Control Theory of Systems Governed by Partial Differential Equations

Control Theory of Systems Governed by Partial Differential Equations EDITORS: A.K. AZIZ University of Maryland Baltimor...

Author: Abdul Kadir Aziz | John Walter Wingate | Mark John Balas

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Control Theory of Systems Governed by Partial Differential Equations EDITORS:

A.K. AZIZ University of Maryland Baltimore County Baltimore, Maryland

J.W. WINGATE Naval Surface Weapons Center

White Oak, Silver Spring, Maryland

M.J. BALAS C. S. Draper Laboratory, Inc. Cambridge, Massachusetts

ACADEMIC PRESSIN

1977

New York San Francisco L?*d n A Subsidiary of Harcourt Brace Jovanovich, Publishers

COPYRIGHT © 1977, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR *TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT

PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

III Fifth Avenue, New York. New York 10003

United Kingdom Edition published by

ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road. London NWI

Library of Congress Cataloging in Publication Data

Conference on Control Theory of Systems Governed by Partial Differential Equations, Naval Surface Weapons Center (White Oak), 1976. Control theory of systems governed by partial differential equations. Includes bibliographies and index. 2. Differential Control theory-Congresses. 1. I. Aziz, Abdul equations, Partial-Congresses. Balas, III. II. Wingate, John Walter. Kadir. Mark John. IV. Title. QA402.3.C576

629.8'312'015 15353

ISBN 0-12-068640-6 PRINTED IN THE UNITED STATES OF AMERICA

76-55305

Contents List of Contributors Preface

REMARKS ON THE THEORY OF OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS J. L. Lions

STOCHASTIC FILTERING AND CONTROL OF LINEAR SYSTEMS: A GENERAL THEORY

1

105

A. V. Balakrishnan

DIFFERENTIAL DELAY EQUATIONS AS CANONICAL FORMS FOR CONTROLLED HYPERBOLIC SYSTEMS WITH APPLICATIONS TO SPECTRAL ASSIGNMENT David L. Russell THE TIME OPTIMAL PROBLEM FOR DISTRIBUTED CONTROL OF SYSTEMS DESCRIBED BY THE WAVE EQUATION H. O. Fattorini SOME MAX-MIN PROBLEMS ARISING IN OPTIMAL DESIGN STUDIES

119

151

177

Earl R. Barnes

VARIATIONAL METHODS FOR THE NUMERICAL SOLUTIONS OF FREE BOUNDARY PROBLEMS AND OPTIMUM DESIGN PROBLEMS 0. Pironneau

209

SOME APPLICATIONS OF STATE ESTIMATION AND CONTROL THEORY TO DISTRIBUTED PARAMETER SYSTEMS W. H. Ray

NUMERICAL SOLUTION OF THE TRANSONIC EQUATION BY THE FINITE ELEMENT METHOD VIA OPTIMAL CONTROL M. O. Bristeau, R. Glowinski, and O. Pironneau

231

265

List of Contributors A. V. BALAKRISHNAN, University of California Los Angeles, California 90024 EARL R. BARNES, IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598

M.O. BRISTEAU, IRIA/LABORIA, Domaine de Voluceau, 78 Rocquencourt, France

H. O. FATTORINI, Departments of Mathematics and Systems Science, University of California, Los Angeles, California 90024 R. GLOWINSKI, IRIA/LABORIA, Domaine de Vol uceau, 78 Rocquencourt, France

J. L. LIONS, IRIA/LABORIA, Domaine de Voluceau, 78 Rocquencourt, France

0. PIRONNEAU, IRIA/LABORIA, Domaine de Voluceau, 78 Rocquencourt, France W. H. RAY, Department of Chemical Engineering, State University of New York, Buffalo, New York 14214 DAVID L. RUSSELL, Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Preface These proceedings contain lectures given at the Conference on Control Theory of Systems Governed by Partial Differential Equations held at the Naval Surface Weapons Center (White Oak), Silver Spring, Maryland on May 3-7, 1976. Most physical systems are intrinsically spatially distributed, and for many systems

this distributed nature can be described by partial differential equations. In these distributed parameter systems, control forces are applied in the interior or on the boundary of the controlled region to bring the system to a desired state. In systems where the spatial energy distribution is sufficiently concentrated, it is sometimes possible to approximate the actual distributed system by a lumped parameter (ordinary differential equation) model. However, in many physical systems, the energy distributions are widely dispersed and it is impossible to gain insight into the system behavior without dealing directly with the partial differential equation description. The purpose of this conference was to examine the control theory of partial differential equations and its application. The main focus of the conference was provided by Professor Lions' tutorial lecture series-Theory of Optimal Control of Distributed Systems-with the many manifestations of the theory and its applications appearing in the presentations of the other invited speakers: Professors Russell, Pironneau, Barnes, Fattorini, Ray, and Balakrishnan.

We wish to thank the invited speakers for their excellent lectures and written summaries. All who were present expressed their satisfaction with the range and depth of the topics covered. There was strong interaction among the participants, and we hope these published proceedings reflect some of the coherence achieved. We appreciate the contributions of all the attendees and the patience shown with any fault of organization of which we may have been guilty. We thank the Office of Naval Research for their financial support of this conference. Finally, special thanks are due Mrs. Nancy King on whom the burden of typing this manuscript fell.

ix

"REMARKS ON THE THEORY OF OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS" J. L. Lions

Introduction

These notes correspond to a set of lectures given at the Naval Surface Weapons Center, White Oak Laboratory, White Oak, Maryland 20910, May 3 through May 7, 1976.

