T]. Applying the maximum principle to (5.48) (cf., for example, [95, Chapter 3, §1]), we conclude that for almost all t E [0, T] we have (5.49)

11~•,(1, ·)IILx(nJ oS;

111/J'"-'(t, ·)IIL=I"I

oS; .. • oS; ll1/;(t, ·)IILxl 1.

.,,

LEMMA 5.4. Let P,;;- be the space (5.60), 21 E (0, 1), p.' > 1. Then the operator of rest·riction 1: c=(TI) -; c=(iJll) to the boundary ()fl can be e:Dfended by . . _.,, ~'21-1/i 0,

IO

E L 2 (r?.),

y~

E

L~:{E)

/>Ci),.y =

6.1. Ua is a convex set of the space Lx (E) \Vith nonempty interior: 41 E C'(B) such that t/!(O,:r) 0 and Uo. The set Uu is closed in the space L2(E).

Ua o"

1./) E Intr,=(~)

'loY= 0,

and Uo satisfies the following condition.

CONDITION

IntL~(>C)

u,

E Uu:

0. There exists a function

=

REI\1ARK 6.1. The assumption that Ua is closed in the space L 2 (E) is essentially used in the proof of the existence of solutions to the problem (6.1)-(6.3) in Theorem 1.4.3. Note that the dosedness Ua in the space L 2 (E) does not follmv from the assumption that Uu is closed in L=(E). For example, the set Ua = {u(:r) E L.x(O,l): lu(:r)l,;; lln:t:l} is closed in L.x(O,l) but is not closed in D,(O, 1). However, if Ua is a bounded subset of the space L.x(~), then instead of the closcclness of Uo in L:2(E), it suffices to assume that Ua is closed in Lx(E).

G. lJACl\WAllD HEAT EQUATlO!'\. DOUNDAllY AND 1!'\ITlAL CONTHOL

lOG

Fr"om Condition 6.1 we can deduce the nontriviality condition for the problem (6.1)-(6.3) which guarantees, in view of Theorem 1.4.3, the solvability of this problem. However, we need more for the deduction of an optimality system. Our next goal is to prove the solvability of the problem (6.2) for at least one u E lnlr.~("'l Uu. \Ve use the results on the exact boundary controllability. Let 0 :( T < To :( T, Q(T.To) = (T, To) >:: n, and :L(T,To) = (T X To) X Consider the boundar;y- value problem

an.

: (t, :r) + !:!.o(!, .1·)

(6.4)

,. ,."

(6.5)

f_,{T,T())

= 0,

a :: = I!

v.'

where g E L::!(O..) is a given function and vis a control. \Ve state the exact boundary controllability problem. Find a control v such that at a given time moment T t.he solution z to the problem (GA), (6.5) satisfies the condition' (6.6)

The following result about the local exact. controllability (cf. [127] aud [17]) can be easily obtained by methods of Chapter 7, where more complicated exact controllability problems arc considered. THEOREM 6.1. Let g E L,(n). Then there e1:i.-IIH~>i = 1.

Denote by PN : L 2 (n) __, L,(n) the orthogonal projection onto the linear space spanned by the first JV functions tp/,:· If 4' E C(E) is a function from Condition 6.1, then 1/J(O,:r) Oandforanyc > Othcreexistsr11 E (O,T) such that [11/'i[C(,_,,,~,,)
:) Ua. 0 By Lemma 6.1, Condition 6.1 guarantees the validity of Condition 1.4.3. By Theorem 1.4.8, there exists a unique solution (fj, fi) E £ 2 (0, T; H 1 (f.l)) x Ua to the problem (fi.l)-(6.3). We set

R= = {u E Lx(I;): ~y(t,x) E L,(O,T;H 1 (f.l)): :iJ

+ !:!.y =

0 in Q, loY= 0, /TY E L,(f.l), Tc;D,y = u}.

COROLLARY 6.1. Let p E L1 (B) and (p, v)(-,;) = 0 for each v E Rx. Then p= 0. PROOF. From the proof of Lemma 6.1 it follows that for each To E (0, T) and E C(~), suppu. C E(ru.r) 1 there exists a sequence VN E L.::x:(E(o,r11 J) such that llvNIIL~(-,; 10 _,,,) ~ 0 as N ~ oo and for any N the function

'U.

'UN= {VN(t,.T), u,

(t,J:) E B(u.r")' (t,x) E B(To.r)

belongs toR=. Therefore, 0 = (p, uN)(-,;) ~ (p, u). Consequently, p = 0.

0

6.2. Boundary control. The deduction of an optimality system. The main result of this subsection is the following theorem.

THEOREM 6.2. Let Cond'ition 6.1 hold. .4 pair (y, fi) E £ 2 (0, T; H 1 (f.l)) x L=(B) is a solution to the problem (6.1)-(6.3) if and only if tlm·e e:rists a function p E C'((O, T] x fi) such that (6.12) (6.13) (6.14) (6.15)

y+!:!.fl=O,

"Yofi = 0, 'Y~Bnfi =

Jj-!:!.p=O in Q, "'(T]J

= 'W

-

/'T!J,

U E Ua, "Y"i'..BnPI:s= o, (p + Nfi, u- fi)cc,- 1 ~ o VuE UvnRx· U,

!i. DACKWAilD HEAT EQUATION. BOUf',;DAft'{ AND INITIAL CONTJlOL

107

PROOF. \Ve prove the necessity. On R=, we introduce the operator A sending a. function v E Rx to the function "lTZ, \vhere .:(t- 1 x) is the solution to the backward parabolic boundary value problem (6.16)

Thus, Av = '(TZ and V(A) = R~. Let u E R~nUa and let z(t, :r) be a solution to the problem (6.16) for v = The functional J defined in (6.1) satisfies the estimate

u.-u.

_l_(J(Y + az, u + a(u- u))- J(fj, u)) ;;, 0. a

Passing to the limit as o· ......., 0 and taking into account that '"'trz = A(n- U), we obtain the inequality

bTY- w,A(u- u))l"l + N(u, u- u)cc) ?o 0.

(6.17)

On the linear space Rx., we consider the functional R.x.. 3 v . . . .,. (F, v) defined by the formula

(F,v) = bTY- w,Av)(n)·

(6.18)

\\Te show that F' can be extended to a continuous functional on the entire space Lx('B). By Lemma 6.1, there exists a function·$ E Rx and a ball B,. = {u. E Lx('B) : II ull L~(OC) < ·r} such that ·$ + B,. C Ua. From (6.17) and (6.18) we obtain the inequality (F, v +

,j- u) ?o -N(u, v +

$- ii:) 1, 1

VuE B,

n R=.

Therefore, sup vEB,.nR=

I(F,v)l=-

(F,v)

inf vER,.nR=

: 0 the function Tf(T,:r)- w(.r) is represented in

y(T,:r)- w(:r) = La.J(f)e~~\J(l'~ol'PJ(:r),

(G.28)

)=1

·where 2:::::.}: 1 a_}(c) < oo, AJ arc the eigenvalues and 'P.i(:r) are the eigenfunctions of the problem (6.8). We set N

(6.29)

9N(.r) = (

~a.J(c)c"•IT~~)'PJ(.1·))

I (t;a.y(c)) · N

1/~

,

·where uj(c) are the coefficients from (6.28} (uniquely determined from this equality). Let (ON, u,y) be a solution to the controllability problem (6.4), (6.5) tor To = T, T = 0 such that vN(L :r) 0 fort E (c, T). Solving the boundary value problem (6.4), (6.5) for T = E, To = T, v 0, g = gN by the Fourier method, we find ~· ,\ 1/~

=

o, (c, .r) =

=

(~a 1(E)cpJ (r))

I (~

n;(c))

On (0, c), we solve the controllability problem (6.4)~(6.6) for T = 0, To = c, g = Taking into account the relation II=N(E, ·)IIL,(I!) = 1 and Theorem G.1, we can choose (zN(t,:r),vN(t,:r)), t E (O,E), such that ON(E, ·).

lluNIIL~I,;I = llvNIIL~(:s,, .• ,l,;;

(6.30)

c,,

where c 1 is independent of JV. It is obvious that VI\' E Rx. Substituting VN in (!3.18) and taking into account that P = p1, AvN = gN, and the functions gN, Tf(Tj)- w admit the decompositions (6.29) and (!3.28) respectively, we obtain the equa.Iity i\

(6.31)

(1J1,u,)l,;l = (fj(T,)- w,g,y)(u) = (

L"7kl)

1/:!

/=l

From (6.30) and (6.31) it follows that 2:;'::, 1 aj(o),;; co, where co is independent of .N and: consequently, 2:::::~ 1 aj(c} < ex). Taking into account the last inequality and the relation (!3.28) \Ve can solve the problem (!3.27) by the Fourier method. Then we conclude that a solutionp(t, .T) to the problem (6.27) exists and p E C 0((E, T] xl"l) for any E > 0. In other words, p E C 0 ((0, T] x D). Let v E Rx, and let :(1, :r) be a solution to the problem (6.16). For any natural number/,: we set zk(t,.r) = o(l -1/l.:,:r), uh(t,:c) = u(t -1/!:,:r), t E (1/1.:, T). For l E (0, 1/1.:) we set zk(t,:r) = 0 and v,,(t,:r) 0. Integrating by parts with respect. to :rand t and using (G.16), (6.27), (G. IS), and Lemma 6.2, we find

=

0=

/

(p( ::,,

+ L'>zk) + zk(Ji- l'>p)) d.r dt

}q(ljk.T)

(6.32)

=

(w- y(T, ·), z(T- 1/1.:, ·))(n) + (p,v,.)(cs, 0.. nl

= (p, vdc:.{l/1.-.Tl)- (PIJ vdcs(l;l,.nl·

Passing to the limit in (6.32) as k __, ex:;, we find that. (p- Jli, v)(c:) = 0 for all I' E Rx. By Corollary 6.1, this equality implies that 1~1' =PI· We obtain (G.L5) [rom the equality (6.18) for F = p 1 = '"t:::.]J and ·u = u- U, where u E Rx. n Uo, and

110

2. OPTII\IALITY SYSTEM FOR Of'Tii\IAL CONTBOL PBOBLEr..TS

the inequality (6.17). The relations (6.12)-(6.14) are valid in view of (6.2), (6.3), and (6.27). Let us prove the sufficiency. B:y Lemma 1.2, it suffices to prove the uniqueness of a solutions to the problem (6.12)-(6.15). We assume that there exist two triples

(!};, u;,p;) E £ 2 (0, T; H 1 (D.)) x L=(l:::) x C 2 ((0, T] x fi),

(6.33)

satisfYing (6.12)-(6.15). We set z = Yt - y,, h. = Ut (6.12 )-( 6.15), the triple ( z, h, q) satisfies the relations i:

(6.34) (6.35)

q

= Pt - p,. By

+ llz = 0, q- llq = 0 in Q,

!ocDnz = h,

(6.36)

u,,

i = 1, 2,

/OZ

+ q, h)(c;),::; 0, TN+ /1'Z = 0.

!c;Dnq = 0,

=

0,

(h

Therefore) 0= =

r

./q (q(z+flz) +z(q-f>.q))dxdt= (!Tq,/TZ)(n) + (q,h)(y;) 1(

0

.,

)

,,

-2 /I!Tqll:i",(Cl) + II/TZIIr".,(CJ) - I hilL,(>::)+ (h + q, h)(")·

Consequently, "fTq = ol "(TZ = 0, h = 0. From these equalities) the relations (6.34)-(6.36), Theorem 1.1.15 about the uniqueness of a solution to the parabolic boundary value problem, and the theorem on the uniqueness of a solution to the backward parabolic problem it follnws that fil = fh, U1 = U2, P1 = P2· 0 6.3. Initial controL We consider the problem with initial control and final observation: 1 . (6.37) 2 11 /TY- w I., l£,(n) + N IIu- Yo II'i,(n) _, mf, 2 (6.38) [J(t,x)+lly(t,x)=g(l.,x), /o:Y=O, /u!f=u, uEUa, where wE L 2 (D.), Yo E L 2 (D.), g E L 2 (Q), N > 0, (6.39)

Ua = {u(o:) E L,(D.):

Ot,::;

u(x),::; a,(x) for almost all xED.},

'Plan=

where a;(x) E C(fi); moreover, there exists a function 0 such that a1(J:),::; 1 + d/2. By Theorem 1.4.2, there exists a unique solution (y, u) E Yt (6.40)

g

E L 2 (Q)

n L,(Q),

Jl

X

L,(D.) to the problem

(6.37), (6.38), where (6.41)

Y1 = {y E L,(O,T;H,j(D.)): 1i E L,(O,T;H~ 1 (D.))}.

LEMMA 6.3. Under the condition (6.40), the component u of the solution (y, E Yr x LAD.) to /.he problem (6.37), (6.38) is a continuous function: u E C(fi).

u)

PROOF. By (6.38), we have (6.42)

y+lly=gEL 1,(Q),

-y,;y=O.

Since yE Y,, for some lr E (D,T) we have 'lt,YE Ht\(D.). By (6.42) and Theorem 1.1.14, we have y E Lo(O, t,; H'2 (D.) n HJ (D.)). Therefore, !t,Y E H 2 (D.) n H,j (D.) C

B1,(D.) for some t, E (O,t 1 ), where (in view of Theorem 1.1.1) llt = 2d/(d- 2)

!i. BACKWARD HEAT EQUATION. BOUNDARY AND INITIAL CONTBOL

111

for d ';, 3 and 11,1 E (0, oo) for d = 2. If d ' ( 8l1; W,;; 1(0, T) ),

the trace operators ''t:~:.P1 ')'~8np,

and "r'tP are

moTeoDeT1

"~"a,p E B,~. - 11 "' (an; w,;;'(o, T)),

1

'YtP E B;;.0(l1)

and far any fu.nctian z(t,x) E c=((j) such that z(T,:rJivn= z(O,xJI,m= 0 the following Green farmnla holds: (Ji- i'>.p, z)(QI =

hrp, "fTZ)()- buiJ, "foZ)(n) - ('Y"8,p,"f"z) 1" 1 + ('Y-;;p,·t"D,z) 1" 1 - (p, i

+ i'>.z)(Q)·

For the proof of this lemma we refer the reader to [113]. Note that the proof of Lemma 6.4 for 1-i/ = 2 is given above (cf. Lemma 4.1 in Chapter 1 and Lemma 4.1 in this chapter). The case of an arbitrary J.L > 1 is treated in a similar way. Using (6.46) and the Green formula, we can deduce (6.43,:,) using the same arguments as in the proof of Theorem 5.3. The relation (6.4,1) follows from (1.28).

