Proceedings of the seventh International Colloquium on Differential Equations: Plovdiv, Bulgaria, 18–23 August, 1996 [Reprint 2020 ed.] 9783112319185, 9783112307915

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Proceedings of the Seventh International Colloquium on Differential Equations

Also available fromVSP Proceedings of the Sixth International Colloquium on Differential Equations Plovdiv, Bulgaria, 18 - 23 August 1995 Proceedings of the Fifth International Colloquium on Differential Equations Plovdiv, Bulgaria, 1 8 - 2 3 August 1994 Proceedings of the Fourth International Colloquium on Differential Equations Plovdiv, Bulgaria, 1 8 - 2 2 August 1993 Proceedings of the Third International Colloquium on Differential Equations Plovdiv, Bulgaria, 1 8 - 2 3 August 1992

Proceedings of the Third International Colloquium on Numerical Analysis Plovdiv, Bulgaria, 1 3 - 1 7 August 1994 Proceedings of the Second International Colloquium on Numerical Analysis Plovdiv, Bulgaria, 1 3 - 1 7 August 1993 Proceedings of the First International Colloquium on Numerical Analysis Plovdiv, Bulgaria, 1 3 - 1 7 August 1992

Proceedings of the Seventh International Colloquium on Differential Equations Plovdiv, Bulgaria, 18-23 August, 1996

Editor: D. Bainov

///VSP/// Utrecht, The Netherlands, 1997

VSP BV P.O. B o x 3 4 6 3 7 0 0 A H Zeist The Netherlands

© V S P B V 1997 First p u b l i s h e d in 1 9 9 7 ISBN 90-6764-233-9

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

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CONTENTS Preface

xi

Identifying the Jordan canonical form and associated non-singular transformations S. Askarpour and T.J. Owens

1

Forward dynamic programming for the numerical solution of parabolic control problems S.A. Belbas

9

Optimal control of Goursat systems over varying domains S.A. Belbas

17

On Newton's problem under side constraints M. Belloni and B. Kawohl

23

Parameters estimation of some electro-elasticity models N.Botkin

31

A nonlinear suspension bridge equation Q.-H. Choi and T. Jung

39

Uniqueness in the Cauchy problem for the heat equation S.-Y. Chung

49

Existence and uniqueness of solutions for a class of abstract nonlinear wave equation M.R. Clark

57

Asymptotic expansions for some classical operators and their use in approximation theory F. Costabile, M.I. Gualtieri and S. Serra

67

Some over-determined boundary-value problems for hyperbolic equations D.E. Edmunds and N.I. Popivanov

75

On asymptotic properties in a neighbourhood of infinity of solutions of elliptic equations A.S. Abd El-Rady

81

Further development of minimum curvature multi-step quasi-Newton methods J. A. Ford and I. A. Moghrabi

85

vi

contents

Some results about boundary conditions for a discrete model in elasticity M. Frontini and L. Gotusso 93 On local and global solvability of nonlinear mixed problems modelling vibrations of a string C.L. Frota and N.A. Lar 'kin

101

Poincare-Bendixson type theorems for n dimensional spaces T. Fujimoto and Y. Fujimoto

109

The effect of heterogeneity in sharing rates and choice of shooting gallery on the spread of HIV amongst sharing injecting drug users D. Greenhalgh

115

Nontrivial solutions for a fourth order O.D.E. with singular boundary conditions M.R. Grossinho and T.F. Ma

123

The limiting problem in a double free boundary problem Y. -M.Ham

131

On a Lienard system with three equilibrium points M. Hayashi

139

Re-examination on validity of analytic solution of Bessel's equation to estimate the magnitude of fluctuation of stem wood temperature in standing Pinus densiflora trees Y. Hayashida

147

Some modernizations the Gronwall-Bellman Lemma and its application to differential equations with delay D. Jama and A.M. Nawrat

155

Numerical analysis of pseudozeros and pseudospectra S. Kado, C.L. Parihar, V.M. Raffee and M. Tsuji

165

Characterization of the holomorphy of infinite dimensional domains by Oka's principle J. Kajiwara, X.D. Li, S. Ohgai and D. G. Zhou

173

Numerical analysis on 8-problem and extension of holomorphic functions from lower dimensional subvarieties J. Kajiwara, C.L. Parihar, V.M. Raffee and M. Tsuji

181

On triple systems and Yang-Baxter equations N. Kamiya and S. Okubo

189

On continuation of Gevrey class solutions of linear partial differential equations A. Kaneko

197

contents

vn

Convergence properties of periodic solutions on a piecewise linear differential equation K. Katori, M. Otake, Y. Manome and T. Mishima

205

Norm estimates for Schrodinger semigroups with magnetic fields M. Loss and B. Thaller

213

On linear differential equations for fractional order F. Mainardi

221

Differentiability for nonlinear parabolic systems M. Marino and A. Maugeri

231

Extension of holomorphic mappings into manifolds with weak disc property Y. Matsuda