In these notes we present a partial survey of some of the trends and problems in the theory of optimal control of distributed systems. In Chapter 1 we present some more or less standard material, to

fix notations and ideas; some of the examples presented there can be thought of as simple exercises. In Chapter 2 we recall some known facts about duality methods,

together with the connection between duality, regularization and penalty (we show this in an example); we also give in this chapter a recent result of H. Brezis and I. Ekeland (actually a particular use of it) giving a variational principle for, say, the heat equation (a

seemingly long standing open question, which admits a very simple answer).

Chapter 3 gives an introduction to some asymptotic methods which can be useful in control theory; we give an example of the connection

between "cheap control" and singular perturbations; we show next how the "homogeneization" procedure, in composite materials, can be used in optimal control.

In Chapter 4 we study the systems which are non-linear or whose state equation is an eigenvalue or an eigenfunction; we present two

examples of this situation; we consider then an example where the control variable is a function which appears in the coefficients of the highest derivatives and next we consider an example where these two

1

J. L. LIONS

2

properties (control in the highest derivatives and state = eigenfunction) arise simultaneously.

We study then briefly the control of free

surfaces and problems where the control variable is a geometrical argument (such as in optimum design).

We end this chapter with several

open questions.

In Chapter 5 we give a rather concise presentation of the use of mixed finite elements for the numerical computation of optimal For further details we refer to Bercovier [1].

controls.

All the examples presented here are related to, or motivated by, specific applications, some of them being referred to in the Bibliography.

We do not cover here, among other things: the controllability problems (cf. Fattorini [1], Russell [1] in these proceedings), the stability questions, such as Feedback Stabilization (let us mention in'this respect Kwan and K. N. Wang [l], J. Sung and C. Y. Yii [1], and Sakawa and Matsushita [1]; cf. also Saint Jean Paulin [1]);

the identification problems for distributed systems, which can be put in the framework of optimal control theory, and for which we refer to G. Chavent [1], G. Chavent and P. Lemonnier [1] (for applications to geological problems), to G. I. Marchuk [1] (for applications in meteorology

and oceanography), to Begis and Crepon [1]

(for applications to oceanography), to J. Blum (for applications to plasma physics); cf. also the surveys Polis and Goodson [1] and Lions [11];

problems with delays, for which we refer to Delfour and Mitter [1] and to the bibliography therein; multicriteria problems, and stochastic problems. For other applications than those indicated here, let us refer to the recent books Butkovsky [1], Lurie [1], Ray and Lainiotis. P.

K. C. Wang [1].

The detailed plan is as follows:

Chapter 1. Optimality conditions for linear-quadratic systems. 1.

A model example.

1.1 Orientation

l.],

OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS

1.2 The state equation 1.3 The cost function.

The optimal control problem

1.4 Standard results 1.5 Particular cases 2.

A noninvertible state operator.

2.1.Statement of the problem 2.2 The optimality system 2.3 Particular cases 2.4 Another example

2.5 An example of "parabolic-elliptic" nature 3.

An evolution problem 3.1 Setting of the problem 3.2 Optimality system 3.3 The "no constraints" case 3.4 The case when

Uad = {vlv > 0

a.e. on

E}

3.5 Various remarks 4.

A remark on sensitivity reduction 4.1 Setting of the problem 4.2 The optimality system

5.

Non well set problems as control problems 5.1 Orientation

5.2 Formulation as a control problem 5.3 Regularization method Chapter 2. 1.

Duality methods.

General considerations

1.1 Setting of the problem 1.2 A formal computation 2.

A problem with constraints on the state 2.1 Orientation

2.2 Setting of-the problem 2.3 Transformation by duality 2.4 Regularized dual problem and generalized problem 3.

Variational principle for the heat equation 3.1 Direct method 3.2 Use of duality

3

J. L. LIONS

4

Chapter 3.

Asymptotic methods.

1.

Orientation

2.

Cheap control.

An example

2.1 Setting of the problem 2.2 A convergence theorem 2.3 Connection with singular perturbations 3.

Homcgeneization

e

3.1 A model problem 3.2 The homogeneized operator 3.3 A convergence theorem Chapter 4. 1.

Systems which are not of the linear quadratic type.

State given by eigenvalues or eigenfunctions 1.1 Setting of the problem 1.2 Optimality conditions 1.3 An example

2.

Another example of a system whose state is given by eigenvalues or eigenfunctions 2.1 Orientation

2.2 Statement of the problem 2.3 Optimality conditions 3.

Control in the coefficients 3.1 General remarks 3.2 An example

4.

A problem where the state is given by an eigenvalue with control in the highest order coefficients 4.1 Setting of the problem 4.2 Optimality conditions

5.

Control of free surfaces

5.1 Variational inequalities and free surfaces 5.2 Optimal control of variational inequalities 5.3 Open questions 6.

Geometrical control variables 6.1 General remarks 6.2 Open questions


Chapter 5.

Remarks on the numerical approximation of problems of optimal control

1.

General Remarks

2.

Mixed finite elements and optimal control

2.1 Mixed variational problems 2.2 Regularization of mixed variational problems 2.3 Optimal control of mixed variational systems 2.4 Approximation of the optimal control of mixed variational systems

5

Chapter 1

Optimality Conditions for Linear-Quadratic Systems 1.