112

2. OPT!!\!ALITY SYS'l"Ei\l FOH OPTl!\L·\L CONTHOL PnODLB.\lS

7. Nonlinear stationary proble1ns In this section. 'iVe derive necessary conditions for au extremum in optimal control problems for nonlinear stationary systems. 7.1. Distributed control and observation. The case of Lp-nonns. Let be a bounded domain ·with boundary On c ex·. \Ve consider the problem

n c JR.d

1

±11!1- wll'i,(n) + ~11u-fll],,(l!)-+ inf, 6y(1:) + b(y(:r)) = u(.1-), '1!1 = 0,

(7.1)

:T(y,11) =

(7.2)

where a> 1, N > 0, wE Ln(ll), f E L2(ll) are given functions, and b(~\) E C 1 (l!l: 1 ) satisfies the follmving conditions:

lb(y)l,::; r(1 + l!!l)", lb'(v)l,::; c(l + lvl)"- 1 ,

(7.3)

"\Vhere c > 0 and x > 1. The simpler case x ~ 1 is not considered here. Let 7 denote the restriction operator to the boundary of the domain n . i.e., "l!f =ulan· For simplicity, we begin with the problem (7.1), (7.2) without the constraint u.. E Ua and clarify conditions on o and r: under which it:. is possible to deduce necessary conditions for an extremum. Let. the assumptions of Theorem 1.6.1 or Theorem 1.6.2 hold. Then there exists a solution (i/, u) E W," (ll) n Ln (ll) x L 2 ( !1) to the problem (7.1), ( 7.2), where r = rnin{2, o/x}.

an

THEOREM 7.1. Let. n ?; 2(x- 1). Then thfTe e:cisls t.riple. Ti, 1i,p sa.tisjie8 lhc system of equations

6i/(.r)

(7A) (7.5)

6p(;1:)

+ b(i/(1:)) = p~·) + f(:r),

+ b'(!i(~·))p(:r) =

(7.G)

/U

p

E L-,(ll) such !.hat. Ihe

= 0,

1

li/(1:)- w(:r)l"~ sgn(i/(:r)- w(:c)),

"yfl

= 0,

p(:r) = N(u(:r)- f(:r)).

nfl

7 f(ll)nLn(D)) x L,(D), PROOF. We use Theorem 1.8. We set X= (W,2(!1) 2 V = L,.(D), Xn = (C (fi)) n HMO) x £ 2 (!1), V, 1 = £ 2 (!1), where 1: = min{2, n/x}, and F(o:) = F(y, u) = 6y + b(y)- 11 for "' = (y, u) E X. The funct.ioual g(.T) defined in (1.47) for the functional .7 in (7.1) is continuously differentiable in X 0 . Since b(~\) E C' 1 (!Pc 1 ), for any .T = (y, u) E Xu and h = (/1 1 ,h 2 ) E Xu the operator G determined from F by ( 1.~17) satisfies the relation

G(:1: +h)- G(1:)- G'(.T)h = b(!i -1- y -1- hi)- !J('iJ + y) -1/(i/ + y)hr, which implies that G : _\" 0 _,. l~l is continuously differentiable because so is the Ncmytskii operator (cf. [6]). We show that the operator G'(O)h = P'(x)h = 6y, +b'(i/)y 1 -n 1 , where h= (.>J 1 ,u 1), transforms Xu= (C 2 (l1)nH1](ll)) x L 2 (ll) to V,, = L,(D). Indeed, for any g E L 2 (!1) there exists a pair (y 1 , ui) E Xo such that 6u,(o:) + b'(i](J:))y 1 (:r)- u 1 (:r) = g(x) since we can take y 1 = 0 and u 1 =g. \Ve set

(7.7)

L.(y, u,p) =

±11:>1- wll/', + ~ llu- filL+ (6y + b(y)1

u,p)(U)·

7. KONLlNEAH.. ST:\TIONAHY PHOBLEI\IS

By Theorem 1.8, there exists p E L,(ll) such that (£~(fj, u, P ), Yr) = (lv- wl''- 1 sgn(y - w), Yr \n)

(7.8)

(7.9)

+ (t.y, + b(fj)y,,p) 1n1 = o Vyr E C''(TI) n H,j(11), (£;,(fj,u,p),ur)

=

N(u- .f,u,)i'lJ- (ur,p) 1111

= 0 Vur E £,(11).

From (7,9) we obtain (7.6). Taking Yr E C'1)(TI) in (7.8), we obtain the first relation in (7.5L which is understood in Lhe sense of distributions. \Ve show that:. the restrictimqp of p(:r) to 8ll is defined and1p = 0, By (7.3), li (fj( :r)) E Ln; (%-1) ( 11), Since p E L,(ll) and a:;, 2(x-l), we have u'(fj(J:))p(:r) E L 1 (.r). By (7.5 1 ), (7.10)

2>p(1:) = .fr(x)

= ly(,T)- w(.r)l"- 1sgn(y(:r)- w(:r))- b'(fi(:r))p(:r) E L,(ll).

We show that for some s E (0, 1) and q > 1, p E 1~('(11),

(7.11)

r/

= qj(q- 1),

where P,(·'(D) is the space (5.60). ForsE (0, 1) and q such that s > djq, from the Sobolev embedding theorem we fiud £ 1 (11) c w,;;·'(l1). By (7.10), we have

t.p

(7.12)

=

.fr(.T) E 1V,('{I1).

Since s > djq, we have s > d(1jq- 1/2) and Ht;(11) C £,(11), which implies L,(D) c (W,;(I1))'. The last inclusion and (7.12) yield (7.11). B)· Lemma 5.4, the restriction/Pis defined. From (5.Ci1) and (7.8) it follows that 1Ji = 0. From (7.2) and (7.6) we obtain (7A). D \Ve consider the problem (7.1), (7.2) with the following additional condition:

u. E Ua,

(7.13)

\Vhere Uu is a convex closed subset of the space £.2(rl). For simplicity, ·we assume that

(7.14)

o = 2x,

1/2- 1/(2x)- 2/d < 0.

J\'iorcover, assume that Condition l.G.l hold. By Theorem l.G.l. this condition guara.ntces the existence of a solution (fj, U) E (lV](rl) n L::.x) XL::. t.o the problem (7.1), (7.2), (7.13). THEOREM 7.2. There e:r:ists a pair (A, p) E (TIL,. x L, (ll)) \ {0} such that. (fj, u, p) satisfies the 'relations

(7.15) (7.16)

2>p(:r:) -1- 1/(fiCr-))p(:c) = ,\lfi(:r:)- w(:r)l'%-l sgn(fj(:c)- w(:c)),

.!

(N,\(uCr)- J(:r))- p(:r))(v(:r) -il(:r:)) rh:;, 0

'IJI = 0,

VuE Ua.

PROOF. We use Theorem 1.5. Assume that Y = I:I'(11) n H,\(ll), U = V = £,(11), F(y, I') = 2>y-t- b(y)- u, and J( u, u) is the functional (7.1) satisfying (7.14). It is obvious that the functional Y : Y x U . . . . ;. IP~ is continuously differentiable in a neighborhood of (fi: U). By the Sobolev embedding theorem and the inequality (7.14,), the embedding H'(ll) C L,AI1) is continuous. Hence; the operator

(7.17)

F~(fi, if) = 2> -1- li (fj) : I:I'

n I:I1\ ~ L,

2. OPTJI\TALITY SYSTEI\I FOil OPTll\IAL CONTHOL PROBLEMS

is continuously difFerentiable. By (7.V12), the embedding H 2 C L2x is compact and the operator h ...._. b'(fj)h is continuous from L2x to £ 2 . Since the operator 6. - l : L:2 ...._. JJ'2 n H inverse to 6. : H'2 n Hr} ...._. L2 is continuous: the operator h-> !';- 1b'(y)h is compact from n to n Hence (7.17) is the Fredholm operator and, in particular, the set ImF~Cfi/U) is closed in £2; moreover 1 codim Im F~(fj 1 U) < oo. Thus, we have verified all the assumptions of Theorem 1.5. Using this theorem, we deduce (7.15) and (7.16). The arguments are similar to those in Theorem 7. L In the use of (7.15) and (7.16), as a rule, it is necessary to prove that ,\ ~ 0. \Ve indicate a number of cases where it is possible to prove this fact. Case I. Let the nonlinear term b(y) in (7.2) satisfy the condition

J

H" HJ

H' HJ.

b'(t) :s; 0 \ft E iF:.

(7.18)

Then ,\ ~ 0 in (7.15), (7.16) (and we can assume that ,\ = 1). Let us prove this assertion. Suppose,\= 0. By Theorem 7.2, p E Lq, (D), where q1 = 2 and li(Y(x)) E Lh/(%-1)(!1). Therefore, b'(Y(x))p(.1:) E Ln(D), where 1/a = 1/q! + 1/2 -1/(2x). From (7.15) for ,\ = 0 it follows that p E w,;(n). By the Sobolev embedding theorem, we have p(.r) E Lq,(D), where

1jq, = ma..x{O, 1/n- 2/d} = max{O, 1/q1 + 1/2- 1/(2x)- 2/d}. By (7.14'2), we have q'l > q1 . Repeating these arguments, '\Ve get 1/qk = max{O, 1/q1.,_ 1 = max{O, 1/q1

+ (1/2- 1/(2x)- 2/d)}

+ (k-

1)(1/2 + 1/(2x)- 2/d)}.

Therefore, we have qk = oo for a sufficiently large k, i.e., p E L=(D). Taking the inner product (in L 2 (D)) of (7.151) for,\= 0 with p(.r), integrating by parts and taking into account (7.152) and (7.18), we get

IIVPIIL( 2 some additional boundary conditions appear in the Euler-Lagrange system. For simplicity1 \Ve consider the problem 1 N ., . 2llu- wlliF!Ill + 2 lla- .fllr",('ll--; mf, 6.y + h(y) = u, ~!Y = 0, 0

(7.23) (7.24)

where b(,\) E C'(iPi) and k is an integer such that

k ~ 2,

(7.25)

k > d/2.

As in Theorem 1.6.3, "\Ve can establish the existence of solutions (fj, U) E H 1'(0.) x L 2 (0.) to this problem. \Ve write the Green formula. for an elliptic operator of an arbitrary order, which will be used in deducing necessary conditions for an extremum in the problems (7.23), (7.2"'u. · vd:r n

\fu.,v E C1f'(!1) .

The last formula is easily proved Ly induction on m using integration by parts.

D

Deduce necessary conditions for au extremum in the problem (7.23), (7.24). THEOREM 7.3. Let CiJ, 'ii) E H 1'(!1) x L 0 (!1) be a. solution to !.he problem (7.23), (7.24), where k is a nat.urol number satisfying (7.25), and let. w E H 1'(!1), f E L"(!l). (7.30)

Then (7.31)

y- w

(

E H:\ (11) ,'' wher·e A = I:.~~~i -1 )j L\.1 ,

andy,

p= N('ii- f),

"Recall lhal 11:((1:1) ~ {c(:r) E H 1·(fl) 'Av(.v) E Lc(fl)}.

together with the function

7. NOr\LlNEAB STATIONAHY PllOIJLEMS

Ill

sati.Sfies the difj'erenlial equations

(7.32)

~P N = f

Mj . + b(!j) -

!:lp + b(fj)p + A(fj- w) = 0,

in fl and lhe bou.nda.Ty conditionS

(7.33) on

an.

·fTi=O,

'r(p+B 1 (iJ-w))=O.

j=2, ... ,k-l,

'IBJ(i7-w)=O,

where B;;' j = 1, ... : k- L are lhe bov.nda.ry opcratoTS from (7.29).

Pn.oOF. \Ve make the change of variables y 1 = !J- w, u 1 = u- .6..w. Then the problem (7.23), (7.2•1) is written in the form

1

211 y, II''11'-('!1 + Nil u,- h'II''Jd 2, 2

then for· a.ny f E L2(Q) there e:Eists a unique solution y E W]· (Q) to the problem

(8.1), (8.2). PROOF. We consider the case (8.4). Let 0

·1 ')

-1

'l

R ,,·-(Q) ={yEW,~ -(Q),'ioU = O,·I~Y = 0},

and let R : L,,(Q) __, l•lr~· 2 (Q) be the operator corresponding to the problem !i - /}.!1 = f, "'loll = 0, ,.../::..!/ = 0. The existence and continuity of the operator R follow from Theorem 1.1.14. Applying the operator R to both sides of (8.1), we obtain the following equation in

llll·:::!(Q):

y+(Rom)y=Rf.

(8.6)

By Theorem 1.1.12, the embedding

~2 (2c.. \-q2c)

(8.7)

w,;·

+

2

(Q) c L 1,(0, T; L,1 (0)) is compact if

(2c, \-p2c) < 1.

Using the 1-Ii:Slder inequality, "\Ve establish the continuity of the operator m Lp(O, T; Lq(rl)) _, L,1 ( Q) of multiplication by a function rn (I, J:) satisfying (8.3) if

1

1

1

1

1

1

q+:;; = :\' p+ ,6 = :x·

(8.8)

If 1 < A < rnin{(J, 1} and (8.4) holds, then there exist p and q satisfying (8.7) 1 .,

1 ')

and (8.8). Consequently, the operator Rom: W.,·- _, W.,·- is compact and (8.6) is the Fredholm equation. \Ve use the theorem on the uniqueness of weak solutions to the problem (8.1), (8.2) (cf. [115, p. 43] and [81]), which is possible here because 0

of the continuity of the operator m:

t-lll· 2

--+

0

LA(Q). By this theorem, a solution

to the problem (8.6) in lTr~· is unique. Since I +Rom is a Fredholm operator and, consequently, has zero index, we conclude that the problem (8.6) has a. solution since the uniqueness has been proved. Thus, we have proved the unique solvability of the problem (8.1), (8.2). Consider (8.5). Taking the inner product (in the space L2(il)) of (8.1) with -i}.y, after simple transformations we find 2

1 oiiV'y(l, ·lllo' + -

(8.9)

j'' llt..y(T, ·lllo, dT IJ

!illu dr,;; c Jo[" ll,nlldYII~-~ IIYIIl dr ,;; c,

Jor' llrniiZ .. II!illi dT + E lur' II vii~ dr.