237

Global instability of natural convection in an elliptic cavity, using a Fourier spectral difference method Y. Mochimaru

243

Capacity associated with an a-parabolic operator M. Nishio

253

Martin boundary at infinity for the heat equation M. Nihsio

261

A general form of a mean value property for poly-temperatures on a strip domain M. Nishio, K. Shimomura and N. Suzuki

269

Cohomology vanishing and q-convex functions on infinite dimensional domains £ Ohgai

277

The existence and the continuation of holomorphic solutions for convolution equations in a half-space in C n J. Okada

283

Solvability and propagation of singularities for some class of microdifferential equations in the spaces of micro-distributions Y. Okada

289

Rapidly varying behaviour of the solutions of a second order linear differential equation E. Omey

295

Stability of Lie groups of the perturbed non-linear wave equation u ( f + e u f = [ / ( x , u ) u j x N. Omolo-Ongati and E.E. Rosinger

305

viii

contents

Application of weighted ranking methods for tie-break simulations K. Oshima, I. Hofuku and K. Horimoto

313

Chaotic mixing of viscous fluids: a marangoni-convection-driven example A. Palanques-Mestre and G. Alba-Soler

321

Numerical solutions to the minimum-time problem for linear second-order conflict-controlled systems V.S. Patsko and V.L. Turova

329

Nonlinear thermoelasticity and symmetry R. Racke

339

Saddle point methods in spatial models of biological systems J. Radcliffe and L. Rass

347

The asymptotic spatial behaviour of a class of epidemic models L. Rass and J. Radcliffe

355

A generalized diffusion equation J. Schmeelk

363

Analysis of a degenerate Hopf bifurcation in a PID controlled CSTR S. Serra and C.T. Possio

371

On the arithmetic properties of the modular function induced from the hypergeometric differential equation of type (3,6) H. Shiga

381

Caloric morphisms and a generalization of the Appell transformation K. Shimomura

389

Boundary behaviors of some families of special functions H.M. Srivastava

395

On a coordinate transformation to express an explicit duck solution in the FitzHugh-Nagumo equation T. Tchizawa

405

On the completeness of the eigenfunctions of elliptic operators T. Tokita

409

An algorithm for high accuracy product in redundant representation K. Tsuji

413

Round-off errors in floating-point solutions for Chebyshev collocation points K. Tsuji

421

contents

ix

Identification of density distribution due to one-codimensional projected figures M. Tsuji

429

Differential inequalities concerning nonlinear systems C. Ursescu

437

Decay for some second order nonlinear systems E. Vitillaro

441

On a priori bound for nonconvolutional integral equations A. Yanagiya

449

A pointwise error estimate for distributed parameter identification Xue-Cheng Tai and P. Neittaanmaki

455

Preface The Seventh International Colloquium on Differential Equations was organized by the Institute for Basic Science of Inha University, the International Federation of Nonlinear Analysts, the Mathematical Society of Japan, Pharmaceutical Faculty ofthe Medical University of Sofia, University of Catania and UNESCO, with the cooperation of the Association Suisse d'Informatique, the Canadian Mathematical Society, Kyushu University, the London Mathematical Society, Technical University of Plovdiv, and it was partially sponsored by the Bulgarian Ministry of Education, Science and Technologies under Grant MM-511. The Colloquium was held August 18-23, 1996, in Plovdiv, Bulgaria. The subsequent colloqiua will take place each year from 18 to 23 August in Plovdiv, Bulgaria. The publishing of this volume is fully supported by the Bulgarian Ministry of Education, Science and Technologies under Grant MM-511. Organizing Committee: H.Adeli, A.S.A. Al-Hammadi, D. Bainov (Chairman), Q.-H. Choi, J. Diblik, S. Hristova, J. Kajiwara (Vice Chairman), V. Lakshmikantham, M. Marino, A. Maugeri (Vice Chairman), E. Minchev (Secretary), K. Nakagawa, N. Popivanov, P. Popivanov, M. Robnik, H.M. Srivastava, Dj. Takaci, R.U. Verma Scientific Committee: D. Bainov (Bulgaria), C. Dafermos (USA), G. Da Prato (Italy), H. Fujita (Japan), R. Ivanov (Bulgaria), T. Kaczorek (Poland), J. Kajiwara (Japan), A. Kaneko (Japan), N. Kenmochi (Japan), V. Lakshmikantham (USA), A. Maugeri (Italy), N. Pavel (USA), N. Popivanov (Bulgaria), P. Popivanov (Bulgaria), M. Robnik (Slovenia), P. Sobolevskii (Israel), H.M. Srivastava (Canada), Fr. Stenger (USA), E. Titi (USA), M. Tsutsumi (Japan). R.U. Verma (USA). The address of the Organizing Committee is: Drumi Bainov, P.O. Box 45, 1504 Sofia, Bulgaria. The Editor