A Model Example Orientation

1.1

We give here a very simple example, which allows us to introduce a number of notations we shall use in all that follows. 1.2

The state equation

Let

S2

be a bounded open set in

Let

A

be a second order elliptic operator, given by n

n

(1.1)

A

i

J

, with smooth boundary

Rn

=l

where the functions

z

i

Ju

(a.-(x) 13

ai

,

a

ai3

a

(x)

L-() ; we introduce

belong to

0

+ ao

J

the Sobolev space H1(S2)

(1.2)

i

ll

J

provided with the norm

(1.3)

- jaj, X2)1/2

1

Ilmll =

where

H _

(1.4)

02dx)1/2 = norm in

L2(S2)

(all functions are assumed to be real valued); provided with (1.3), H1(S2)

is a Hilbert space; for o, y ( D2aii Ni

(1.5) We assume

(1.6)

A

to be

as dx

H1(Q)

we set

+L J9ai aax ,y dx + f9a04'dx

H1(q) - elliptic, i.e. 2

, a >0 , vo E

7

H1 (c )

I

J. L. LIONS

8

The state equation in its variational form is now:

a(y,*) = (f,*) + fr v*dr

(1.7) where

(f,*) = f f*dx

f given in

,

"control variable" v is given in We recall that of

,y

space

on

r

V4rcHI(52)

L2(52)

L2 (r)

, and where in (1.7) the

.

one can uniquely define the "trace"

it is an element of

;

H112(r))

VyeH1(s2)

L2(r)

(actually of a smaller

and the mapping

y --- * I r

is continuous from

-

H1(s2)

L2(r)

.

Therefore the right hand side in (1.7) defines a continuous linear form on

H1(52)

so that, by virtue of (1.6):

,

Equation (1.7) admits a unique solution, denoted (1.8)

by

y(v); y(v) eH1(s2)

is affine continuous from

y(v)

v

and the mapping L2(r)

-

1(2)

.

The interpretation of (1.7) is as follows: Ay(v) = f

(1.9)

ay(v)

(1.10)

= v

in

sZ

on

r

,

A

where

as =

a

cos(v,xi) , v = unit normal to

r

directed

J

A

toward the exterior of aid eLW(52),

52

;

of course, under only the hypothesis that

(1.10) is formal; in case

aid eWl-(,Q)

as (i.e.

(1)

ax

A-0) V

k)

,

then one can show that

k

H2(Q)

00 a

2

' axi ax &L2 (2) J

y(v)eH2 (S2)

1) (

and


(1.10) becomes precise.

In the general case one says that

solution of (1.7) is a weak solution of (1.9)(1.10)

.

9

y = y(v)

p

We shall call (1.7) (or (1.9)(1.10)) the state equation, y(v) being the state of the system. The cost function.

1.3

The optimal control problem.

To each control v we associate a cost J(v) defined by (1.11)

where

J(v) = fr Iy(v)-zdl2dr + N fr v2dr

zd

Let

is given in v

L2(r)

and where

belong to a subset

Uad

N

of

,

is a given positive number.

L2(r) (the set of admissable

controls); we assume (1.12)

Uad

is a closed non-empty convex subset of

We shall refer to the case

L2 W)

Uad = L2(r) as the "no constraint"

case.

The problem of optimal control is now (1.13)

1.4

find inf

J(v)

Standard results.

,

vE Uad

(cf. Lions [1])

Problem (1.13) admits a unique solution u (the optimal control). This optimal control u is characterized by (J'(u), v-u)

>_ 0

VV&Uad

(1.14)

Up Uad

where

(J'(u), v) = d

J (u4 v)Ir=0 (this derivative exists).

The condition (1.14) which gives the necessary and sufficient condition for

u

to, minimize

J

a Variational Inequality (V.I.).

over

Uad

is (a particular case of

J. L. LIONS

10

An explicit (and trivial) computation of

J'(u)

gives (after

dividing by 2)

Ir (Y(u)-zd) (y(v)-y(u)) dr + N fr u(v-u) dr (1.15)

Vve Uad, uc Uad.

#

Transformation of (1.15) by using the adjoint state. In order to transform (1.15) in a more convenient form, we introduce the adjoint state p defined by

A* p = 0

in 2

,

(1.16)

aA*=y - zd where we set

on

r

y(u) = y, A* = adjoint of

A

The variational form of (1.16) is

(1.17)

a* (P,l,) = Ir (Y-zd)* dr

V*eH1 (52)

where we define

(1.18)

a* (,,4r) =

Let us set X = fr (Y-zd) (y(v)-Y)dr

by taking

y(v)-y

in (1.17) we obtain

X = a* (p,y(v)-y) = a(y(v)-y,p) = (by using (1.7)) = Ir (v-u)p dr

and (1.15) becomes

fr (p+Nu) (v-u) dr > 0 (1.19)

ue Uad'

ve Uad'


We can summarize as follows the results obtained so far: control

the optimal

of (1.13) is characterized through the unique solution

u

{y, p, u}

of the optimality system given by:

=

(1_2nt

,

-u

=y- zd

av

on

r,

Ir (p+Nu)(v-u)dr > 0 VvE Uad uE Uad

1.5

Particular cases. 1.5.1

Uad = L2(r)

If

(1.21)

The case without constraints.

p + Nu = 0

, the last condition in (1.20) reduces to

.