By the Poincare inequality and the estimates for solutions to elliptic boundary value problems, we have

From this inequality and the estimates (8.9), (8.11) it follows that

(8.12) lly(t,.) II? +

r' II!!( T,.) II~ dT ,;; c(i"' llfiiG dT + Jor' llm(T,. liiL,IIv( T,.) IIi rlr).

lrl

(J

By (8.12) and the Gromvall lemma, we have ·l

lly(t, ·)IIi+

i

lly(r, ·)II~ dr,;; const,

• II

which implies the existence and uniqueness of a solution y E y,Tl ~·~ (Q) to the problem (8.1), (8.2). In the case 1 > d, it is easy to prove (8.11) using the embedding II 1 c Hdf-, c Lq, the Cauchy inequality, and the estimate (3 > 2. After that we can complete the proof as before. 0

8.2. Distributed control and observation. Consider the problem (8.13)

(8.U) (8.15)

1

~IIY- wllc., 1Q 1 + ~ llu- JII'L 1Q 1 _, inf, [J(t.,:z:)- L'>y(l,:r) +b(y(t,:r))

=

u(t,:z:).

'(-,cy=O, "(oy=yn,

u. E Uu,

where a> 1, N > 0, wE Lu(Q). f E L,(Q), the function b(A) E C 1 (!P1 1 ) satisfies (7.3) for x E (1, a) and Un E L,(Q) is a convex closed subset satisj'ying Condition 1.7.1. Recall that for J/n E iJ;Il-l/ 0, wE £,(0.),

.f E L,(Q), Yo E H,\(0.) are givenllmctions and

II(vuliiLdnJ
0 the numbers x+l (3=!= (d+2+c)/2 < - x-1 satis(y (8.4) in view of (8.39 2 ). Therefore, (8.41) implies (8.40).

0

We deduce an optimality system of the problem (8.32)-(8.3,1). We begin with the case dim 0 = d ~ 2. THEOREM 8.3. Let d;, 2, and let (y, u) E Xr.n x L,(Q) be n. solution to the problem (8.32)-(8.34) satisfying the estimate (8.37), let b(,\) sa.tisfij (8.38), and let %satisfy (8.39). Then there e.1:ists a function p E L 2 (Q) snch thn.tp+D.p E L,,(Q), where a= ~i~~ll)) and the triple (fj, U,p) satisfies the relations

7j(t,.r)

(8.42)

+ D.p -1/(y(t,:r))p = 0,

JQf (N(u( t, .r) -

(8.43)

J"r(p+ fj) = w,

~(-,:;)! = 0,

f(t, .1:)) - p(t, .1:)) (v(t, 1:) - u(t, :z:)) d.nil ;, 0

Vv E Ua.

PRooF. We use Theorem l.G. Set Y = {u E X1.1 1 n L%+ 1(Q) : 1>:!1 = 0}, U = V = V,, = L,(Q), l"[, = {y E L,(O, T: H 2 (0.)) : [J E L,(O, T; H 0 (0.)), 'loY= 0, {-,;1/ = 0}, F(y, -u.) = : 2 is equivalent to the condition x < 3, and the assumptions (8.39) for d ) 2 lead to more restrictive conditions on x. Thus, we have verified all the assumptions of Theorem 1.6; moreover, we can t.) and x sotis.f)J (8.38) and (8.:39) respectively. Then the1·e e:Dists o Junction p E L:J(Q) s·uch that ]i + b.p E L,,(Q), where a = "\:;~;), and (i}, il, p) sa.tis.fies (8.42), (8.43).

PROOF. \Ve define all t;he objects except 1"(1 and 1~ 1 in the same \vay as in Theorem 8.:3. We set 1'(1 = L:1; 2 (Q) and}~,= {y E w~:;;(Q): /n!l = 0,-f~ = 0}. Note that in this case, the inclusions }'(J C Y and 1~ 1 C 1/. do not hold. By Theorem 1.1.11, !I E 1(1 C L 1,(Q) for all p > 1. By (8.:39), we have x < 5. Therefore, li(y) E L(%-H)/(%-J)(Q), (x + 1)/(x- 1) > 3/2. Consequently,

111/(i} + Oy)ui!L,,.,(Ql

~

cllb'(i} + llulllr.,,+n:r"-" llull>:,·

Therefore 1 as in the proof of Theorem 8.3, \VC conclude that. G(y, 11) : 1(1 xU ~ 1~1 is a continuous and conLinuonsly differentiable mapping. The relation 1 (0, O)Yi1 = 1'rl follows from Lf~mma 8.1. \Ve prove that (U: v) = (0, 0) is a solution to the problem (8.4G). It sufHces to prove I. hat if a pair (y, n) E r(, x U satisfies G(y, u) = 0,

c;

!:::!G

2. OPTII\T:\LITY S"'{STEM FOil OPTJ!vlr\L CONTT10L PBDRLEl'dS

then !f E ~YuJ n L%+ 1 (Q). Since Yl1 C L:;-:+l (Q), it suffices to verii~r the inclusion y E Xt.o· From G(y, u) = 0 and F(fj, u) = 0 it follows that (8.49)

.~-

6.y = u- (l1(fj + y)- b(YJ),

Since y E Y(, C Lp for all p with y and find that.

> 1, we

"'loY= 0,

"Y~U = 0.

can take the inner product (in Lo(Q)) of (8.49)

(8.50) Let Q, = [0, T- c] x fl. The inclusion fj E XJ.ll implies that fj E H 1 (Q,) C Lp(Q,) 0 for all p > 1. Therefore, y E (Q,) in view of (8A9). Since G(y, u) = 0 and F(fj, u) = 0, we find that

wJ·

d (~ dt y + !J ) - 6. (~ y + y ) + b(~ y + y)

(8.51)

'io(fi + y) =Yo,

'dY + y)

=

u~ + u,

= 0.

Taking the inner product (in £ 0 ( Q,)) of (8.51) with (T- t- E) (y + y) and arguing as in (1.7.16), we deduce that

1

1

. . ·1"' ;; (T-t-E) 1.y+y-d:rdt

Qe -

(8.52)

,;; c( Eo+

+

l,.I'V(!i + y)l dxdt 0

k. (w(!i+y)+~(T-t-c)lul")dJ:dt).

By (8.50) and the inclusions y E Xr.o C £ 2 (0,T;H 1 (f1)) and fj E Lx+r(Q), y E Lp( Q) for all p > 0, the right-hand side (8.52) is bounded by a constant independent of E. Let E-; 0 in (8.52). Then ..j(T- t).ii

,; ; (.{(TX

n-(1-o)

'/+

£ ll.·;,(·t," ·)II,'·__

""+''!)

II :PI fill n'·+:l/2+1, ("\. ' -.::. ..,, I :( Cui liP II n·"-;-:1/~ .::. --

This estimate and lhc !:race theorem imply the continnity of the operator G 1 IJ-•+m(r 1 ) for all s E Rand all natural numbers m.

FF(ro) _,

::1. THE SOLVADlLIT'{ OF DOl;NDARY VALUE PH.ODLE!\IS

Since the operator A is positive definite, for any c > 0 and arbitrary u 0 E H 0 (ro), equation (1.23) has a unique solution q E H 0 (f 0 ). Since the operator (1.22) is continuous, we have q E Hk+l(fu) if u 0 E H"+ 1 (f 0 ) in (1.23). Consequently, equation (1.23) is uniquel:y solvable in the space Hh+l (r 0 ). Using similar arguments, it is easy to establish that the spectrum of the operator A : H"'+ 1 (fu) ~ Hk+l (fn) consists of the eigenvalues .\_i. . J'vioreover, the corresponding eigefunctions eJ are infinitely differentiable: e1 E c~(fo). Let uu E H"'+ 1 (fn) have the form IV

(1.24)

L uPei"

u.o =

JV

< ·XJ.

p=l

Then the solution q E H 1'~+l (fo) is given by the formula N

_

(1.25)

~

EHp

q - - L---ew ,\I;+c p= 1

By (1.25), we have N

(1.26)

l!qiiH'+'(r,) ~ L \clnPI_IIcPIIw+•(r,) ~ 0, p=l

... p

+c

E

~ 0.

From (1.26), (1.16), (1.13), and (1.14) it follows that for uo of the form (1.24) there exists a sequence of functions Y=:(a:) that are harmonic in rl, satisf~y the condition -tu8y, = 0 and (1.27) By (1.27), we can show that the solvability set for the problem (1.8) is dense provided that the system of eigenfunctions {e1 } of the operator A: H"+ 1 (f 0 )--;. J:[h·H(ro) is complete in J:[k+l(f 0). Assume the contrary. Then there exists a vector wE H-(l.•+ll(fo) such that (1.28) where Ex- is the set of all finite linear combinations of e.i. By [114, Chapter 2, Theorem 6.7], we deduce (cf. footnote 4 on p. 133) that the problems (1.17), (1.18) are uniquely solvable for any q E H·'(r 0 ), wE H'(f!), and s E JR. The operator (1.21) is extended to an operator

(1.29)

A: H-("+'l(ru) ~ I:f-(H'I(ro).

The adjoint operator A': Hk+ 1 (r 11 ) ~ H'-+ 1 (r 0 ) coincides with the restriction of the operator (1.29) to H 1'+ 1 (r 0 ). The last assertion is proved in the same wa~y as the assertion that the operator A : H 0 (f 11 ) ~ H 11 (r 0 ) is selfadjoint. It is obvious that AEx C E=. By (1.28), we have ( 1.30) Since Ex is dense in L"(fu) and AwE L"(fu) by (1.22), we conclude that Aw = 0 in view of (1.30). Consequently, w = 0. 0 2. The approximate controllability of parabolic equations \Ve prove the approximate controllability of the heat equation for controls of various classes. \Ve use a. method that is close to the method presented in 1.1.

2. THE APPT10:'\Ti\IATE C'ONTllOLLABTLJTY OF PAFI:ABOLlC' EQUA'I'lONS

J:Jfi

2.1. Local distributed control. Let ll C IR" be a bounded domain with boundary an of class and w a subdornain of D. such that w c nl ·where w is the closure of w. For the heat equation we consider the following mixed boundary value problem with control:

c·x

(2.1)

y(t,.1:)- D.y(t,>:) = .f(t.>:)

+ u(t,J:).

(2.2)

111 ,;= 0,

(2.3)

y(t,a:)l 1 ~ 11 = !ln(:c),

(t,.c)

E

Q,

where Q = (0, T) x ll, I; = (0, T) x all, T > 0 is a given number, f E L 2 (Q), if= ayjat, Yn E L,(ll) are given functions, and u(l,:r) is a control. Assume that the controlu(t-, x) is concentrated in (0, T) x w. J'vlore precisely, we introduce the space of controls

(2.4)

Uw = {u E L,(Q): suppu(t,.10) C (O,T) x w}.

Let Yt E L,(ll) be given.

DEFINITION 2.1. The problem (2.1)-(2.3) is said to be approximately controllable with respect to the class of controls Uw if for any E > 0 and Yt E L,(ll) there exists a controlu E Uw such that a solution y to the problem (2.1)-(2.3) with control u. satisfies the condition (2.5)

ll:t~(T,·)-

y,(-)IIL,(nJ < c:.

THEOREM 2.1. The pmblern (2.1)-(2.3) is a.ppro.1:irnately controllable with respect to the class a.f controls Uw defined in (2.4). PROOF. \\.Te reduce the situation to the case of the problem

(2.6) (2.7)

i:(t,x)- D.z(t,:c) = u(t,:c),

zl ,= 0,

z(t,:r)l t~u= 0.

We set

(2.8)

y(t, a:) = y(t, x)

+ z( t, :c),

where y(t,x) is a solution to the problem (2.1)-(2.3) with u(t,.'E) o= 0. By Theorem 1.1.16, a solution fj(t,x) to the problem (2.1)-(2.3) for u o= 0 exists and belongs to the spaceY= {y E L,(O,T;H1\(ll)); .Y E L,(O,T;H- 1 (11))}. We assume that the problem (2.6), (2.7) is approximately controllable. Then for any y, E L,(ll) and f > 0 there exists u E Uw such that the solution o to the problem (2.6), (2.7) satisfies the inequalit:r (2.9)

II z(T, ·) - (V(T, ·)

+ Y1 (·))II t, (II)

oS;

E.

Then the function y defined in (2.8) satisfies (2.1)-(2.3) and (2.5). Therefore, the problem (2.1)-(2.3) is also approximately controllable. Let W be a subset of the space L,(ll) sweeped by the traces TrZ = z(T, ·) of the solution z(t, x) to the problem (2.6), (2.7) as the control u runs over the entire space Uw. We prove the approximate controllability of the problem (2.6)-(2.8) by contradiction. Assume that the linear manifold lV is not every-where dense in L~(rl). This rneans that there exists a nonzero element v:'o E £2(0) such that

(2.10)

('Ptio o(T, · ))(11) = 0

where(·, ·)(ll) is the inner product in £ 2 (11).

Vz(T, ·) E W,

:J. THE SOLVAJJ[LtTY OF DOUND:\BY VALUE PHCHJLE..\[8

];\()

\Ve consider the bouncla.ry value problem for the baclnvard heat equation (2.11)

\'J(I,:r)+Z>y(l,.r)=O,

iJI ~= 0,

(2.12)

(t,:r)EQ,

y(t, :r) lt~T= iJo(.c).

rviaking thE' change of the variable T = T- f, \Ve reduce this problem to the mixed Dirichlet problem for the heat r:~quation. By Theorem l.l.lG, there exists a unique solution iJ E Y to the problem (2.11), (2.12). Moreover, iJ E C=(Q) in view of Theorem 1.1.14. Take the inner product (in L,(Q)) of equation (2.11) with the solution: to the problem (2.6), (2.7) and use the Green formula (1.1.40). the relation (2.10), the boundary conditions (2.7), (2.12), and equation (2.6). We obtain the equality

!.~o(t,:r)u.(t,.?:)rLrdt

= 0

VuE Uw,

.q which implies that (2.13)

(!,1:) E (O,T) x w.

y(l,:r) := 0,

Lei: w, be a ball contained in w. By (2.13), y(t,:c) is a solution to the Cauchy problem for equation (2.11) in (0, T) x (fl \ w) with zero Cauchy data

Yl

((U')xa:.v 1 =

0,

Bn:PI (o.T)xiJ:.v 1 = 0.