7th Int. Coll. on Differential Equations, p.p. 1-7 D. Bainov (Ed) ©VSP1997

I d e n t i f y i n g t h e J o r d a n Canonical Form and A s s o c i a t e d N o n - S i n g u l a r Transformation S . A S K A R P O U R AND T . J . Department

OWENS

of Electrical Engineering

and Electronics,

Uxbridge, Middlesex

UB8 3PH, United

Keywords:

canonical

Jordan

Brunei

University,

Kingdom

form

INTRODUCTION At present all existing methods for determining the Jordan canonical form of a matrix are ambiguous in identifying which chains of equations lead to the correct selection of characteristic vectors that achieve the Jordan canonical form through similarity transformation. Golub and Wilkinson [1] discuss several methods for computing the Jordan canonical form but do not discuss a numerically stable method. It is easily shown that only some linear combinations of characteristic vectors can be employed in order to determine the Jordan canonical form leading to limited freedom in the selection of characteristic vectors. Gantmacher [2] has described some of the possible operations that can be performed on the transformation matrix without altering the shape of the Jordan form. However, the operations discussed by Gantmacher [2] are performed only on complete Jordan blocks and no proofs are given that the operations do not change the shape of the Jordan form. In this paper six theorems are presented relating to which linear combinations of characteristic vectors can be employed to achieve the Jordan canonical form. From these theorems two rules are deduced concerning the freedom available in the choice of the transformation matrix for the Jordan canonical form. A seventh theorem is presented which provides conditions that enable a new method for determining the Jordan canonical form and associated non-singular transformation to be developed. The new method does not possess the ambiguities associated with existing methods. The method is demonstrated through a seventh order example.

S T A T E M E N T OF T H E P R O B L E M Square matrices of dimension n which have repeated eigenvalues may be transformed into the Jordan canonical form using a similarity transformation. That is there exists

2

5. Askarpour

and T. J.

Owens

a transformation V, obtained by using t h e characteristic vectors of the original matrix, such that: J=V^AV

where J has t h e Jordan form. Construction of a matrix V is achieved by searching for linearly independent eigenvectors from t h e standard eigenvalue-eigenvector problem (A — I \ ) v i = 0 which produces u < n linearly independent vectors and completing the n dimensional basis by using generalised eigenvectors of Jordan chains. To clearly express the chaining process t h e following notation is introduced: Let ulxyz be an eigenvector or generalised eigenvector relating to eigenvalue A; where x shows t h e level of t h e vector in the chaining process e.g. vlQyz denotes an eigenvector and v2yz denotes a second chained generalised eigenvector.

T h e suffix y indicates the

Jordan block with which t h e vector is associated. T h e maximum value that y takes is the geometric multiplicity of A¡. T h e suffix z indicates t h e number of generalised eigenvectors chained off the vector, e.g. i/g30 denotes the third eigenvector of A( which has no following chain and i/^n denotes the second chained generalised eigenvector of the first Jordan block which has one generalised eigenvector chained off it. Now from the definition of eigenvectors and generalised eigenvectors we may write: (A - / A j ) ^ , = 0 (A - I\l)vlx+lyz-x-l

=

¿xyz-x

and (A - I h ) x

x + l

(A - I \ i ) v

l

^

xvz-x

x

= 0

= ulQyz

From the above it is clear t h a t the nullity of (A — IXi) is equal to the number of eigenvectors belonging to A; and difference between t h e nullity of (A — I \ i ) 2 and t h e nullity of (A—IXi) is equal to the number of first chain generalised eigenvectors belonging to A; and so on. Thus, it is conceptually simple to determine the Jordan form. Determination of a transformation matrix for the Jordan form is, however, a challenging problem. This presentation will address the construction of a transformation matrix for the Jordan form. Firstly, however, rules t h a t enable many similarity transformations t h a t achieve the Jordan form to be determined once one transformation is known will be developed.

N O N - U N I Q U E N E S S OF T H E T R A N S F O R M A T I O N MATRIX It is well known t h a t the Jordan canonical form is unique up to the size and the order of the Jordan blocks. T h e following six theorems determine which combinations of eigenvectors and generalised eigenvectors can be used in the transformation matrix without altering the Jordan form.

3

Identifying the Jordan canonical form.

T h e o r e m 1. Any generalised eigenvector can only be chained off one eigenvector or generalised

eigenvector.

T h e o r e m 2. Any multiple of an eigenvector of a repeated eigenvalue added to any multiple of another eigenvector associated vrith the same repeated eigenvalue with a smaller or equal number of chained vectors can replace the latter without altering the Jordan T h e o r e m 3. No linear combination eigenvector

of an eigenvector

form.

of a repeated eigenvalue and an

associated with the same repeated eigenvalue with a longer following

can replace the latter without altering the Jordan

chain

form.