Then one solves the system of elliptic equations:

Ay = f, A*p = 0

in

0

(1.22)

aA+Np=O,ap=y-zd on and

u

is given by (1.21). 1.5.2

In case

Uad = {vi v >_ 0 a.e. on

Uad

r)

is given by 1.5.2, the last condition (1.20) is

equivalent to (1.23)

u

0, p + Nu >_ 0, u(p+Nu) = 0

i.e.

(1.24)

.

u = sup (0, N) =

N p

11

J. L. LIONS

12

Then one solves the system of non-linear elliptic equations:

Ay=f,A*p=0 in St, (1.25)

a Np =0, and

y

a

zd

on

r

is given by (1.24).

u

Remark 1.1

By virtue of the way we found (1.25), this system admits a unique solution

{y,p}

.

Remark 1.2

We have two parts on

r

r- = {xI xer, p(x) s 0}, r+ = {xl xer, p(x) > 0} (these regions are defined up to a set of.measure u = 0

r+

on

.

The interface between

and

r

as a free surface or as a commutation line.

0

r+

on

r ) and

can be thought of

#

Remark 1.3

For interesting examples related to the above techniques, we refer to Boujot, Morera and Temam [l]. 2.

#

A non invertible state operator. 2.1

Statement of the problem

In order to simplify the exposition we shall assume that A = - o

(2.1)

but what we are going to say readily extends to the case when

A

self-adjoint elliptic operator of any order (or to a self-adjoint system).

We suppose that the state y

qy=f

- v

in

(2.2) on

r

.

2

,

is given by

is any

OPTIMAL CONTROL OF DISTRI6UTED SYSTEMS

But now if

A

denotes the unbounded operator

-A

13

with domain

= 0 on r), 0 e spectrum of A so that A is not

jpe Hl(52), A eL2(S2),

invertible; but a necessary and sufficient condition for (2.2) to admit a solution is (2.3)

(f-v,l) = 0

and then (2.2) admits an infinite number of solutions; we uniquely define

y(v)

by adding, for instance, the condition

(2.4)

M(Y(v)) = 0

where

M(') = TiT f9 dx,

,

js2j

= measure of

2

Summing up: we consider control functions v which satisfy (2.3); then the state y(v) of the system is given as the solution of (2.2) (2.4).

#

The cost function is given by

(2.5)

3(v) = frly(v)-zdl2 dr + N f2v2dx

We consider EUad = closed convex subset of L2(r) and of the (linear) (2.6)

set defined by (2.3)

and we want again to solve (2.7)

2.2

inf J(v), ve Uad

The optimality system.

One easily checks that problem (2.7) admits a unique solution which is characterized by (we set

y(u) = y ):

fr(y-zd)(Y(v)-Y)dr + N(u,v-u) > 0 (2.8)

ucUad

#

yve Uad'

u,

14

J. L. LIONS

We introduce now the adjoint state p as the solution of

Ir(y-zd)dr

-op =

a = y-zd M(p) = 0

on

in

s

r

.

We remark that (2.9) admits a unique solution. If we take the scalar product of the first equation in (2.9) with

y(v) - y , we obtain

(-op, y(v)-y) = Inl 1r(y-zd)dr (1, y(v)-y) = 0

(by virtue of (2.4)) = - Jr- (y(v)-y)dr +

+ (P, -6(Y(v)-Y)) _ - 1r(Y-zd)(Y(v)-Y)dr + (p,-(v-u))

(the use we make here of Green's formula is justified; one has just to think of the variational formulation of these equations). Then (2.8) reduces to (2.10)

(-p+Nu,v-u) ?. 0

Vvc Uad, uE Uad

Summarizing, we have: the optimal control u , unique solution of (2.7), is characterized by the solution {y, p, u} optimality system:

I -ny 1

= f-u, -AP =

a = 0, M(y) = 0,

tr(Y-zd)dr

= y-zd

on

in

r

M(p) = 0,

(-p+Nu, v-u)

0

Vve

Uad

Ut Uad

2

of the

OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS 2.3

Particular cases

Let us suppose that (2.12)

Uad =

{v

I

(v,1) _ (f,l)}

i.e. the biggest possible choice of

Uad

Then (2.10) is equivalent to

(2.13)

p + Nu = c = constant and

up- Uad

i.e.

-(p,l) + N(u,l) = c (2.14)

c = N M(f)

1521

= N(f,l)

i.e.

.

Then one solves first the system:

-Ay + N = f - Mf ep =

1r(y-zd)dr

(2.15) av

= 0,aV= y - z d

on

r

,

M(y) = M(p) = 0 and then

(2.16)

u=Mf+N'

#

Let us now suppose that (2.17)

Uad = {vJ v >_ 0 a.e. in g,

under hypothesis (2.18)

Mf > 0

(v,1) = (f,1)}

15

J. L. LIONS

16

which implies that

Uad

is not empty (case

Mf < 0) .

does not reduce to

{0}

(case

Mf = 0

)

or

Then the solution of (2.10) is given by

u=N+Mf+r-Mr

(2.19) where

r= (N+Mf xFR, x being a solution of

(2.20)

x = M(r)

fr(y-zd)dr

(actually unique if k =

+ Mf

0)

admits a solution

x = M(r)

Indeed, let us check first that

at least assuming that

,

L-(2)

N if we set p(x) = 0

p(x) = M((x-x) ) then for

p(x) is an increasing function, for

- c, p(X) = x - M(x) = x - Mf

x

x

enough, hence the result follows; let us notice that (2.19) does not depend on the choice of now check that

(2.21)

u

satisfies (2.10).

x

M(u) = M(f)

,

large

defined by

satisfying (2.20); let us

We can write

u = N + Mf - x + r = (N + Mf We have

u

0

and

+ u, v-u) = (Mf - x + r, v-u) = (r,v-u)

(-

N but (r, u) = 0

hence

+ u, v-u) = (r,v)

(-

N hence the result follows.