I3y the Holmgren theorem on the uniqueness of solutions to the Cauchy problem, we have y(t,1:) = 0 for (t,:r) E Q. Consequently, iJu(J.·) = 0. 0 Theorem 2.1 actually establishes the L,-controllability of the problem (2.1)(2.3), i.e. 1 the validi!Jy of the e~:;timate (2.5) in the £:2-norm. It is of interest. to consider the approximate controllability of the problem (2.1)-(2.3) with respect to other norms. To investigate this question, we generalize Dellnition 2.1. Let .N be a normed space of functions defined in D., and U a nonned space of functions defined in Q and supported in (O,T) x w. We assume that N C LI(fl) and U C Ll(Q). Let the following condition hold. CONDITION 2.1. The data .f E L 1 (Q) and Yo E Lr (fl) are such that for any u E U the problem (2.1)-(2.3) has a unique solution y E Lr (Q); moreover, the trace 7TY E JV of this solution b defined. DEPJNJTJON 2.2. The problem (2.1 )-(2.3) is said to be N -appm:rimately controllable with respect to the class of controls U if for any E > 0 and Yt E j\l there exists a control u E U such that a solution y to the problem (2.1)-(2.3) with control u satisfies the condition (2.14)

The following genera.li?.ation of Theorem 2.1 holds. THEOREM 2.2. Let 1 < p < ,YJ, f E Lp(Q), !In E L1,(fl), Uw.p = L 1,(Q) n Uw, where Uw is the space (2.4). Then the problem (2.1)-(2.3) is Lp-appro:rinwtcly controllable with respect t.o the space of co'nlrols U.._·.p· PH.OOF. The arguments are similar to those in the proof of Tlleorem 2.1. 1\Toreover, the element iJo orthogonal to the set lV C L 1,(fl) (i.e., satisfying (2.10)) is assumed to belong to the space Lq(Q), where 1/q + 1/p = 1. 0

'.:. TilE

Al-'1-'llOX!;..!AT!~

CONTHOLL.-\n!UTY OF PAI1AUCJL!C

EQL;XnO~S

J;Ji

The question of the H 1'-controllability of the problem (2.1 )-(2.3) is not so trivial. It \Vill be considered in the next subsection.

2.2. The case of approxhnate noncontrollability. \Ve show tha.t the problem (2.1)-(2.:3) is not. H"-approximately controllable for a Sobolev space H 1'(0.) if k ~ 3. \Vc introduce sonw function spaces. Let ej(:r): j = 1, 2, ... , br:~ the eigenfunctions, and let 0 < ,..\ 1 ~ ,..\::! ~ • • · ~ /\ ~ ..• be the corresponding eigenvalues of the spectral problem -D.c(.T) = .\c(:c),

(2.15)

:r E 0.,

It is knmvn that {cj} is an orthogonal basis for the space L~(r!). Suppose that this basis is orthonormal. The space Gn(O.), n E IR, is the set. of functions X

o(:r) =

(2.16)

L

o,;e,;(:r),

Zj

E IR,

j=l

with the finite norm

ll=llf:,"

(2.17)

=

L X/l=.il~. j=l

His obvious that G 0 (0.) = H 0 (0.) k, G"(O.) is a subspace of H 1'(fl). LEl\·UviA 2.1.

= £ 0 (0.).

By Lemma 2.1 (below), lor a natural

The following equalities hold:

(2.18)

_ {- J'~k+l(") · AJ-1 - O1 J· - 0 11 1 • • • 1 h'-} 1 G 21c+l(n) ~L-~E1. ~~.w'~Dn-

(2.19)

G 21'(0.) = {o

E

H"'(O.): t,..izlm 1=0, j = O,l, ... ,k-1},

where k is a na.tuml number, Ll 0 .:: = .:: and .6_) ~ j = 1, 2, ... , is the j t.h degr·ee of the operator D.. The norm II ·IIG' of the space G"(O.) defined in (2.16) is equJvalent to the norm of the Sobolev space H 1'(0.). PROOF. By the theoremt:i on the regularity of solutions to elliptic boundary value problems, the eigenfunctiont:i ej(x) of the problem (2.15) with .;\ = /\i is infinitely differentiable in TI. Therefore, from (2.15,) and (2.15~) it follows that. 6-e.J(:r)i ml= 0. Applying the operators to both sides of (2.15 1) and using the induction, we find

(2.20)

D."eJ(:r)l 11 "=0,

k=0,1,2, ... , .i=1,2, ....

Let. us prove that the norms II·IIG"' and 11·1111"' are equivalent in the space G"' (0.) for odd rn (the case of even m is trca.ted in a similar way). Instead of the functions (2.1G), we begin with the function N

(2.21)

zN

(:r) =

L

z,;e1 (:c).

)=I

It is obvious that functions of the form (2.21) for any k satisfy· the boundary conditions (2.20). Therefore, using (2.17) and the orthogonality of the basis {CJ} for

:J. THE :-JOL\',\BILITY OF BOUNDAilY VALUE PHODLEI\IS

1V

(2.22)

II=NIIb""'

=

N

(2::;A]·+lzje1 .I:;A:··z;e;) )=!

=

i=l

0 ( .. )

= (C.6.''zN6.'-zN)((n) "'GllzNII~I"''I!!)•

\vhere Cis independent of JV. Using the known estimate for solutions to the elliptic boundary value problem

L:."z(a:) = f(:c),

6.j =I an= 0,

j

=

0, 1, ... , k- 1,

and the Poincane-Steklov inequality (cf. (1.1.6)) and (2.22)), we find

llzNIIfi"+' "'C\IIC."ziiii' = GJ(II'V6."zNIIL + ll6."zNIILJ "'Gcii'V 6."zNIIII" =Co llzNIIbnH,

(2.23)

where the constant C'.!. is independent of iV. To prove the estimate (2.23) for 1.: = 0, it suffices to use only (1.1.6) and (2.22). Since the constants C and G, in (2.22) and (2.23) are independent of J\T, we pass to the limit in these inequalities as ]\T-o- oo and prove that the norms II · llo,,~, and I · I If"+' are equivalent. To prove the validity of the boundary conditions in (2.18) for any function z E G'21~+l (0), 'iVe note that these boundary conditions hold for each function of the form (2.21). Therefore, from Theorem 1.1A and (2.23) we find k

(2.2'1)

2::; lha 0 in (1.4.16), i.e., the differential equation in (1.4.16) is the heal equation, can be easily clone using Theorem 1.1.14 on the solvability of the mixed Dirichlet problem for this equation and the Sobolev embedding theorem. Therefore, we consider t.he case u < 0. I'vla.king the change of the variabk~ h = u( t - TL 1.ve can write t.he constraints (1.4.16), (1.4.17) in the form (2.40)

iJ,,:(ti,:r)-b.:=m(ti,:r),

'1>:::=0,

Tr,z=n,

11EUn,

where :(t,, .1:) = y(T + t,fu), !It(!,, .1:) = g(T + tJ/11)(11, T, = -11T. For simplicity, we use the notation t 1 ~ I, T 1 ~ T, and g 1 ~ g in (2.40). We write (2AO) as follows: iJ1 :(i,:r)- b.z = g(t,:r),

(2.41) (2.42)

TrZ E

'(o;Z

= 0,

ui).

To verify Condition 1.4.2, it is necessary to prove the existence of a function (2.43) satisfying the relations (2.41) and (2A2), where Ua is one of the sets (2.38) or (2.39). \Ve begin with the sc:.~t (2.39) under the assumption that there exists E such that (2.44)

n, (:r)

+ f,;;

lf:r E :0,

a,(:r)

o 1 (.1:) E C(TI),

i = 1. 2.

On the right-hand side g(t.,:r) we impose the additional condition (2.45)

g(l.,:r) E L1,(Q),

Let us prove the (2.46)

existence:.~

zEZ

11

> (d + 2)/2.

of a function.:.: satisfying (2.41), (2.42) such that

={: E W ;''(Q): 'I":= 0}, 1

I'> (d + 2)/2.

It is obvious that Z C Yl. This fact is sufficient for the verification of Condit.ion 1.4.2. If z E Z satisfies (2.41) and (2A2), then (2.41 0 ) implies that the following necessary compatibility conclition holds:

(2.-17)

bnno1 )(:r'),;; ll

=·,aniJrz,;; bunn,)(.7:'),

:z:'

E iJfl.

t.t:!

:1. THE SOLVABILITY OF DOUNDABY VALUE PRODLEJ\TS

On o· 1 and

o:~

\Ve impose the following stronger condition:

(2.48)

< 0 < hanrr2)(:r') ':1.7:' E ()fl.

(/'o 1/p).

It suffices, for a given 6 > 0, to find an initial condition ·a E H ..z~~h (Sl), 1

(2.50) such that the solution z to the problem (2.41), (2.42) satisfies the inequality

II'ITZ- ztll 11 .,;-c;,,(ll)

(2.51)

By Theorem 1.1.1 and the condition ll

,::;

0.

> (d + 2)/2, we have

IITr- z,llc(ITj ,::; ciiTrZ- ztllw,;·'/"(!!l ,::; co.

(2.52)

By (2.49) and (2.52), for a sufficiently small owe find"' (:r) < ')'rz(x) < a2(:r) for 2 all1: E fl. Thus, it suffices to verify the W,~- / 1 '(11)-approximate controllability of the problem (2A1), (2.50) with respeet. to the space of controls u E W,~- 2 ;,, (rl). As in the proof of Theorem 2.1, the situation under consideration is reduced to the caBe g(t, :r) "" 0 by replacing z(t, :r) = y( I, x) + y(t, :r), where TJ( I, :c) E W~· 2 ( Q) is a solution to the problem 8/fj- !::J.fj = g, ""'l::.fi = 0, 1o!J = 0, \Vhose existence is guaranteed by Theorem 1.1.1L1. Arguments of Theorem 2.1lead to the adjoint problem (2.11), (2.12) for 'Pn E 1''(11))'. By the Sobolev embedding theorem, fork> (4 + d)/2 the embed211 ding c '(11) is continuous, where 1' is the space (2.16), (2.17). Let ex· be the space of finite linear combinations of the eigenfunctions ei of the problem (2.15). We show that is dense in L 1,(rl). Indeed, is dense in en with respect to the norm (2.17) of the space e". If a:;, d(1/2-p.), then, by Theorem 1.1.1, ex is also dense iu cc._ with respect to the £ 11 (0)-norm. Since C 1J(rl) c ca, the space an is dense in £ 11 (0.), \Vhich implies the density of G= in L 11 (D.). By

(w,;-' e" w,;-

e

ex

ex

_..._ _...._ ----·) ')I ex is dense in lV 7(Sl). Since H'17(D) is dense in T-V1~-- 11 (!1) and 1 w,; c w,;- 2 ;,, (rl), the space e'' is dense in w,;-o;,, (rl). Identifying

Theorem 1.1.5,

ex c e"

c

the space L~(Sl) with the dual space, we obtain the following chain of continuous and dense embeddings:

e'·

c

w,;- ;''(11) c L,(n) c (w;;- ;"(rlJr c e-". 2

2

2

Consequently, the problem (2.11), (2.12) is solvable for .1, :( · · · be the eigenvalues of the operator A. V\.Te set N

Ex= {v LVJCj: v EIR}, =

1

.J=I

where JV depends on the norm

u,

and let the space

vn,

n E iR. be the completion of B.x in

(4.11)

The inner product in i'n is denoted by (·, ·)n· One can show (cf. Lemma 4.5 below) that for ct = 0, 1, 2 the space vn coincides with the spaces (4.6)-(•1.8). By the definition of 1/ 0 \ the operator (4.12)

vn

and -vn-:1. is continuous and defines an isomorphism between the spaces The norms (.4.11) satisi~' the well-known interpolation inequality given in the following lemma. LEwiMA

4.1. Let -co< k

,s; l ,s; rn d/2.

First, we prove the lemma in the case where (except, possibly, one) are positive. Let (4.19)

Pi ;;, 2,

2,;; q < oo,

Lq(fl) if d(1/2- 1/q) ,;; a,

d(1/2- 1/p;) ,;; s;, i

Si

+ Sj > 0 for all i-:/:.

= 1, 2, 3,

1/p 1 + 1/p2

j, i.e., all s:i

+ 1/p:J = 1.

Since L Sj ~ d/2 and all s.i (except, perhaps, one) are positive, there exist finite numbers p1 , j = 1, 2, 3, satisfying (4.19). Applying the Holder inequality for Pj to the right-hand side of (4.15) and using the embedding theorem (4.17), we obtain the inequalities

lb( v., v, w) I ,;; llu I L,,, 1111 II Vvll L,,, ( d/2. Using the embedding theorem (4.18), we obtain the inequalities

lb(u,v,w)l,;; lluiiL,(n)IIVviiL,(D)IIwlll~~ln),;;

Cllull,, llvll,,+tllwll,,

which proves the lemma.

0

Integrating by parts on the right-hand side of (4.15), we find (4.20)

b(u,v,w) = -b(u, w,v).