T h e o r e m 4. Any multiple of an eigenvector of a repeated eigenvalue added to a generalised eigenvector

associated with the same eigenvalue with a shorter or equal length of

chain can replace the latter without altering the Jordan T h e o r e m 5. No linear combination and an eigenvector

form.

of a generalised eigenvector of a repeated eigenvalue

associated with the same repeated eigenvalue

without altering the Jordan

form.

T h e o r e m 6. No linear combination with another generalised

can replace the latter

of a generalised eigenvector of a repeated eigenvalue

eigenvector

associated with the same repeated eigenvalue can

replace the latter without altering the Jordan

form.

The six theorems can be summarised in two rules: R u l e 1. No linear combination

of a generalised eigenvector of a repeated eigenvalue and

any other vector associated with the same repeated eigenvalue can replace the latter without altering the Jordan form.

This also means that a generalised

eigenvector

can not be

replaced by multiples of itself. R u l e 2. A multiple generalised

of an eigenvector

eigenvector

of a repeated eigenvalue

or multiple of an eigenvector

can only be added to a

associated with the same repeated

eigenvalue to replace the latter without altering the Jordan block if the length of the chain of the former is greater than or equal to the length of the chain of the latter.

D E T E R M I N A T I O N OF A T R A N S F O R M A T I O N MATRIX A condition is needed satisfaction of which ensures chaining can take place. condition is:

Such a

T h e o r e m 7. Any vector vlxyz satisfying the condition of a further z generalised

eigenvec-

tors to be chained from it must vl

satisfy: e

- ni)' n

- i\i)x+l.

4

S. Askarpour and T. J. Owens Recall that: x gives level of the vector in the chaining process, y indicates the Jordan

block with which the vector is associated, z gives the number of generalised eigenvectors chained off the vector. Remark 1. To develop a computational algorithm for determining a similarity transformation for the Jordan canonical form based on Theorem 7 the following propositions are required: Proposition 1. JR(>1) Pi 9t(B)

can be determined from.

— B] where A and B are

matrices with an equal number of rows. Proposition 2. If Au2 =

v\ and V2 £

(B),

where A and B are matrices with an

equal number of rows and A is square, then all possible vectors 1/2 can be obtained from K[AB - i/i}. Remark 2. Propositions 1 and 2 mean that all the subspaces needed to determine a similarity transformation to achieve the Jordan canonical form can be obtained from basis vectors of null spaces.

A COMPUTATIONAL ALGORITHM FOR DETERMINING A SIMILARITY TRANSFORMATION FOR THE JORDAN FORM The following algorithm constructs the generalised eigenstructure associated with the Jordan blocks of a repeated eigenvalue A¡. It enables the entire generalised eigenstructure of a matrix with repeated eigenvalues to be determined. Step 1: Determine the nullity nl of {A — I \t) and a set of basis vectors NL

which

span the nullspace of (A — /Aj) using the matlab null command. Determine successive higher powers of (A — /A/) and the nullities of the powers up to the point where the nullity of the power is equal to the algebraic multiplicity of the eigenvalue. Step 2: Determine the shape and size of the Jordan blocks of A; from the nullities of step 1. Step 3: Determine an eigenvector UQyz belonging to a largest Jordan block of A1 for which the eigenvector is unknown using the result of Proposition 1: Determine a vector u^

e H([NL

— (A — /A() z ]) then

= NL *

vector containing the first nl elements of

: nl) where ul0yz(l

: nl) represents a

ul0yz.

Step 4-' Determine the next generalised eigenvector in the chain using the result of Proposition 2 with A = (A — /A t ) and B = (A - I A,)* : Determine ulxyz G N([(,4 - IXt)'+1 element of

ulxyz.

Then

ulxyz

= (A

- i>£_ lyz+1 ]) — IA/)z

*

wlxyz(l

anc*

wxyz =

u>,xyzh

where 7 is the last

: n) where n is the dimension of A.

Step 5: Follow step 4 until all the vectors in the chain are determined. Step 6: Follow steps 3 to 5 for the remaining blocks until all the vectors are determined remembering that all eigenvectors chosen in step 3 must be linearly independent.

Identifying

the Jordan canonical

form.