0

since

r

and

v

are > 0


The optimality system is given by

-AY =f - (N+Mf-x)+, -op = T2T 'r(y-zd)dr (2.22)

a = 0,

= y-zd

on r

X = M {(N + Mf - a) Remark 2.1

)

,

M(y) = M(p) = 0 ,.

.

#

Regularity of the optimal control.

It follows from (2.20) or (2.21) that (2.23)

ue H1(Q)

since

,

+ Mf - x eHl(S2)

.

N

Let us also remark that if does not improve (2.23)

zd a H1/2(r)

then

p e H2(12)

but this

#

.

Remark 2.2

One can find (2.19) (2.20) by a duality argument.

(cf.

Chapter 2 for the duality method). 2.4

Another example.

As an exercise, let us consider the state equation -Ay = f

2

in

,

(2.24)

1

a=v

on

r

which admits a set of solutions {y + constant} iff (2.25)

- fry dr = fnf dx

We define the state (2.26)

M(y) = 0

y(v) .

.

as the solution of (2.24) which satisfies

17

18

J. L. LIONS

If we consider the cost function

(2.27)

J(v) = frIY(v)-zdl2 dr + N frv2 dr

then the optimality system is given by

-AY = f, -AP = n fr(y-zd)dr

av

= u,

= y-z

av

on

d

in

s2

r

(2.28)

M(y) = M(p) = 0

fr(p+Nu)(v-u)dr

where

Uad

0

Vve Uad, ue Uad

is a (non-empty) closed convex subset of the set of v's in

L2(Q) which satisfy (2.25). 2.5

An example of "parabolic-elliptic" nature Let us consider now an evolution equation

(2.29)

at - Ay = f-v

fr veL2(Q)

Q = St x JO, T[

in

,

with boundary condition

(2.30)

a = 0

on

z = r x ]0, T[

and (2.31)

y(0) = y(T)

on

2

(where y(t) denotes the function x-y(x,t))

.

The equations (2.29) (2.30) (2.31) admit a solution (and actually a set of solutions y + constant) iff (2.32)

fQ v dx dt = fQ f dx dt

.


19

Let us then define the state of the system as the solution

y(v)

of (2.29) (2.30) (2.31) such that (2.33)

fQ y(v) dx dt = 0

.

If the cost function is given by J(v) = fQly(v)-zdl2 dx dt + N f v 2 dx dt, N > 0, zdeL2(Q)

(2.34)

Q

and if

Uad

is a (non empty) closed convex subset of the v's in

L2(Q)

such that (2.32) holds true, the optimality system is given by

a - Qy=f - u -

k - op = Y-Zd at

a (2.35)

- TQ7fQ(Y-zd) dx dt

on

z

y(0) = y(T), p(0) = P(T)

,

fQydxdt=fQpdxdt=0 fQ (-p+Nu) (v-u) dx dt > 0

Vve Uad

ue Uad

3.

An evolution problem. 3.1

Setting of the problem.

We consider now an- operator A as in Section 1

(cf. (1.1)); we

use the notation (1.5) and we shall assume there exists

a

and

a>O

such that

(3.1) a(o,m) + X14,12

--

allmll2 V4e Hl(S2)

(this condition is satisfied if that

a0, a3 eLm(2)

and

aij&L (St) such

J. L. LIONS

20

n

i j - a1E i2 , a1A

a..(x)

E

i,j=1

We consider the state equation:

(3.2)

at + Ay = f

(3.3)

a = v

(3.4)

Y(0) = yj

Q = S2x]o,T], feL2(Q)

in

vcL2(Z),

on

E

(1)

on

,

2, y0cL2(2)

-

This problem admits a unique solution which satisfies y cL2(O,T;H1(S2))

(3.5)

.

(cf. Lions [1] [2] for instance, or Lions-Magenes [1])

The variational formulation of this problem is

(L, *) + a(y,i,) = (f,*) + fr v$dr VyeH1(s2)

(3.6)

with the initial condition (3.4). Let the cost function

J(v) 2

(3.7)

J(v) = fE Iy(v)-zdl

and let

Uad

be given by

dE + N fE v2dz, zdeL2(2), N > 0

be a (non empty) colsed convex subset of

L2(z)

consider the problem of minimization: (3.8)

inf J(v), ve Uad '

3.2

Optimality system.

Problem (3.8) admits a unique solution, say u, which is characterized by

(1)

We write

a

av

instead of a

ev A

,

.

We


21

(J'(u), v-u) > 0 Vve Uad, ue Uad

(3.9)

i.e. (where we set

y(u) = y):

f (y-zd) (y(v)-y)dZ + N fz u(v-u)dz > 0 (3.10)

Vve U

ad'

uc Uad

In order to simplify (3.10) we introduce as in previous sections the adjoint state p given by

(3.11)

aP* = y - zd

[P(T) = 0

on

(7)

z

on

Then

fE (y-Zd) (y(v)-y)dz = jr, p(v-u)dE

so that (3.10) becomes (3.12)

j

yve Uad, ue Uad

(p+Nu) (v-u)dz > 0

The optimality system is given by

at I

(3.13)

1

+Ay=f,-

at + A* p = 0

av,d = u

=

y

-z

on

on

y(0) = y0, p(T) = 0

fF (p+Nu) (v-u)dz > 0

(1)

We write av* instead of as A

;

in Q,

,

2 ,

dve Uad, uc U d . a

J. L. LIONS

22

The "no constraints" case.