'l. THE NAVlEH.--STOh:ES EQUATIONS

151

By Lemma 4.2 and (4.20), the form (L1.15) is continuous on the space F·~l x \f 8 :J x \f 82 + 1 if s J satisfy the assumptions of Lemma 4.2. Lemma 4.2 and (4.20) imply the continuity of the operator (4.14) in the spaces

(4.21)

B : vsl

X vs::!+l --;. v-8~]'

(4.22)

B : vsl

X vs~l

--;. F-s::!~]'

where s 1 , s:2, and s:1 satisZr the assumptions of Lemma 4.2. \Ve will use the notation

(4.23) 4.3. Reduction to an abstract equation. Let us show that the problem (4.1)-(4.4) for the Navier-Stokes equations is reduced to the Cauchy problem for ordinary differential equations with operator coefficients

y(t)

(4.24)

+ Ay + B(y)

= .f(t),

y(t)l,~o= 1Jn,

(4.25)

where y(t) = y(t, ·), .f(t) = f(t, ·),A is the operator (4.10), B(y) is the quadratic operator corresponding to the bilinear operator (4.14):

(4.26)

B(y)

a ). =" ( L" vJ. a.;vJ J=l

\\Te assume that

(4.27)

f(t) E

Lg,

Yo E

V1•

We will look for a solution y(t) to the problem (4.24), (4.25) in the space

V 1•2 (Q)

(4.28)

=

{y(t) ELi: i;(l.) E Lg}.

Let f and Yo satisfy (4.27). A function y E V 1•2 (Q) is a solul:ion to the problem (4.24), (4.25) if and only if there exists a jimction p( t, .1;) E £ 2 (0, T; H 1 (rl)) such that the pair (y, \lp) is a solution to the boundary value problem ('1.1)-(4.4). THEOREM 4.1.

PROOF.

It is known (cf., for example, [14 7]) that the embedding

V 1 •2 (Q)

(4.29)

c C([D,T[;V 1 )

is continuous. By the continuity of the operator (£1.21), for s 1 = s 2 = 1 and s:1 = 0 we have ·T

(nO)

IIB(y)lllg,:; C

J,

llv(t)lliiiY(t)lli dt,:; llvlli'·([n.TJ;F'IIIvllil,:; Ctllvll~n.'(QJ·

Let y E V 1 ·2 (Q) be a solution to the problem (4.24), (4.25). The solution y satisfies (4.4) in view of (4.25) and satisfies (4.2), (4.3) in view of (4.29) and the definition (4.7) of the space V 1 • Taking into account (4.24), (4.27) and the definitions (4.10), (4.26) of the operators .A, B respectively, \Ve find

fJ + (y, \l)y- ui'>.y = !i + B(y) + Ay +(I- rr)((y, 'V)y)- ''(I- rr)i'>.y =

f +(I- rr)((y, 'V)y- ui'>.y),

:.1.

IG'2

THE SOLVArHLITY OF UOL::-.JDAUY V,\LC:E

PHOJJLf~l\IS

where I is the identity operator. Since y E V 1 ,0 (QJ C £ 2 (0, T: H 2 (rl)), we have -vD.y E L,(O, T; L0 (fl)), Since

yE

V""(Q) C {: E L 0 (0,T;H 0 (fl)):

zE

D2 (0,T;L 2 (rl))},

we prove that (y, \i)y E L 2 (Q) by the same method as in Lemma 4.2 and (4.:30). By the Weyl decomposition (4.9), there exists p E £ 2 (0, T; H 1 (rl)) such that

(I- ;r)[(y, v)u -I/D.u] = Vp(t,:r).

(ci.:J2)

From (4.31) and (4.:32) it follows that the pair (y, \ip) satisfies (4.1). Let (y,p) E 1'1 ,"2 x L 0 (0,T;WJ(rl)) be a solution to the problem (4.1)-(4.4). As was shown earlier, -vD.y E L 2 (Q) and (y, \i)y E L 2 (Q). Applying the operator 11 to both sides of (iLl) and t.aking into account that 71lj = .11: 11f = f: 11(\lJJ) = 0: we conclude that u satisfies (4.24). D LEMMA 4.:1. The solution if lo the pmblcm (4.24), (4.25) satisfies t.he energy

incqna1ily

llu(I.JIIi, +

(4.33)

./o!' IIY(T)IIi dT,;; llunlli> + ./o!' llf(T)II',l dT.

PROOF. Taking the inner proclud. (in 11°) of (4.24) \vith y(l), taking into account that ( Ay, if )n = II.~ II j in view of the definition (4.11) of the norm of vn, and using the equalities (B(y), y) 0 = (B(if, y), if)u = 0, which are valid because of ('1.15) and (4.20), we find

1 d

'I

'I

2 dt llu(tlllo + llu(tJII; =

(.f(t), v(l)),;;

1

'I

1

.-,

:zll.f(tJII:..l + :zllv(IJII;.

Integrating the left- and right-hand side of the last relation with respect to t and taking into account (4.25): we obtain the inequality (4.33). D REMARK 4.1. It is known (cf., for example, (49] and [138]) that the boundary value problem (4.1)-(4.4) can be reduced to the Cauchy problem ('1.24), (4.25) considered in spaces of smaller regularity than those in Theorem .:1.1. \Ve do not formulate these results because we will not nse them later.

4.4. An estimate for the trilinear form. \Ve \Vill prove an estimate for the trilinear form b(n, 'I.J, w) \vhich generalizes Lemma 4.2. \Vc will need this estimate in Chapter .:1. \:Ve begin by estimating the product of two periodic functions. \:ve consider the torus Tc:t {:r = (:r 1 , ..• ,:rrt); -a/2:::;; .rJ ~ n/2, :r.i = -a./2 is identified with .r1 = n/2, j = 1, ... , d}. A function 1/!(:r) defined on the torus ·~~~is naturally identified \Vith the periodic function ·1/'(:t:t, ... ::r11) of period a with respect to each variable :rJ, j = 1: ... , d. The function -~~~ can be expanded in the Fourier series

=

where

z:; = and

{~= (~,, ... ,~,): ~./ = 2rrkifa, kj is an integer,}= l, ... ,d},

·L THE NAVIEI\--STOh:ES E:QUATlONS

are the .Fourier coefficients of the function lj;. The Sobolev space J-J'S(Td), s E J!:ll, of functions on the torus is equipped 1vith the norm

11~~11;

(4.34)

=

L

(1

+ l~l 0 ri0(0I 2

~Ez~;

It is easy to show that the norm (4.34) is equivalent to the norm (1.1.3) of the Sobolev space H'"J(T1~1 ) introduced in Chapter 1, §1. LEl\IT\-IA

~1.4.

Let- n ): 0.

There e:rists a constant c

>

0 independent of r.p E

H 3 (T") nnd 1/' E H 0 (T") such that (4.35)

ifeither/3> o, 'I> o, ;3-i-"1? a+d/2 or(3? n. 'I? n, ;3-i-'1 > n+d/2. PROOF. Let (3 > a ? 0, 1 > a, and (3 +'I = o + d/2. From (4.34) and the inequality (1 +1~1°)' ,o; c[(1 +I~- '71')-" -1- (1 + 1'71')'], where s ? 0 and cis independent of (, 17 E z:~, we find

(4.3G) where

\Vc consider the operator

L e'nand (3- n +'I= d/2). By the Parseval identity, the relation (4.37), and the Sobolev embedding theorem, \Ve have

SimilarlJ', we can prove the inequality (4.39) From (4.3G), (4.38), and (4.39) we obtain (4.35). Let f3): o:, 7 ): o:, and ;3 +~I> o: + d/2. It suffices t:o consider the case \Vhere either f3 = et or ~~ = Ct. Let, for example, 1 = n. Then ;3 > d/2. By the Young inequality,

Let a= 'I< d/2. As in the case (4.38), we find (4.41)

J,

,o; cll,lu\cpiiL, ,o;

c, IIA'PIIc~;o--, 111/>11-, ,o; c, II'PILJII 1/2, from the Cauchy inequality we have

IlL:

(u(T)- u,(,))

'"II, ~ L:

~ u~: (I + ITI'J _, rlT

.l

cllu- u,llti(., ..Y.l')

0.

~

_,

llu.(T)- u,(,)lh

+ IT I' r IIU:(T) -

(1

(1

+ ITI'fi'(I + 1,1')-'i' dT

u, (T Jill

dT))

I/O

This relation and the condition (cJ.54) for u, imply that u satisfies (4.54) and, consequently, belongs t.o lF0 (s, X, Y). Since H·(-J{.'i', _y, Y) is a closed subspace, the quotient space W(s,X,Y)/H~ 1 (s,X,Y) is a Banach space (cf., for example, [6]). The mapping (4.51) generates the mapping (L55)

in a natura 1 \Yay. The mapping (4.55) has the zero kernel and is surjective because the mapping (4.51) is surjective. By the Banach theorem on the inverse operator, 1 is an isomorphism. As in .J.2, we denote by F" the completion of Ex in the norm (~!.11) for a:= k, where k is a natural number and V 1' is equipped with the norm 11·1[,., from (4.11), whereas V" is the set V" with the norm II · IIH'(ll)· By part (a) of the proof of the lemma, the spaces 1'1' and V" are the same space \Vith equivalent norms. Consequently, W(s, V 1 , V 0 ) = TV(s,

lV(s,

V 1·,

0

V )/Hi1(s, V

1

,

V

0

)

=

v'·, V

0

),

W(s, V", V )/Wo(s, V1'', V 0 ), 0

and the norms of these spaces are equivalent. Since the operator (4.55) is an isomorphism, the spaces [Fk 1 V0 ]I;(:.:s) and[\!", F 0 ]t/(:.!sl coincide and the norms of these spaces are equivalent. From the definitions it. follows that (4.56)

' [1;1• li"J l/(..:!e) 1

= ]fk(l-1/(2.;)). 1

moreover, the space on t.he right hand-side is equipped \vit.h the norm (4.11) for cr = k(!- 1/(2s)). As was shown in [114], the norm of the space [\1", V0 h;p,) is equivalent to the norm of the Sobolev space H 1•(l-l/l 2·'11(1l). To prove the assertion of the lemma for an arbitrary noninteger positive number a, we take any natural number 1.: >nand finds from the equality o = k(I-l/(2.s)). \Ve haves = [2(1- n/k)]-l > 1/2. Therefore, the above arguments prove the req11ired Assertion.

-!. TI-lE K:\VIEH STO!\ES EQUXrlOI\S

l:i7

:3. 'We consider the case a E (-1/2,0). Since C!)"(\1) is dense in FJ-'(\1), .s E (0, l/2), we have H·'(\1) = Hi!(\1) for s E (0, 1/2). Therefore, (4.57)

llvlllf-'("J =

sup

(u,tp)( 3/2, p = 2. Take ·r = 3/2. Using (5.16) instead of (5.15), we obtain estimates similar to (5.20)~(5.24). These estimates imply (5.14). D RE:rvlARK 5.1. The set Fy 0 is every·where dense in some other topologies as well. For example, in [131] it is proved that for Yo E 11·1 the set Fyu is everywhere dense in the topology of the space £,(0, T; Lq(r1) 3 ) provided that s, q E (1, oo) and 4 < 2/s+3jq.

Lg.

REI\·! ARK 5.2. The set F 110 is open in This fact follows from the \Veil-known results on the continuous dependence of solutions !IE 17L::!(Q) to the Navier-Stokes equations on the initial data. (.f,yu) E L8 x V 1 (cf., for example, [94] and [147]).

li. CO.ivB-IENTS

lfl:~

6. Comments The density in C(ro) x C(ru) of the Cauchy data for which the Cauchy problem for the Laplace operator is solvable is established by Mergelyan [120]. The proof of Theorem 1.1 fork+ 1 = 0 based on the Hahn-Banach theorem was obtained by Lattes and J .-L. Lions [100], whereas the constructive proof fork+ 1 = 0 belongs to Romanovich and Fursikov [126]. The general ca.-,e of Theorem 1.1 was established by Fursikov [62]. Results on the approximate controllability· of parabolic equations "\vere first obtained by Fattorini [40] and J.-L. Lions [109]. Recently, some results concerning the approximate controllability of the Stokes system were obtained by J.-L. Lions [111], Fursikov and Emanuilov [65H67], Diaz and Fursikov [25], J.-L. Lions and Zuazua [116]. The approximate controllability of semilinear parabolic equa.itons with nonlinearities satisfying the Lipschitz condition is studied by Fabre, Puel) and Zuazua [38, 39]. The results presented in 2.1-2.4 on the approximate controllability are an adaptation of the results of Fursikov and Emanuilov [66, 67] to the case of the heat; equation. The results described in §3 were obtianed by Fursikov [62]. The results of §4 are well known (cf., for example, [49], [108], [147], and [138]). Lemma 4.4 is proved in [55] and a special case of Lemma. 4.5 can be found in [147]. The material of §5 follows F\1rsikov [51]-[53].

CHAPTER 4

The Problem of Work Minimization m Accelerating Still Fluid to a Prescribed Velocity In this chapter, \Ve study an optimal control problem for the flo\v of incompressible viscous fluid in a bounded domain n. The role of a. control is played by the density of external forces defined in the entire domain 0. Using a control of the above type, we must acceleta.te the still fuid up to a prescribed velocity performing minimal work. In this chapter, we investigate this problem in detail. In particular) we analyze the existence of a solution to this problem. The existence of a solution depends on certain relations bet'\veen the data of the problem. The investigation depends on whether the problem is regular (the notion of the regula.rity of a problem is also introduced in this chapter). The problem of work minimi:za.tion for the Navier~Stokes equations is regular in a two-dimensional domain. \Ve derive necessary and sufficient conditions for a minimum, prove the uniqueness of a solution for large values of the parameter J\f (an upper bound for the £:2-norm of the control), and construct the asympt.otics of the solution (y, u) as A/ .......,. CX\ where ,11 is the state function and u. is the optimal control. The problem of work minimization is a problem with rigid control. To solve this problem, we introduce a certain aw{iliary optimal control problem "\Yith compromise cost ftmdional, ·which is also studied in detail in this chapter. 1. Statement of the problem and the existence theorem

In t:.hi:::; section, we formulate the problem of accelerating still Huid t:o a preficribed velocity v(:r) performing minimal work; we also formulate mathematical analogfi of this problem. The resulting optimal problem is a problem with rigid control. Therefore, in order to guarantee the solvability of the problem, we impose a constraint on the control (which is the density of the external forces) (d. Chapt.er 1. §5). Then we study the existence of a solution to this problem. The existence of a solution depends on relations between the data of the problem. \Ve also study the solvability of a similar problem with compromise functional. 1.1. Statement of the problem. \Ve describe the problem of Work nummization in accelerating still fluid to a prescribed velocity. Assume that a fluid of cln.ss and the occupies a bounded domain fl c !Rrl' d ):: 2, with boundary velocity y(t, :r) is given by the Navicr~St.okes equations

an.