5

EXAMPLE Consider the 7th order matrix: -1

1

-1

-1

0

1

-1

0.5

-0.5

0

-2

0.5

0.5

0

0

-1

0

3

0

-1

1

0

0

0

-1

0

0

0 -1

-1

-1

-1

0

-2

0

0.5

-0.5

2

5

0.5

-2.5

2

0.5

0.5

1

1

0.5

-0.5

0

with seven eigenvalues at —1 and Jordan form: -1

1

0

0

0

0

0

-1

1

0

0

0

0 0

0

0

-1

1

0

0

0

0

0

0

-1

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

-1

0

0

0

0

0

0

0

-1

The condition number of A in the sense of the ratio of the largest to the smallest singular value of A is 62.3144. Applying the above algorithm to determine a similarity transformation for the Jordan form gives: V = Columns 1 through 3 9.965710984760412e - 02

—6.106349859081298e+00

1.034450927331190e — 01

- 1 . 1 5 3 1 2 2 4 0 0 3 0 5 9 6 2 e - 16

9.96571098476038le — 02 -6.206006968928888e 4-00

- 9 . 9 6 5 7 1 0 9 8 4 7 6 0 3 9 4 e - 02

6.106349859081298e 4-00

6 . 6 0 9 7 3 4 4 2 6 9 2 0 5 6 4 e - 17

9.586912696208919e-02

0

0

- 2 . 9 8 1 5 5 5 9 7 4 3 3 5 1 3 7 e - 18

—9.965710984760381e —02

1.211493659975772e 4-01

- 9 . 9 6 5 7 1 0 9 8 4 7 6 0 3 8 3 e - 02

6.106349859081298e 4-00

-5.813060503866741e 4-00

- 4 . 5 2 2 4 7 8 3 1 1 9 8 0 1 5 9 e - 17

0

-6.008586740676437e 4-00

Columns 4 through 6 - 1 . 2 1 0 5 1 6 0 4 1 3 6 5 6 0 4 e 4-01

4.309822056027898e-01

— 1.771408807344095e 4-01

1 . 0 1 5 5 1 1 0 1 2 9 0 3 9 9 3 e - 01

-5.035309947468366e-16

3.281031771563433e —01

7 . 4 8 3 2 9 5 4 8 6 9 8 1 0 0 4 e - 02

-4.309822056027889e-01

2.343571382500273e 4-00

9 . 9 6 5 7 1 0 9 8 4 7 6 0 3 9 7 e - 02

2.909728334617533e-16

-1.026385132017130e-14

- 1 . 2 7 3 6 6 5 6 4 5 0 2 0 3 2 7 e - 01

1.028790284464509e — 01

1.086922180115206e 4-01

1 . 1 7 5 9 0 3 7 8 4 0 0 2 3 0 4 e - 01

—5.338612340492395e —01

6.516763095132537e 4-00

- 5 . 8 7 9 3 6 7 7 7 5 9 9 0 3 2 2 e - 02

-1.028790284464512e-01

4.070312684185825e-)-00

6

S. Askarpour

and T. J.

Owens

Column 7 7.896658308256629e —01 —9.018596958641767e—16 — 1.513142880156455e —01 4.964561747811125e—16 —4.488544555234243e —01 —3.408113753022363e —01 -1,894970872865920e - 01 which transforms A to: J =

Columns 1 through 3 -9.999999999973138e- 01

9.999999998204316e- 01

2.293010226139813e —11

3.913067548819219e —14

-1.000000000002642e+00

1.000000000000640e+00

7.104776836263623e —16

-4.485474491833230e- 14

—9.999999999999921e —01

-1.618216441070205e- 27

-6.632476534735134e- 16

—7.595152795467902e —27

-3.357545837229238e- 14

1,945887895260512e —12

3.944344850737025e —13

-6.700578002036070e- 17

4.362482597386474e —14

3.191891195797325e— 16

—5.683230036341138e —15

1.304512053934559e —12

-7.838174553853605e- 14

Columns 4 through 6 —4.269400111223121e— 11

1.229467916363802e- 11

— 1.276924788129463e —09

— 7.339165082956289e —13

1.818202706449501e —13

— 1.905840762983502e —11

9.999999999999996e —01

3.330249891558917e —15

-3.139154536175963e- 13

—9.999999999999999e —01

— 1.987387723286967e —20

—3.056296636131730e —15

-4.900363968773913e- 13

— 1.000000000000146e -1- 00

—3.005007267325999e —14

-4.923362065256676e- 16

—9.999999999998127e —01

— 1.146157821430016e —13

-2.572941859568800e- 14

5.115963208623953e —12

Column 7

1.000000000017093e+00

2.318017999769495e —11 3.324940149596056e —13 6.079984695933130e —15 4.942214969267663e—18 —3.316973432032633e—13 —3.93565388612238 le—16 -1.000000000000035e+00

CONCLUDING REMARKS T h e largest error in J is -1.276924788129463e - 09 in element (1, 6). This error is large for such a well-conditioned A matrix. Much of it may be associated with the error in the inversion of V as the condition number of V is 6.6305e + 07. However, it might be possible

Identifying the Jordan canonical form

7

to greatly reduce the error by using rules 1 and 2 to construct from the V delivered by the algorithm an "optimally conditioned" transformation matrix. Alternatively, the error might be reduced by constraining the algorithm to deliver characteristic vectors t h a t result in a relatively well-conditioned transformation matrix.

REFERENCES [1]

G. H. Golub and J. H. Wilkinson, 111 conditioned Eigensystems and t h e computation of the Jordan Canonical Form, SIAM

[2]

Review, 18 (1976), 578-619.