3.3

If we suppose that (3.14)

Uad = L2 (E)

then (3.12) reduces to p + Nu = 0

(3.15)

.

Then one solves first the system in {y, p}

+ Ay = f, (3.16)

a Y(

and then

u

aP a

+ A* p = 0

+ U'p = 0, ap* = y - zd av

in on

Q

E

0) = y0, p(T) = 0

is given by (3.15).

Remark 3.1

We obtain a regularity result for of

u = - N p

L2(0,T; H1/2(r)) (and one has more, since

M eL2(Q) 3.4

,

if we assume more on

Tha case when

zd ).

;

u is an element

peL2(O,T; H2(12)) and

#

Uad = {vfv > 0 a.e. on Z}

In the case when (3.17)

Uad = {v1 veL2(E), v >_ 0 a.e. on E}

then (3.12) is equivalent to (3.18)

u

0, p + Nu > 0,

u(J+Nu) = 0

on Z

i.e.

(3.19)

u = N p-

lnen the optimality system can be solved by solving first the non-linear system in {y, p} given by


a+Ay=f, -at+A*p=0 in

a

(3.20)

p- = 0, a2 = y -zd

-

on

23

Q

z

N

y(0) = y0, p(T) = 0

on

2

and by using next (3.19). Remark 3.2 We obtain

(as in Remark 3.1) the regularity result on the

optimal control: ue L2(0,T;H112(r))

(3.21)

3.5

.

Various remarks.

Remark 3.3

For the "decoupling" of (3.16) and "reduction" of the "two point" boundary value problem in time (3.16) to Cauchy problems for non linear equations (of the Riccati-integro-differential type) we refer to Lions [1] [3] and to recent works of Casti and Ljung [1], Casti [1] Baras and Lainiotis [1](where one will find other references) for the decomposition of the Riccati equation.

We refer also to Yebra [1],

Curtain and Pritchard [1], Tartar [1]. Remark 3.6

We also refer to Lions, loc. cit, for similar problems for higher order operators A, or operators A with coefficients depending on x and on t ; also for operators of hyperbolic type, cf. Russell [2], Vinter [1], Vinter and Johnson [1]. 4.

A remark on sensitivity reduction

4.1

Setting of the problem Let us consider a system whose state equation is again (3.2),

(3.3), (3.4) but with a "partly known" operator A let us consider a family

A(t)

of operators:

.

More precisely,

24

J. L. LIONS

(4.1)

A(r)m = - E aal

where

e R

3O) + E

axe + a0(x,K) 6

we suppose that

;

Iaid,

a0 c Lm(2xR)

ail

(4.2)

Then for every

(4.3)

a z 1i 2

r} n.J

z ai

the state

,

at + A(4)y = f

in

. ate- = v

F.

AN = y0

on

on

a>O,tlr, eR is the solution of

Q

s2

The cost function is now (4.4)

fE ly(v,?.)-zd12 dE + N fz v2 dE

We know that

A(,Y)

is "close" to

A(r0)

, and we would like to obtain

an optimal control of "robust" type, i.e. "stable" with respect to

changes of A(d "around" A(40)

.

p

A natural idea is to introduce a function

(4.5)

f

such that

P is > 0, continuous, with compact support around (C)dr. = 1

(of course the choice of about the system). (4.6)

p

will depend on the information we have

We now define the cost function 2

J(v) = fR

fEty(v,e-)-zdl

The problem we want now to solve is (4.7)

inf J(v), ve Uad

dE + N fE v2 dE


denotes a (non empty) closed convex subset

where, as usual, Uad of

25

L2(z)

4.2

The optimality system.

Problem (4.7) admits a unique solution u , which is characterized by

dz + N f u(v-u)dE

fP(d& fz (y(u,t.)-zd) (y(v,r) (4.8)

vv e

Uad

ue Uad

be the solution of

Let

-

p=0

+ A*

ate- _ y()

(4.9)

p(T) = 0

on

-z

2

d

in

Q

on

E,

.

Then multiplying the first equation (4.9) by y(v,s) 0 = - Iz

so that (4.8)

(v-u)dz

dz + Iz

educes to Nu) (v-u) dz

(4.10)

0 Vve Uad, ut Uad

Summarizing, the optimality system is given by

+ A(n)y = f, - a +

(4.11)

aV

= u, A(Y,)

a v( ) A* c

y(0) = y0, p(T) = 0

and (4.10).

we obtain

in

0

= y - z d on z

,

Q

0

J. L. LIONS

26

Remark 4.1

For numerical applications of the preceding remark, as well as for other methods of reduction of sensitivity in the present context, we refer to Abu El Ata [1] and to the bibliography therein. Remark 4.2

y

If

mass + 1 at K,0) in the weak star topology

&,)d

of measures (i.e.

support) and if we denote by

v

uP

continuous with compact

the solution of (4.7), then one can

show that (4.12)

-+ u in L2(E) weakly

u

P

where (4.13)

5.

u

solves inf

ve Uad

Non well set problems as control problems 5.1 Orientation.