(1.1)

:W, :r) + (y, \i)y- D.y + \ip(t, :r)

(1.2)

div !J = 0,

( 1.3)

Yl,;= 0,

( 1.4)

ult=o= o, J(jfi

= u(l., :r),

c=

lOG

"l. \VOHK 1\:IINfl\.fiZATION IN ACCELERATING STILL FLUID

where \lp(t,.r) is the gradient of pressure, u(t,.1;) is the density of external forces, which, in the case under consideration, is a control, I: = (0, T) X denotes the lateral surface of the cylinder Q = (0, T) x rl and T > 0 is a given time moment. \\le assume that we can act on the fluid b:y volume forces of density u.(t,.1;). It is required to accelerate the fluid from the still state {1.4) at the moment t = 0 to the prescribed velocity v(.r) at the moment T:

an

(1.5) performing minimal work. To define the functional to be minimized, we write the work of volume forces of density u(t,x) for the motion of the fluid with velocity y(t,x). Using the relations (1.1)-(1.5), we compute ·T

j Jnr(u(t,x),y(t,x))dxdt u

1'

(L6J

.

f I (il + (y, v)y- !:>.y + vp, yJ d:c dt

Jn ./n = -l ~· 2.n

lv(J:)I'd:r +

iT l I'Vul'dx

.o.n

1 dt = -llviiG 2

+

;,·To lly(t, ·)llidt,

where IIYIIi = IIVYIILull is the norm (3.4.11) for a = 1. We use the notation introduced in Chapter 3, §4. The mathematical formalization of the problem of work minimization in accelerating the still fluid to a prescribed velocity has the form

,.

(1.7) ( 1.8)

Jl (y) =

G(y) =

r llu(t)llidt _, inf,

lu

y + Ay + B(y) = u(t),

(1.9)

"'loY= 0,

(1.10)

"YtY = v,

(1.11)

{T llu(t)llgdt

lo

~ M',

where M > 0 and v(x) E V 2 (rl) are the data of the problem (we write the NavierStokes equations in the abstract form as in Chapter 3, Subsection c1.3.) REMARK 1.1. Since (1.7)-(1.11) is a problem with rigid control, it is necessary to introduce the constraint (1.11) on the control u if we want to obtain the solvability of the problem (in accordance with the results of Chapter 1, §5). We will show that the problem (1.7)-(1.10) (without the constraint (1.11)) actually has no solutions. To understand better certain mathematical patterns related to optimal control problems it is useful to consider, instead of the problem (1.7)-(1.11), the following more general problem: Find a vector field (y(t,:c),u(t,x)) satisfying the conditions (1.8)-(1.11) such that T

(1.12)

ln(Y) =

{

.fu

liy(t)ll~dt _, inf,

l. STATE!'v!ENT OF THE PH.Ol3LEl'vf AND THE EXISTENCE TI-IEOR.E?\.f

1U7

where o E [0, 2] is a fixed number and v in (1.10) is a given vector field such that

(1.13) In addition to the cases of the domain the case d = 4:

(1.14)

2

~

n of dimensions d =

2, :3, we 1vill also consider

d = dim n ~ 4.

Besides the problem (1.8)-(1.12), we also consider the following extremal problem. It is required to minimize the functional

J(y, u.) =

(1.15)

j·To i!y(t)ll~dt + N ./o(

llu(t)1!6dt _, inf

over all (y, u.) satisfying (1.8)-(1.10), where v E V'(D) and N > 0 are given. This problem is amciliary with respect to the problem (1.8)-(1.12). However, it is of independent interest.. Recall that functionals of the form (1.15) are called compromise functionals. They often occur in the study of optimal control problems. The problem of minimizing the functional (1.15) can be regarded as the problem of minimizing the functional (1.12) under the condition that the integral llu(t)lludl. is "not too large." The value of this integral is controlled by the parameter JV.

J:·

1.2. The existence theorems. Using Theorem 1.2.4, 1ve study the existence of solutions to the problems (1.15), (1.8)-(1.10) and (1.8)-(1.12). We set

U

(1.16)

= Lg,

Y1

= {y E L~ n Li : iJ E L:Jd, loY= 0}.

THEOREM 1.1. For v E V' and N > 0 there ezists a solution ('if, u) E Yi to the problem (1.15), (1.8)-(1.10), where }'i and U are the spaces (1.16).

X

u

PROOF. \Ve use Theorem 1.2.4. Assume that

Y =La'

F

=

D"id

X V(l-d)/:!'

Y1 and U are the spaces (1.16), Ua S

=

U,

L(y, u)

=

(y

+ Ay- 11, /TY),

= {s(t) E C 1(0, T; \fd+J) : s(O) =

s(T)

F'(y)

= (B(y), -u),

= 0} x

j!(d-ll/',

Y-1 = L~, and J is the functional (1.15). To prove the theorem, it suffices to veri!}' Conditions 2.3, 2.4, and 2.5 in Chapter 1, i.e., the conditions of nontriviality, coercivity, and compactness. \Ve set

The pair (y, u) satisfies the relations (1.8)-(1.10). Therefore, to prove the nontriviality condition, it suffices to verify the inclusions

vt

y= T EYI,

v vl (vt) ELg.

u=T+AT+B T

The first inclusion is obvious and the second follows from (1.13), (3.4.13), and (3.'1.21). If (y,u) E A11 = {(y,u) E !2l: I(y,u) ~ R}, where 2l is the set of admissible elements of the problem (1.15), (1.8)-(1.10), then

(1.17)

IIYIIi"'' ~ R,

llullig ~ R/N.

·!. WOHK l\IINJlviTZATIOI'\ IN ACCELEHXl'lNC: STiLL FLCID

HlS

By the energy inequHlity (3.4.:33), we have

(1.18) Expressing iJ from the differential equation (1.8) and estimating the right-hand side of the obtained equality using (1.17), (1.18), and (3.4.21), we find

(1.19) By (1.17)-(1.19). wee conclude that. the coercivity condition holds. If s = (s,,s2) E S: y E Yj: then Lemma 3.4.2 and the rela.tion

(F(y),s) = (B(y,y),s,)q + (v,s2)o = -(B(y,s,),uh!J + (v,s2)u imply that the functional y --+ (F(y), s) is extended b}·· conLinuity to the space Y~1 = Lg. The compactness of the embedding Y1 C Y~b where Y __ ! = L8 and Y1 are the spHces from (1.16), is well known (cf. [26] and [147]). Thus, we have verified all the assumptions of Theorem 1.2.:1. From this theorem we obLa.in the required assertion. D We pass to the study of the existence of a solution to the problem (1.8)-(1.12). The characteristic feature of this problem is the constraint u. E Ua. In this case 7 Uo = {u E Lg : II u II L!i ~ M} is the ball of radius M. The existence of a solution depends on a certain }·elation between the data v and lH. THEOH.El\.1 1.2. There c:rists a posil'ive numbcr1\Io = Afo(v) such that fm· AI> Mn(u) the pmblern (1.8)-(1.12) ha.s n. solution (fj, ii) E 1·, xU, whc1-e Y, and U are the spaces (1.16), whereas forM< AI0 (v) the problem (1.8)-(1.12) has no solutions. PROOF. All spaces 1 operators, and so on (except Ua and Jn) are introduced in the same way as in Theorem 1.1. \:Ve set Ua = {u E Lg: \luiiL!: ~AI} and .J = lr_.., 1 ·where .In is the functional (1.12). Fix a vector field v E \f'2, 'I~ # 0. \Ve prove the following assertions: (a) for a sufficiently large JH the nontriviality condition holds for the problem (1.8)-(1.12); (b) for a suHiciently small M > 0 the nontriviality condition fails for the problem (1.8)-(1.12). (a) For a sufficiently large AI the pair

ut v (y(t), n(t)) = ( T, T

I.

+TAo+

2

1 ) T" B(v)

belongs to the set 2t of admissible pairs; this can be established in the same 1.vay as in Theorem 1.1. (b) Assume t;he conLrary1 i.e .. for each small 1\J > 0 there exists au admissible pair (.IJM,uM) E 2l for the problem (1.8)-(1.12). It. is known (cf. [147]) that for a sufficiently small 1\I = II"·MIIL~ there exists a unique solution w E LJ to the problem -

w+Aw+B(w)=aM(t), moreover,

/_L,

wl,~ 0 =0,

E £~. In addition,

(1.20)

where cis a constant independent. of AI E (0, M).

IE[O,T];

1. STATBivlENT OF TI-lE PHOBLEJ\1 AND THE EXlSTEKCE Tl-IEOHE2\'l

B)· Lemma 3.5.1 on the uniqueness of a solution, we have YM (t) (1.20) and the trace theorem, we arrive at a contradiction:

0
l(QT) n £ 2(0, T; V 1\(S1)) be a solution to the adjoint equation (5.11). Then the

G. OF'Tf?dAL C'ONTHOL FOfi- NONSTATIONAHY FLUD FLOW

distribution C..w defined in Cff'(O)nV 0 (0) can be e:ctendcd to a .functional defined by the formula

(5.31)

L

C..w- hdx =

_Ia,, (('vw) + (\lw)'1')n- hds- 3

L

V(w): V(h.) dx

fo·r a:ay h. E ex (G) vanishing in a neighborhood of (0, T) x (88 \ 80). Furthermore, the evoluUon ·version of (5.31) a.lso holds, i.e.,

I. 1(-:l/ T

,o

(.5.32)

C..w · hdxdt =

~-

. (I

T ;· •

nn '1'

-2 /

lo

((vw) + (\lw) 1 ')n- hdsrll: •

f

it~)

V(w): V(h.) dxdt

for· any h E c=((O, T) x G) vanishing in a neighborhood of (0, T) x (88 \ 80). F'orrmtlas (5.31) and (5.32) renwin va.hd 'If we replace w with q.

PROOF. For w = wU-) E Lx (0, T; V 2(8)) the equality (5.32) is the well-known Green formula (for the proof see (1.5) and the subsequent formulas). Substitute in (5.32) the solution wll:) to the problem (2.1). (5.23) constructed in the proof of Lemma 5.3. By this lemma, wll:) __,win Vi 1 1(QT)- Therefore, ~-

·T

ju lerV(wlkl): V(h)dxdt __,I, n ./erV(w): V(h)dxdt,

1.: __,

CXJ.

B;y Theorem 3.4: w f-+ '"'I(C'Vw) + (\7w)'1}n is continuous as an operator from the set {wE Y: w satisfies (2.1)} to the space L 1 (0,T;B;; 11"(80)), 1 < o < 2. Consequently, T

1 ·T

t r ((vwlk)) + (vwl"lf}n. hdsdt __, lo lan .u

_;.,·,-,((vw) + (vw)')n. hdsdt

as k-:. oo. Substituting w = w(k) in (5.:32) and passing to the limit on the righthand side as 1-.: -:. CXJ 1 we obtain the required extension of the distribution ~'W defined by formula (5.32). Formula (5.31) is proved in a similar way. The case of the distribution ~q is treated similarl,y. D 5.3. The strong form of the optimality system. We write the optimality system (2.1), (2.3), and (5.1) in the form of a system of partial differential equations "\Vith boundary conditions on the boundary of the cylinder QT = (0, T) X n. Recall the notation in !32: V 0 (0) = (v E (£ 0 (0)) 0 : divv = 0},

v::(o) = the closure of (C;-;'(0))" n V 0 (0) in the norm of (£ 2 (0))". We will use the well-known Weyl decomposition (cf. [94] and [138]) (Lo(0)) 0 = V:i(O) 8 (vH 1 (0)), where vH 1 (0) = {\lg : g E H 1 (0)}. We have Vu(O) = V~(O) ·Jl (vHrr). where YHrr = {\lg: g E H 1 (0), C..g = 0}. Indeed, by the Weyl decomposition~ for every 'W E V 0 (f2) we have w = Wa + \7w ... , where Wa E Vg(f2) and w" E H( (!"1). Taking the divergence of 'W, we find that ~w ... = div w-div Wa = 0. Now we can interpret the weak form of the optimality system as a system of partial differential equations with boundary conditions on the entire boundary of the cylinder QT = (0, T) x 0.

5. OPTTI\IALITY SYSTEJ\·1

THEOREM 5.2. Lei w E Y be a solution to Pmblem 2.1, and let q E V(l I ( Q·r) be the vector field defi:ned in Theornn .5.1. Then there e:rist p E L, (0, T; Lh'"(fl)) and r E L, (0, T; LU'' (5:1)) such that the tuple (w, p, q, r) satisfies the .following partial differential equations (·in the sense of distribut:ions):

w- C.w + ((w + v=) · 'V)w + 'Vp =

(5.33) (5.34)

0 in (0, T) x 0,

\7 · w = 0 in (0, T) x 5:1,

-q- C.q + q · ('Vw) 1

(5.35)

-

(w · 'V)q- (v= · 'V)q + 'Vr = C.w- 'Vp,

'V . q = 0 in (0, T)

(5.36)

n,

X

the ·in·itial and .final conditions:

w(O, ·) = wo(-) in V 0 (5:1),

(5.37) (5.38)

q(T, ·)

+ ~wu(T, ·)

V~(rl),

= 0 in

where Wa is the pmjection of w(T:.) on vg(O) ~ and the bmmdary con diU on on the lateral surface of the cyl-indel'

(5.39)

AIm·eo uer:

yw E Ll(O,T;B:;'/"(85:1)),

(5.40) ~t[((Vw)

(5.41)

+ ('Vw)~") ·n]

E Ll(O,T;B;;- 11"(85:1)),

/P E L 1 (O,T;B,:;- 11"(85:1)),

(5.42)

;· p = 0, r)ll

(5.43)

~t[(('Vq)

(5.'14)

T~"

+ ('Vq) 1 ) · n]

E Ll(O,T;B:;'/"(85:1)),

E L 1 (0, T; B,:;-'1"(85:1)),

;· rds

=

0,

' r)il

where 1

(5.45)

< o: < 2

1

and the following bounda:ry conditions hold:

2NTW- A(w)- T(w,p)n- T(q, r)n

= ~(t)n,

where (.5.46)

(5.'17)

T(w,p) =-pi+ 2V(w), A(w) = /" ( !:lY.1 w

T(q, r) = -ri + 2V(q),

1 ( (w + v=) · n ) w + 4n lwl' ) , + v= I'" --.,( w + v= ) + 2

r1(t) = - ;· A(w). nds/

(5.48)

au

f_

f-m

ds.