F. R. Gantmacher, The Theory of Matrices,

Chelsea Publ., 1959.

7th Int. Coll. on Differential Equations, pp. 9-15 D. Bainov (Ed) ©VSP1997

FORWARD D Y N A M I C P R O G R A M M I N G FOR T H E N U M E R I C A L SOLUTION OF PARABOLIC CONTROL PROBLEMS S. A . BELBAS

Mathematics Department, University of Alabama Tuscaloosa, AL 35487-0350, USA

Abstract. The numerical solution of parabolic control problems is carried out by using an implicit or semi-implicit finite-difference discretization and a corresponding type of dynamic programming that deserves the name "forward dynamic programming". By contrast, the standard dynamic programming equations are solved backwards in time. We present here the essential features of forward dynamic programming.

1.

INTRODUCTION

We are interested in optimal control problems for parabolic systems. A model problem consists of a controlled parabolic equation dv(t at

'

x)

= C(t, x)y{t,

x) + / ( ( , x, y(t, x), V y ( t , * ) ) + j ( t , x, u ( t ) )

(1.1)

with appropriate initial and boundary conditions, where £(t, x) is a second-order uniformly elliptic operator with sufficiently smooth coefficients, y is an ]R n -valued function, and u(t) is an IR m -valued control function. The spatial domain f2 is an open bounded set in 1R3 with sufficiently smooth boundary, and the temporal horizon is (0, T]. An optimal control problem associated with (1.1) concerns the minimization of a functional J of the form J = [T ( F(t, Jo Jn

x, y(t, x), Vy(t,

z ) , u(t))dtdx

+ f $ ( T , x,y(T, Jn

x))dx

(1.2)

Of course, other more general functionals can also be considered. Many applied problems lead to mathematical problems of the type (1.1)-(1.2). For example, problems of atmospheric pollution (transport and diffusion of aerosols), coastal pollution (transport and diffusion of hydrosols), and groundwater pollution (flow in porous media), lead to problems of the type (1.1)-(1.2); cf. [1, 2, 3, 4, 5]. The numerical solution of problem (1.1-2) relies on three steps: first, the discretization of the system (1.1) and the functional (1,2); second, an algorithm for the approximate

S. A. Belbas

10

solution of the discretized version of (1.1); and third, an algorithm for the approximate solution of the corresponding discretized optimal control problem. The first two steps have been extensively studied in the research literature; in our opinion, relatively less work has been done for the third step. In this paper, we are concerned primarily with the third step. After a discretization in both space and time, we obtain a system in discrete time, with discrete time-variable still denoted by t, where now t takes nonnegative integer values. If z(t) represents a discrete approximation to y(t, x), x e fi, where Q is the set of interior grid points in ft, so that z takes values in I R m , where N is the cardinality of ft. If an explicit scheme is used, then the discrete system for z has the form z(t + l) = g(t,z(t)Mt))

(1-3)

If a fully implicit scheme is employed, then the corresponding dynamics for z are obtained as z(t) = g(t + l,z(t + l),u(t + 1)) (1.4) Finally, if a semi-implicit (e.g., Crank-Nicolson) scheme is used, the control system in discrete time can be written in either of the two forms below: z(t) = g(t + l,z(t),z(t

+ l),u(t),u(t

+ 1)

(1.5)

z(t + 1) = 7(t, z(t), z(t + 1), u(t), u(t + 1))

(1.6)

For the purpose of applying the ideas of dynamic programming to semi-implicit discretizations, it is useful to approximate (1.5) or (1.6) by (1.7) or (1.8) below, respectively: z(t) = g{t + I, z(t + l),u{t),u{t z(t + 1) = 7 (t,z{t),u(t),u(t

+ 1)) + 1))

(1.7) (1.8)

The functional J given by (1.2) is discretized as J = ^G(i,Z(i),u(i)) + i(T,Z(T)) t=o

2.

(1.9)

BRIEF OVERVIEW OF CLASSICAL D Y N A M I C P R O G R A M M I N G FOR DETERMINISTIC DISCRETE-TIME SYSTEMS

The method of dynamic programming for discrete-time systems has been beautifully explained in the classical work of Boltyanskii [6], as well as in several other books and papers. Here, we shall give a very terse summary, entirely for the purpose of paving the way for the rest of this paper; our notation and terminology are different from [6]. Consider a discrete dynamical system (recursive formula) y(t + 1) = f ( t , y(t), u(t)), y(0) = y0 = given

(2.1)

where t is discrete time-parameter, t = 0,1, 2 , . . . , N. The function u(t) is the control. The problem is to minimize N

J = Y,F(t,y(t),u(t))+^(y(N+ t=o

1))

(2.2)

Forward Dynamic

11

Programming

Define a parametrized system Vsxit + 1) = f(t,

y(t), u ( t ) ) , y„(s)

= x,t

(2.3)

= s,s + l,s + 2,...,N.