Let us consider the following (non-well-set) problem (this problem arises from a question in medicine, in heart disease; cf. Colli-Franzone, Taccardi and Viganotti [1]): in an open set

with boundary

Q

r0Ur 1, a function

satisfies an

u

elliptic equation (5.1)

Ay = 0

,

and we know that

(5.2)

a

-

=O

rl

on

A

and we can measure (5.3)

y = g

on

S

Figure 1


If

g

is precisely known, this uniquely defines

27

but, as it

y

is well known, in an unstable manner.

The problem is to find

y

on

ro

Formulation as a control problem

5.2

Ay(v) = 0

(5.4)

in

y(v) = v

on

of our system as the solution of

y(v)

Let us define the state 2 ro ,

aA(v)=0 on rl (we assume that the coefficients of

A

are such that this problem

admits a unique solution).

We introduce

Uad

as the smallest possible closed convex subset

of

L2(ro) which "contains the information" we may have on the values

of

y (the "real one") on

ro

;

in general it will be of the form

Uad = {vI veL2(r0), mo(x) < v(x) < ml(x)

on r o

,

(5.5)

mo

and

ml

given in

L'-(r o)}

We introduce the cost function (5.6)

J(v) = IS ly(v)-gl2 dS

and we want to solve the problem (5.7)

If

Uad

(5.8)

inf J(v), ve Uad

has been properly chosen (i.e. not too small)

inf J(v) = 0

which is attained for

v = the value of y

on

r0

.

But. of course, this is again an unstable problem and following the idea of Colli-Franzone, Taccardi and Viganotti, loc. cit. we are

28

J. L. LIONS

now going to regularize the above problem of optimal control. Remark 5.1

Another approach to the problem stated in 5.1 Is given in Lattes-Lions [1] via the Quasi Reversibility method. 5.2

Regularization Method

There are a number of methods available to "stabilize" (5.7). Following Colli-Franzone, et al., we introduce the Sobolev space and the Laplace-Beltrami operator

on

r0

Uad = {vI ve H2(r0), m0 0

a(m,o)

(3.3)

is the linear form associated to

Remark 3.1

The result below readily extends to higher order elliptic operators A .

M

We assume that feL2(Q) (actually one could take (3.4)

where

H-1 (2)

u0eL2(Q)

.

= dual of

feL2(0,T,H-1(Q)


41

We define

(3.5)

U = (kf 4sH1(2x] 0,T[), 4, = 0

on

z,

By virtue of (3.3), A is an isomorphism from whose inverse is denoted by

(3.6)

A-1

J(0 = JO [1 a($) +

+

.

(x,0) = u0(x)

Ho (2)

onto

on

H-1(2)

We now set

a (A 1(f

-

)) - (f,o)] dt +

Im(T)12

where we have used the notation (3.7)

a(.*) - a(.k,,o)

.

We are going to check that inf J(o) = J(u), u

(3.8)

solution of (3.1)

,

qscU

the inf in (3.8) being attained at the unique element

u

Proof: we set

where

*

spans the set of functions in

H1(Qx)0,T()

such that

*=0 on E,*(0)=0. We have

(3.9)

J(.) = J(u) + K(*) + X(u,$)

(3.10)

K(V)

fO [- a(p) +

a

(A-1 at)] dt +

au X(u,*) = JT [a(u,*)-a(A 1(f - at)' 0

+ (u(T), 4'(T))

A-1

-

I41(T)I2

a )-(f,*)] dt +

s2}

42

J. L. LIONS

But from the first equation (3.1) we have

A-1

and

a(u, A-1

(f

at) = u _ (u, A A-1

at

X(u,*) =

)

= (u,

)

, so that

at)-(f,$)]dt +(u(T) , *(T))

JT 0

But taking the scalar product of the first equation (3.1) by

it

X (u,$) = 0 so that J(o) = J(u) + K(ey)

(3.11)

since

K(*) > 0 3.2

and

iff

K(ey) = 0

,p = 0 , we obtain (3.8)

Use of duality

Let us define

(3.12)

F(¢) =

a(,k)

on

HO(B)

r Then the conjugate function

(3.13)

a(A-1

F*(m*) =

F*

*)

of

F

is given on

H-1 (2)

by

,

and

JW = IT 0

(3.14)

4EU

CF(O)+F*(f - -al) - ] dt +

Iu0I2

-

gives


It follows that Iu012

Iu012

and that JW =

(= J(u)) iff

2

F(,k) + F*(f - at

_< 0, f ->

i.e.

= derivative of

i.e.

=

i.e.

A-1 (f

- 1)

F*

at

f - at

43

Chapter 3

Asymptotic Methods 1.

Orientation

The aim of asymptotic methods in optimal control is to "simplify" the situation by asymptotic expansions of some sort. This can be achieved by one of the following methods: (1) simplification of the cost function - this is, for instance the case when the control is "cheap", cf. Section 2; (ii) simplification of the state equation, by one of the available asymptotic methods: (j) the most classical one is the use of asymptotic expansions in terms of "small" parameter that may enter the state equation,

i.e. the method of perturbations, in particular the method of singular perturbations; we refer for a number of applications in Biochemistry or in Plasma Physics to J. P. Kernevez [1], Brauner and Penel [1], J. Blum [1] and to the bibliography therein; cf. also Lions [7].