Aforeover. lhe following compat.ibilily conditions hold: (5.49)

(5.50)

bw) ·

rl,~,= o,

G''"" + 2N~tw n)) ~~~T= ·

o,

where T is the unit tangent vector to [)[). and 'Wr. ( (: ·) is the primitive function of the pmjection of w(t, ·) on \7 H, determined on GD. by the condition

(.5.51)

;· w"(l,·)ds=O . . on

G. OPTil\'iAL CONTROL FOB. NONSTATIONAHY FLUID FLOW

PROOF. Since wE v(li(QT) is a solution to Problem 2.1, it satisfies (2.1) and (5.37). As was shown in (3.27), from the De Rham lemma it follows that there exists p E £ 2 (0, T; Lb0 '(Sl)) such that (5.33) holds. The relation (5.49) follows from the inclusion w E Y and Remark :u. By (5.12), for the adjoint function q E V(ll(QT)nL,(O,T;VG) there exists r E L 2 (0,T;Lb"'(Sl)) such that (5.35) holds. The relations (5.34) and (5.36) follow from the inclusions wE vOI(Q·r) and q E V(ll(Qr), whereas (5.39) follows from the inclusion q E L 2 (0,T;VA(Sl)). By Theorem 3.5 and Lemma 5.1, the traces of the functions p and r belong to the space L 1 (0,T;B,-; 11 "(8Sl)) for 1 0 is independent of :r: E TI. On a part I' 1 of the boundary of the domain condition

(1.2)

D, .., u(:r')

+ iJ(x')u(x')

n,

E

lffi.d,

we impose the boundary

:r' E r~,

= 0,

where [3 E c=(r,) and D,.,u(:t:') is the conormal derivative of A, i.e., d

(1.2')

D,"' u(.T') = ~ a.;,(x)n;(:r:')D;u(:r:)L,="·" i,j=l

where n(:r:') = (n 1 (:r:'), ... ,nd(x')) denotes the vector field of outward normals to In addition, on r 1 we impose the condition

r 1.

(1.3)

1 ' r,

'[' ,

[u(:r)- uo(x) -dS < o-,

where tto E L2(I't) is a given function and t: > 0 is a given number, \Vhich, according to the meaning of the problem, is assumed to be sufficiently small. In formula (1.3) and below, ciS denotes the volume element on the manifold over which the integration is taken (we assume that all manifolds considered belmv are embedded in IP:.rl; therefore, the volume elements dS are defined.) On the manifold f 2 , we impose the condition

(1.4)

f

Jr".!

[u(x')i"rlS

~

M 2,

where ]\f > 0 is a given number. We also impose the following additional conditions on the problem (1.1)-(1.4). CONDITION 1.1. The domain n is diffeomorphic to the cylinder r X [0, To], where '1\) > 0 is a. number and r c ~d is a closed (d- 1)-dimensiona.l manifold of class c=.

The spherical layer{~· E JR:"; r 1 < [:~:[ < ro} is a typical example of a domain satisfying ( 1.1) Let I'' c n be a. manifold that is diffeomorphic to r 1 and divides the domain n into parts n' and r!" such that an' = f 1 u f' and an" = f' U r:]. On f': we require the boundary condition (1.5) to hold.

n(.T') = 0,

:r' E

r'

!. STJ\TEI\JENT OF THE PHOBLEI\1

~:~;:;

CONDITION 1.2. The boundary value problem (1.1), (1.2), (1.5) has only the zero solution in L, ([l).

Let

(1.6)

u(:r:') = 0,

:r:' E f

1

U

r,.

The next condition is close to Condition 1.2. CONDITION 1.3. The problem (1.1), (1.6) has only the zero solution in L,(rl). The following condition postulates t:he solvability of the problem (1.1)-(1.4). It is typical for ill-posed boundary value problems. 11

CONDITION 1.'1. Given uo E L,(f:), M > 0, and£> 0, there exists a function E H 1i 2 (rl) satisi'ying (1.1)-(1.4). As before, H'(G) denotes the Sobolev space (cf. Chapter 1, §1).

RE:tviARK 1.1. In the problem of data interpretation in the electrical prospecting, f 1 is the surface of the earth and --u(x) is the potential of an artificially created electrostatic field inside the earth (in rl). In this case, equation (1.1) has the form C.u = 0, where C. is the Laplace operator, and !:he boundary condition (1.2) has the form al!u(J; 1 ) = 0: where n.(J:1) is the vector field of outward normals to rl. Denote by un(J:) the value of the potential u(:z:) measured on the surface of the earth. lvleasurements provide only an approximate value of u(J;); this value is given by (1.3), where£ is the accuracy of measurement. The estimate (1.4) for u(x) is known from ph:ysical considerations. Finally, the existence of a potential u(:r:): i.e., Condition lA, follows fron1 the physical statement of the problem. In physical questions leading to well-posed boundary· value problems, for example 1 to the Dirichlet problem

(1.7)

C.u(J:) = 0,

J.· E rl,

the boundary condition uo(.r') is also detennined by measurements and. consequently, is approximate. Hmvever, in this situation, we can ignore this fact in the mathematical statement of the problem considering the boundary condition from (1.7) instead of (1.3). This is possible because the problem (1.7) is solvable and the solution depends continuously on the boundary condition. However 1 such a simplification is impossible for the problem (1.1)-(1.'1) because the restriction to f 1 of any function u(:z:) satisfying (1.1), (1.2), and (lA) belongs to a rather small subset of the space L::!(ri). The function Hu(x) is given as a result of measurements and, consequently, with some error. As a rule, it does not belong to this subset. Passing li"Oln ( 1..3) to the boundary condition in (1.7), we actually pass from a problem that is solvable from the physical point of view to a problem having, generically: no solutions. The formulation of problem (1.1)-(1.4) is significantly different from well-posed boundary value problems of mathematical phy·sics. Therefore 1 it is necessary to specify what. we mean by the solvability of the problem (1.1)-(1.4). The feature of the condition (1.3) is that. the existence of at least. one function satisfying (1.1)-(1.4) usually implies the existence of infinitely many solutions to the problem (1.1 )-( 1.4). Therefore: the precise determination of the required solution~ which is a natural question to be asked about \vcll-posed boundary value problems 1 makes no sense for the problem (1.1)-(1A). However, by (1.3), the restriction tor, of any function

0. COL\"DIT!Ot\ALLY WELL-POSED FOH.i\IULATION

satisfying (1.1)~(1A) is close to u 0 in the L2 (rl)-norm. If this fact implies the closeness of every pair of functions satisfying (1.1)-(lA) in the L~(S1')-norm 1 where Sl' is a subdoma.in of n, then the question of finding an approximate solution to the problem (1.1)-(1.4) in rl' becomes meaningful. In this case 1 everything is reduced to the development of methods of constructing an approximate solution to the problem (1.1)~(1.4). Thus, by solving the problem (1.1)~(1.4) we mean the following. (a) To establish the estimate llwll ~((E), where ((E)-+ 0 as E-+ 0 and 11·11 is the nann over nor a subset of n, the function ((c) is explicitl:y given, w = ul- '11.:2, and U.J, u.~ are arbitrary functions satisfying (1.1)-( 1.4). (b) To propose a constructive method of determining one of the functions satisf'ying (1.1)~(1.4). By (a), this function is close to the required function u(1:) and, consequently·, is an approximate solution. We set = {n E H·'(rl) : A-u E L,(rl)}, where A is the operator from (1.1) and s E [0, 2]. This space is similar to the space (1.1.20) introduced in Chapter 1, Subsection 1.3. 1L is known (cf. [114]) that the operator 'I of restriction to the boundary 8S1 is defined for 11. E H:\; moreover, "/'U. E Iis-l/'2(8S1) and "(8nu E H·~-:v~(8rl), --iVhere 8n denotes the differentiation along a vector field a transversal to Therefore, the restriction of a function u E H'i 2 (rl) satisfying (1.1) belongs to £ 2 (r, ), .; = 1, 2. Consequently, the integrals in (1.3) and (1.4) are defined and the equality (1.2) is understood in the sense of the space H~ 1 (rl).

H::,

ul,,

an.

1.2. Examples. \Ve give examples of boundary value problems satisfying Conditions 1.2 and 1.3. EXAMPLE l.l. We set A = -L!., where L!. is the Laplace operator, and f3C1:) Then Conditions 1.2 and 1.3. hold. EXAMPLE 1.2. We set b1 = 0 and c ~ 0 in the operator (1.1) and;3(x) the boundary condition (1.2). Then Conditions 1.2 and 1.3 hold. ExAMPLE

~

= 0. 0 in

1.3. Let the coeflicients b; and c of the operator (1.1) sa.tis(v the

inequality

.i=l

and the boundary condition (1.2)

(ll(:z:'),n(.,')) + 2(3(x')

~

0 'ch:' E r,.

Then Conditions 1.2 and 1.3 hold. Since Examples 1.1 ancll.2 a.re special cases of Example 1.3, it suffices to prove the validity of Conditions 1.2 and 1.3 onl~'( for the boundmy value problem from Example 1.3. Let u(:z:) be a solution to the boundary value problem (1.1), (1.2), (1.5). Multiplying (1.1) by u.(:r), integrating over rl', integrating by parts, and using (1.5), we obtain the equality

~'[d ai:iB,uiJJu + .In·

_2::

l,j=l

(

2:: a1&J(.T) )u]2 d.,;

c(.T)- :_;:J"

J=l

./r, ~'[·(cJ,_, u)u.- :_;:(b, 1 n)tc''] dS =

0.

:2. t\ 1\IODEL PHOBLE?\'1

Expressing the term Dn . , u. in the integral over (1.1'), we obtain the inequality

J',[:v.t(iJJu)'+ 'll

j=l

r1

from equality (1.2) and using

(c-~tiJJuJ)u']d:r+ { [;3+~(b,n)]u'ds,;;o, .i=l

. 11

which implies a( :c)= 0. Condition 1.2 is verified in a similar way. To conclude the section: \Ve ·write formulas we \Vill need later. Let n E H1 and U) E H 2 (D.). Then the following Green formula for the operator (1.1) holds (cf. [114]):

.l

(1.8)

[Au· w- A"'w · u] d.T

.!:n [(b, n}uw + u.Dur~ w- wD,/.-~ u]rlS,

=

1

where the operator .11"' is formally adjoint to the operator A: d

(1.9)

rl

L iJJ(a,JiJ,w)- LiJJ(brw) + cw,

A'"w = -

i,j=l

j=l

iJ,,, is the conormal derivative defined in (1.2'), and (b, n) = '£ Vjllj· From (1.8) we easily obtain the Green formula for the problem (1.1), (1.2), (1.5): (1.10)

/' [Auw-A'w·u]d:r= /' [u(iJ,,w+[(b,n)+i3]w)-w(iJ,,u+/lu)]dS'

.In'

./rl

+

!

[uDn .. ,w- wDn ..\u

+ (b: n)u.w]dS.

'i''

2. A model problem. Formulation of the theorem on the main estimate In this section: we consider the siniplest variant of the Cauchy problem (1.1)(1.4). Based on this model example: we continue our discussion of conditionally well-posed formulations or the Cauchy problem. At the end of the section: we formulate the theorem on the closeness of functions satisfying (1.1)-(1.4). 2.1. The 1nain estimate for a model problem. \Ve look for a function

u(t, :r) in the strip

I'h

=

{(l.,x): IE [0, T],:z: E !R:}

that satisfies the Laplace equation

(2.1) in the class or functions periodic in

(2.2)

u(t,.T

;T:

+ 27r)

= u(/;,.1:).

The condition (1.2) is replaced with the condition

a,u(t,:z:)l,~o= 0,

(2.3)

whereas the conditions (1.:1) and (1. 0. Therefore, the matrix 8/fi is

(3.30) for any x' E l-1/,, '"" E [0, Tu]. Note that the matrix a depends on the set lh of the covering of the manifold r on which this matrix is defined and 1 consequently, a(:r) = a,(ki(.I). We set

(3.31)

and find :rd(

n from the conditions Drxd(t) >

(3.32)

2D?1:rd(t)au(.Id(t))

o,

Dz,xc~(t) ~

o,

2

+ (D,xd(t)) ar(:rd(t))

~ 0.

We set

a+ (xc~ (t ) ) = max { 0,

a.r(:rd(l.))} ( . ( )) , a.o xd t

a_(xc~(t)) = a+(xd(t))-

ar(:c,(t)) (" ( )) a.o .rd t

and replace the last inequality in (3.32) with the equation

,, . .,ar(:rd) ( ., 2D[1:rd + (D,xd)--(-) =a+ :rd)(D,xd)-. a.o :rd Then (3.33) since :rd(O) = 0. Exponentiating (3.33) and passing to inverse functions, '\Ve find

which implies t(x,J) =

l.

• 0

x,, exp ( --1 2

f.' 0

a_(()d(

) I!'"'' ( j'' d~

' 0

exp - a_(x) dx 1 2 ()

)

d~

(the constant Ddt(O) is chosen in such a way that t(To) = 1). From (3.34) it follows that Ddt(xd) > 0 and D,JA:rc~),:; 0. Hence

Consequently, the transformation :cc~(t) (and t(x. 1·u.hr1S+ { Ahpdx=O

./r'2

./n

.l

Au· pd:r,

U. CONDITJON:\L!,). \VELL-POSED FOIU\"lULATION

where

ll

is a solution to the problem (5.1)-(5.4). By (2.1.52), we have

,\o ;:, 0,

(5.19)

A, ;:, 0,

,\, (bouiiLr;~ - 11I') = 0.

\Ve prove that /\o ::/::- 0. Assume the contrary. Let /\o = 0. By Condition 1.3, for any q E L2(rl) there exists a unique solution hE H 2(rl) to the boundary value problem Ah = q, ~11h = 0, 1oh = 0. Substituting h in (5.18), we find

l,

qp d.r = 0

\fq E L,(fl),

which implies p = 0. 1°\·om (5.18) we obtain the equality Al(-y2u,~l,h)t,(Co) = 0

(.5.20)

!/2

\fh E H4 (fl).