Define J3X =

u ( t ) ) + * ( y « ( J V + 1))

y„(t),

(2.4)

t=s

and V(s,x)=infJM

(2.5)

the infimum being taken over all admissible control functions u(-). For simplicity, assume that u takes values in a compact set K, and then (assuming continuity of /, F, $) V(s, x) = min Jsx

(2.6)

Compare V ( s , x) to the value of Jsx obtained by using an arbitrary value a € A" of the control u(s),

and using an optimal u from time s + 1 until the final time N, starting with

the value ysx{s + 1) = f(s,

x, a):

V(s,x)

+ V(s + l , y « ( s + l))

< F(s,x,a)

(2.7)

or V(s, x) < F(s, x, a) + V ( s + 1, f(s,

(2.8)

x, a)).

If a happens to be the value of an optimal control u* at time 5, i.e., if a = a' =

u*(s)

where u ' ( - ) is a minimizer of Jsx, then V(s, x) = F(s, x, a') + V(s + 1 ,f(s,

(2.9)

x, a'))

T h e formulae (1) and (2) can be combined into V ( s , x) = m i n { V ( s + 1, /(s, x, a ) ) + F(s, x, a ) } ; a* =

argmin{Vr(s a

(2.10)

+ 1, f(s, x, a)) + F(s, x, a)}

(2.11)

where "arg m i n a " denotes a value of a that minimizes the expression in brackets.

In

addition, from the definition of V and J3X, we have V(N

= $(x)

+ \,x)

(2.12)

because, for s = N + 1, JN+I,X

=

*

(

V

N

+

U

N

+

!)) =

(2-13)

using the convention that a sum over an empty set of indices is zero, so that, for s = N

+1,

the sum part of Jsx vanishes. T h e system of (2.10-11) is a backward recursive relation (starting f r o m time N + 1, and moving, at each step, from s + 1 to s). This yields the values of an optimal control. T h e advantage of dynamic programming is this: suppose A" is a finite set with k elements. In the minimization problem in (2.10), the variable a takes k values. It is readily seen from the definition of J that the variable of minimization is ( w ( 0 ) , « ( 1 ) , • • •, it(AT)}, which takes kN+l

values. Thus, by dynamic programming, a minimization over a

dimensional variable is reduced to N + 1 minimizations, each with respect to a

kN+l-

fc-dimensional

variable. T h i s implies serious reduction in the amount of computations. T h i s reduction is possible because of the particular structure of the dynamical system and the functional J.

5. A. Belbaa

12

3.

T H E F O R W A R D D Y N A M I C P R O G R A M M I N G E Q U A T I O N S A N D RELATED EQUATIONS

When the discrete-time dynamical system has the form (1.4) and the functional J has the form (1.9), then we define a parameterized functional Jtx=

t ^2G{s,ztx(s),u(s)),

for 0 < t < AT;

(3.1)

3=0 N W

=

ZN+u(a),

u(s))

- ( - a) = l;/(t,*,a) =

Ay(i,i,a)>0; y€w(t,x,a)

£

\v(t,x,a)y

(3.6)

ySw(t,x,o)

Then the function V is approximated by V, where V solves V(t + \,x) = i n f { G ( i , i , a ) + j/€a>(t,x,a) £ \y(t,x,a)V(t,y)}

(3.7)

and V has the same initial conditions as V. Estimates for the error \V — V\ are provided in the next section. Let N be the dimension of x. In order to be able to find A's satisfying (3.5), cj(t, x, a) should contain at least n •+-1 points; on the other hand, if oj(t, x, a) contains at least N + 1 points, then the determination of the A's is generally not unique. For computational convenience, particularly with the prospect of parallel implementation, it is useful to have a grid Q and corresponding sets u>(t,x,a) of the following type: points y of Q have the

Forward Dynamic Programming

13

form xo + /ije 1, for every choice of A > 0 and C. Hence every concave solution of the minimum problem (10) must be a trapezoid. But more can be said. T h e o r e m 3 . 1 : (see [1]) Let n = 1 and U be a concave solution of the minimum (10). Then we have

| u r ( r ) | g (0,v/2) for a.c. r € (0,1).

problem

(12)

Moreover, if L < \ / 3 . then

L =l JI uir) W \-V2(l-r) and if L > V3, then u(r) -

^ [ 0 , 1 - ^ ) ifr€ [1-^,1]

L2 - 1(1 - r) for every r £ [0,1].

For the moment, we only know that there exists "at least" one concave solution of (10). In fact, if n = 1, we have given examples of minimizers that are nonconcave. But if we restrict our attention to higher dimensions we have the following result. C o r o l l a r y 3 . 2 : (see [1]) For n > 2 every solution of (10) must be concave.