Since the mapping ~1 2 : H;("(fl) ~ L 2(fl) is 1m epimorphism (cf. [114, Chapter 2, §7.3]), from (5.20) it follows that either ,\ 1 = 0 or ~1,'" = 0. The second case implies the first in view of (5.19}. However, )11 = 0 contradicts the Lagrange principle asserting that the Lagrange multipliers /\ 0 , /\ 1 ) and p cannot vanish simulteneousl:y. Thus, /\n #- 0 and \Ve can assume that /\ 0 = 1. For simplicity, we set /\ 1 = A. From (5.18) for h E C1f(rl) it follows that the second equality in (5.12) holds in the sense of distributions. By the Lagrange principle and (5.12L p E G~~l· = {p E H 0 (fl) : A'p E L 2 (fl)}. Therefore (cf. [114]), tlw traces ~fjp E H- 112 (r,;), '"""/j8n . 1 JJ E II-:Jf:l (rJ ), j = L 2, of p arc defined and the following Green formula holds: (5.21)

/" (Ahp- hA'p) d.r =

./n

/"

Jr1ur'2

In (5.21), we substitute the expression of

[hiJ,Ap- piJn., h

+ (u, n)ph]dS.

J Ahpd:c obtained from

(5.18) for h E

H 2 (fl) nH;('(fl). Using (5.12) and the equality ~1 1 iJ,, h = -7I.Bh for h

E

H 1('(fl),

we find /" (iJ,,p-1- [;3+(u,n)]p+ (u- u 0 ))hd8

(5.22)

./r1

+

.!;, [(

iJ,_,p + (b, n)p + ,\u)h- piJ, "' h JdS = 0.

Taking hE H 0 (fl) n H;('(fl) such that 1oh = 1oiJ,"'h = 0, from (5.22) we obtain the second relation in (5.13). The first equality in (5.13) coincides with (1.2). The equality (5.14) is deduced from (5.22) in a similar way. The relation (5.15) follows from (5.19). D COROLLARY 5.1. Let u E H let u 0 E H'' (r 1 ), " ;:, 0. Then u E c=(n

1 0 / (fl), p

uri),

E G~1 .,

,\

E !P: sul.isfiJ (5.12)-(5.15), and

P E c=(n uri) n H''+"i'(n).

If,\> 0, then u E c=(\1). PROOF. By (5.12) and the regularity of solutions to second order elliptic equations, we have u E c= (fl) and p E c=(fl). By (5.12,) and (5.13 1 ), the function u(:z:) is infinitely clifl"erentiable up tor,, that is, u E c=(fl U r, ). This inclusion, (5.12o), (5.l:lo), and (5.141) imply that p E c=(fl U ro) n IJ''+"/'(fl). If~\ > 0, then the inclusion p E c=(fl U r,) and the relations (5.12t), (5.131), and (5.14 2 ) imply that II E c= (TI). D

li. THE CONSTHL:CTlON OF A QU.-\SISOLFTlON

Using Corollary 5.1, it is easy to establish the uniqueness of a. solution to the problem (5.1)~(5.4). PROPOSITION

5.2. A soi'ution o E H /'(rl) to thepmblern (5.1)~(5.4) is Hnique. 1

u,

u,

PROOF. Let o/o be solutions to the problem (5.1)~(5.4). By Theorem 5.2 and Corollary 5.1, we have u.i E c·x(o U I' 1 ). By the CalderOn theorem on the uniqueness of a solution to the Cauchy problem (cf. (121, Chapter -1, §2]), \Ye have 11"1 ofo /'lu,. It is obvious that (u, + u,)/2 E 2l (cf. (5.6)). Since the squared Hilbert norm is strictly convex, we have

'l

ll(l',u, +i'Juo)/2- ·unllr.,(r, 1
0 and increase ,\otherwise. The iteration process terminates when J1J'2 -ll"/2ui1L(r:!) = 0. 1

7. Properties of a quasi-inverse operator 7.1. Some applications of a quasi-inverse operator. By a qururi-inverse operator of the problem (1.1.)-·(1.4) we mean the operator Q that. assigns to an initial condition u 0 a quasisolution -u to the problem (1.1)~(1.4), i.e., a solution u to the problem (5.1)-(5.4):

(7.1) In this section, \VC study properties of the operator Q. Recall that the problem (1.1)-(1.4) was investigated under Condition 1.4. However: this condition is not used in the construction of quasisolutions in §§5 and G.

7. PIWPEH.TIES OF :\ Qt.ASI-INVEH.SE OPEIL·\TOR

::!(il

A quasi-inverse operator can be used for the verification of Condition 1.4. Indeed, the following assertion holds. PROPOSITION 7.1. Condition 1.4 holds

if and

only

if

(7.2) PROOF. If (7.2) holds, then a quasisolution satisfies (1.1)-(1.-1) and, consequently, ConclitionlA is satisfied. The converse assertion was proved in Proposition 5.2. 0

Note that Condition lA is not: satisfied for all Vo, J.U, and£. However, for any ~ Mu the triple u 0 , AI,£ satisfies Condition lA. This assertion is true because the set of functions '"'( 1 u, where u. E H 1i 2 (f.l) satisfies (1.1). (1.2), is dense in H 11 (f!). In the case where A is the Laplace operator and ;3 = 0 in (1.2), this result was proved in Theorem 3.1.1. However, the same arguments remain valid for the general problem (1.1), (1.2). Using a quasi-inverse operator, we can deduce an estimate that is more precise than (2.28).

E

> 0 and u 11 E H 0 (fi) there exists 1110 such that for each J\1

PROPOSJTION 7.2. Let (7.2) hold and let EJ > 0 be detennined by the formula 2

-II"IIQuo- uuiiL(r,J = Ej. for w = u- Q·u 0 , whe'l·e u E J:[Ii 2 (f.l) E

Then the estimate (2.28) remains val'id after replacing 2£ with PROOF. It

satisfies (1.1)-(lA),

E1-

suHices to prove the inequality

(7.3)

Consider the set /I2l = {·1 1u, u E Ql} in the space J:Ill(rJ), where 2l is the set (5.G). The set "/12t is convex. Arguing as in the proof of Theorem 5.1, it is easy to show that /I2l is closed in H 0 (ri). From the definition of the operator Q it follows that '"'/t Qu 0 is a minimum point in the problem infve11 ·x \lv- unll;-fo(r!l· Consequently, -II Q is the orthogonal projection from H 0 (r I) \•1 12l onto "ti2l. Hence ("/!'Ill: Uo- "(!Qu.o)LC!(l'rl ( 0. By the S(Xoncl cosine la-w,

lhiwll\',(r, 1 ,;;; lluo- ~ll"IIL(r, 1 - lluo- ~II QuoiiL(r, 1 2 ,;;; E -lluo- "IJQuoll' = cj. This completes the proof.

0

7.2. The continuity of a quasi-inverse operator. \Ve discuss the properties of the quasi-inverse operator Q that are useful in the numerical construction of quasisolutions. First of all, we mean the continuity and the regularity of Q(u. 0 ). Let BM =

{u E H 0 (f"): llviiH"(r·,J < J1J}.

\Ve consider the set RB;H, where R is the operator (6.7). Pn.OPOSITION

7.3. The opera loT '"1' '1Q(un) is discontinuo·us at Ho E REM.

G. CONDITION ALL'{ \VELL-POSED FOH.I\-IULATION

PROOF. Let ·uo E REM. By the definition (G.7), (6.1) of the operator R, the problem 'It (8, .., u + (Ju) = 0,

Au.= 0,

'ftU

= uu

has a solution u E c=(O); moreover, II'Y2 1iiiH'(l',J < M. Since the triple (u,p, ,\) = (u, 0, 0) is a solution to the problem (5.12)-(5.15), we have u =Quo. Consequently,

(7.4) On the other hand, for any

{u

E H

0

E

> 0 we have

(f,): llo- ouiiH"(r,J < c} \ RB,u

io 0.

0

Therefore, there exists a sequence vJ E H (r 1 ) such that (7.5)

vJ __, u 0 in H 0 (ft);

Vj vJ if_ RBu.

Therefore 1 (7.6) From (7.4), (7.5), and (7.6) we obtain the required assertion.

D

Unlike 12Q, the operator -y 1Q is continuous.

PROPOSITION 7.4. The following inequality holds: (7.7) PROOF. As was shown in Proposition 7.2, 11Q is the orthogonal projection operator from H 0 (ft) \'tl!2l onto 'Y 1!2l. Therefore, the estimate (7.7) holds (cf. [93, Chapter 1, §2]. D

Denote by 'Yl+t• t E [0, 1] the operator of restriction of u E H;(2(0) to the manifold Xt:

'Yt+t: H;('(O) __, L 2(x1 ), where x 1 is defined by formula (2.27). THEOREM 7.1. For any t E [0, 1] the operator· 'lt+tQ : H 0 (f,) __, H 0 (x,) is continuous and satisfies the following estimate: 21 1 (7.8) II'II+tQ'Pl- 'Yl-t-tQ'PoiiH"(%,) ,Uolan = 0.

As in §1, we assume that a solution ~(t,x) E W 1·2 ( 2 1(Q) to equation (2.2) with u(t,x) = 0 and the right-hand side g E L2(Q) is given; moreover, ·~(t,:r) satisfies the boundary conditions (2.3), (2.5) and the estimates

11~(0,·)- ¢u(·)IIH'(ll) < E,

(2.8)

where c > 0 is small enough. The local exact controllabilit~y problem consists in constructing a controlt1(t,:c) E L 2 (Q), suppu C Qw, such that the solution ·,U(t,x) to the boundary value problem (2.2), (2.3), (2.5), (2.6) satisfies the condition

~·(t,x)l,~r= ~(t,x)I,~T·

(2.9)

We look for a solution (2.10)

~;(t,

x) in the form

,U(t,:c) = w(t,J:)

+ ;J;(t,x),

where 1v is a new unknown function. Substituting (2.10) in (2.2)-(2.6), we obtain the follmving equation for w: (2.11)

81 ( -t.w(t, "'))

+ t. 2 w + B( ,j; + w, w) + B(w, t'f;)

= u(t, :c),

where (2.12)

and the boundary and initial conditions

( -t.w)l~= 0,

(2.13)

wl,;= 0,

w(l,J:)I,~n= wn(:r),

(2.14)

where wu(J:) = 1/•o(x)- ~(O,:z:) and (2.15)

in view of (2.10), (2.7), and (2.8). In §§2-7 we will prove the following result. THEOREM 2.1. Let fu co·incide with aD. Suppose that~ E W 1 •2 ( 2 i(Q) is a function sal:isfying (2.2) with u = 0, (2.3), (2.5), and let w0 E H"(D) be the in:itia.l condition satisfying (2.15) j'o·r a sufficiently small E > 0. Then !.here is a. contml 2 11. E £2( Q), supp 'II. C (0, T) X w, such that a. solution w E W'-'( 1( Q) l.o the problem (2.11)-(2.14) exists and satisfies the eq·ual-ity w(l, :r)I,~T= 0.

(2.16)

2.2. The linear controllability problem. To prove Theorem 2.1, we use the following theorem on the right inverse operator. THEOREl'vi

2.2. Let ..Y and Z be Banach spaces, and

(2.17)

A:X_,z

a continuously differentiable nwp]Fing. Suppose that for .Lo E X (2.18)

A(xo) = zo,

and the de'f'i-valive (2.19)

A'(.,o): X___, Z

1

.:o E Z we have

::!7:2

7. CONTftOLLA131LlTY OF THE FLO\V OF JNC'CH!PflESSlLJLE VlSCOlTS FLUlD

of the operator A al the point :r0 is an. epimorphism. Then for a sufficiently small > 0, there is a mapping M(z): B,(zo) _,X,

f

B,(zo) = {z E Z: //:- zo/lz < c},

such that: (2.20)

A(M(:)) = :,

z E B,(zo),

and (2.21) for some k

1/M(:)- :rollx,;; k[[A(o.·o)- =liz,

: E B,(:o),

> 0.

Theorem 2.2 is a simple consequence of a generalization of the implicit function theorem (cf. the proof in [6]). In our case, the space ~Y consists of pairs (to u) and the operator A(:r) is defined by formula (2.11): 1

(2.22)

A(o:) = ( -81L>w + L>'w + B(0 + w, w) + B( w, 0) - u, wi,=U)

(the condition w/t=T= 0 and the boundary conditions for w "\Viii be included in the definition of the space X), whereas the set of pairs of the form (2.22) defines the space Z. For :ro and zo we take the zero elements :r 0 = (0, 0) and zu = (0, 0). For these :rn and =o the equality (2.18) is obvious. To verify that the operator (2.19) is an epimorphism: ·which is the most difficult condition of Theorem 2.2, we write the equation A'(:ro):z:: = z. This equation takes the form

(2.23) where u =

Lw - u :t~"/U 1 :t~·

= 81 ( -L>w) + L> 'w + B( t):, w) + B( w, tj;) -

f,

is the function (l.Hi),

wl~= .6.wl~= 0,

(2.24) (2.25)

u=

wii.=D= ·wn, wjt=T= 0. and z0 = (0, 0) the function 1!: from

Note !_hat lor :r 11 = (0, 0) (2.23) coincides with 1j1. However, we prove the solvability of the problem (2.23)~(2.25) for any 1/; E Hrl ,:2( 1l (Q). This will allow us to generalize Theorem 2.1 ( cf. H.emark 0.2 below). Let us give a precise definition of the spaces X and Z corresponding to the problems (2.11)~(2.14) and (2.23)~(2.25). We set (2.2ri)

where,\> 0 is a parameter (it will be specified in §3 below) and ;3(o>·) E C 0 (rl) is a function such that (2.27) (2.27')

'V;J(:r) f 0 \f.r E 0.\w'. ;J(:c) ?o ln3

\f:c E l'l,

(v(J(o:), n(:r)),;; 0 \f.r E 80.,

min/J(:r) > :rEf!

~ max;J(:c). .:1 xED"

Here w'