This follows from an ispection of an inequality due to Riesz. When n > 1, a result analogous to Theorem 3.1 holds as follows. T h e o r e m 3 . 3 : Let u be a radial solution of the minimum problem (10), with n > 1, and suppose that u is piecewise of class C 1 . Then u must be concave and we have

|Vu(x)| g (0,V2) and

for a.e.

x€Q

at the free boundary.

Can we compute a solution of (10) if n = 2? Now the Euler-Lagrange equation reads - r " - 1 — ^2 + = c , (13) (1+"?) y/l + «» where A is a Lagrange multiplier and c is an undetermined constant. Note that this equation holds (in the weak sense) for every r for which ur(r) ^ 0, because

On Newton's Problem Under Side Constraints

29

t h e n we can calculate t h e first variation of

JoTT^rdr XUo^ +

X M i r - L ) .

Since u is concave, equation (13) holds on an interval from a to 1, where a m a y be zero or positive. If (3.10) holds in 0 and u is of class Cl, then c = 0. T h i s might serve sis a justification to choose c = 0. In fact, as noted by Legendre in t h e case n = 2, choosing c = 0 simplifies f u r t h e r calculations, because t h e n t h e equation becomes Tit rur Ur = XAt this point Legendre s t o p p e d his calculation. We continue by noting t h a t A > 0 and (14)

i. From (14) we have «(r)-«{l)=/7(j)

Ì / 8

- l dr

a n d noting t h a t u ( l ) = 0, we obtain u ( r ) = 12A 2 {2z - 1) (2 2 - 2) ' / 2 - log

+ 2z - l ) (r/A)2/3

(15)

a n d obviously the solution depends on A. Since by Theorem 3.3 ur(a) = —y/2, f r o m (14) we have t h a t a = A\/27. But from the arc-length constraint we o b t a i n

so

Since L is a m o n o t o n e decreasing function of A, it is invertible as A = A ( L ) . S u b s t i t u t i n g in (15) we have the explicit form of Legendre's solutions. So, if for instance L = 2, then A = 1/27 and a = l / \ / 2 7 (see Figure 6).

F i g u r e 6: Legendre's solution in the case L = 2

30

M. Belloni and B. Kawohl

R e m a r k 3.4: It is interesting to note, that for every L < oo there exists a solution to (10) given by (15) with a nonempty flat region. This is in contrast to t h e case n — 1. References [1] M.Belloni & B.Kawohl, [2] M.Belloni & B.Kawohl, [3] Buttazzo,G., V.Ferone (1995). [4] V.Ferone &c B.Kawohl,

Forum Math., to appear. in preparation. & B.Kawohl, Mathem. Nachrichten. 173. pp.71-89 in

preparation.

[5] H e s t e n e s , M . R . , Calculus of variations and optimal control theory, J o h n W i l e y &

Sons, Inc., New York - London - Sydney (1966). [6] Legendre,A.M., Mémoires de L'Académie Royale des Sciences, Année MDCCLXXVI - Paris, pp. 7-37 (17SS). [7] Newton,I., Philosophiae Naturalis Principia Mathematica (1686). [8] R.T.Rockafellar, Convex analysis, Princeton Univ. Press, Princeton (1970). [9] Tonelli,L. Fondamenti di Calcolo delle Variazioni, Zanichelli, Bologna (1923).

7th Int. Coil, on Differential Equations, pp. 31-38

D. Bainov (Ed) ©VSP1997

Parameters estimation of some electro-elasticity models NIKOLAI BOTKIN

Institute of Applied Mathematics and Statistics of Technical University of Munich, Dachauer str. 9a, D-80335 Munich, Germany ABSTRACT

An algorithm for the estimation of unknown parameters of a nonlinear electro-elasticity model is proposed. Distributions of the displacement observed with an error are input data of the algorithm. Finite element approximations of the input data and of the model are used. The algorithm implements the idea of the direct minimization of the residual of equations [1, 2, 3, 4]. The paper is illustrated by computer simulations. K e y w o r d s : Nonlinear

thin plates,

Piezoelectric

actuators,

Estimation

of

parameters.

1. PROBLEM SETTING

We consider a nonlinear system describing oscillations of thin plates excited by patches made of piezoelectric ceramics PU +

div V ( 7 A £ ) - ^

-

(jajÇxp)

-

v i t ) G ^

x

J s

P

) , (1)

a

puctt ~

= v(t)K*(lSp)

- -v(t)G^Is

P

,

a = 1,2.

Here £ and ua, a = 1,2, are vertical and longitudinal displacements, A is the Laplace operator, V is the operator of the gradient, v(t) is the voltage applied to the piezoelectric patch, Isp is the indicator function of Sp. The summation over repeating indices is assumed in (1). The coefficients p., K. and G are constant; p, 7, and rap are discontinuous functions defined as follows:

~2 (¿11 + &pd22), SB

j(d2

2

^

dug

+