(jj) the homogenization method for operators with highly oscillating coefficients;cf. Section 3; (JJJ) the averaging method of the type of Bogoliubov-Mitropolski [1]; we refer to Bensoussan, Lions and Papanicolaou [1]; (iii) simplification of the "synthesis" operator by the choice of a particular feedback operator (in general on physical grounds) ;

We do not consider this aspect here; we refer to Lions [4], Bermudez [1], Bermudez, Sorine and Yvon [1];.it would be apparently of some interest to consider this question in the framework of perturbation methods. 2.

Cheap control.

An example.

2.1 Setting of the problem.

With the notations of Chapter 1, Section 3.1, we consider the state equation given by

45

46

J. L. LIONS

(2.1)

at + Ay = f

in

a = v

z

on

Q = 2x]O,T[

A

y(0) = y0

2

on

We consider the cost function

(2.2)

Je(v) = IE Iy(v)-zdl

where

c > 0

is "small"

2

dz + e IE v2 dE

.

This amounts to considering the control v as "cheap" - a situation which does arise in practical situations, where one often meets the case where acutally Let let

ue

Uad

e = 0

.

be a (non-empty) closed convex subset of

L2(z) , and

be the solution of

Je(uE) = inf

J&(v), ve Uad

(2.3)

ueeUad We want to. study the behavior of

ue

as

0

We shall see that this question is related to problems in singular perturbations. 2.2

A convergence theorem.

Let us set (2.4)

uc

y(ue) = ye

is characterized by

IE (Y,-ad)(y(v)-y e)dE + e IE ue(v-ue)dz >_ 0, Vve Uad (2.5)

u

e

U

ad


We define

0(v) = y(v) - y(O) (where here y(0) denotes the solution y(v) of (2.1) for v = 0) ; we have

{v) + Ac(v) = 0, in Q (2.6) avA A

0(v)It=O = 0 on 2

If we set (2.7)

0(uE) = 0e

(2.5) can be written

f

0E(0(v)-0E)dz + e fZ uE(v-u6)dz >_

(2.8)

'- fZ (zd-y(0))(0(v)-0E)dE

Let us consider the case when (2.9)

r = a2

is a

and let us write the Set which are zero for

(2.10)

t.;:: 0

- + A0 = 0

variety ,

C"

in

I

of all distributions

m

in

9xJO,T[

and which satisfy

52x3-m

T[

One can show (cf. Lions-Magenes [1], Vol. 3) that one can define, in a unique manner

{di (2.11)

- 1E } e D' (E) x D' (E) a J

D'(E) = space of distributions on

z ,

47

48

J. L. LIONS

the mapping

0 -

with the topology of

,

{O1z

at- iE } being continuous from a

D' (2x]-',T[)) -

f

(provided

D'(E) x D' (z)

We then define

(2.12)

K = 10 I ocj , 41z &L2(E), a

2=

I+ib

eL2(E)I

A

which is a Hilbert space for the norm (f2[[2+(b)2]dL)1/2

(2.13)

We define next (2.14)

Kad

Kad = (ol oeK, M,e Uad

is a closed convex subset of

K

With these notations, (2.8) is equivalent to

eKad

4. FE

(2.15)

(de9 0-06) 2 + L (E)

e)) 2 L (E)

(I"l0e)

(zd-y(0), 0-0 e) 2 L

Vme Kad

.

(E)

We can now use general results about singular perturbations in Variational Inequalities; using a result of D. Huet [1], we have:

me y 0o

where mD

in

L2(2)

as

e - 0 ,

is the solution of

(2.16)

(40,0--00) > (zd-y(D),m-0D) VOe Tad ,

#0FKad where


Kad = closure of

Kad

49

K

in

(2.17)

K - {ml 001.01Z

eL2{z)}

But if Proj_ = projection operator in

K

on %d , we have

Kad

(2.18)

00 = Proj_

(zd-y(0))

Kad

and going back to (2.19)

yE

one has:

yE -. Y(0) + Proj

_

(zd-y(0))

in

L2(z)

Kad

Remark 2.1

One deduces from (2.19) the convergence of

UE

in a very weak

topology. 2.3

Connection with singular perturbations

Consider now the "no-constraints" case. (2.20)

Then (2.18) reduces to

00 = zd-y(O)

so that

(2.21)

yE -. zd

in

L2(z)

which was easy to obtain directly. But since in general, considering not have

zd

to be smooth, one does

the convergence (2.21) cannot be improved

zdIt=O = yo'r , (no matter how smooth are the data) in the neighborhood of z

.

There is a singular layer around

t = 0

on

t - 0

on

z .

The computation (in a justified manner) of this type of singular layer is, in general, an open problem. We refer to Lions .[8] for a computation of a surface layer of

similar nature, in a simpler situtation, and for other considerations along these lines.

50

J. L. LIONS

3.

Homogeneization

A model problem

3.1

Notation: We consider in

Rn functions

y - ai .(y)

with the

following properties:

(3.1)

Jaij &L-(R n)

,

aij

is Y-per iodic, i.e. Y = ]O,y0 (x... X]O, Yo

a1

is of period

yo

in the variable

and

yk

E aij(y) riej ' a z qZ , a > 0 ,.a.e. in y ; for

e>O , we define the operator

Ac

by

n

A&o

(3.2)

az (aij(E)

)

.

J

Remark 3.1

The operator

is a simple case of operators arising in the

AE

modelization of composite materials; operators of this type have been the object of study of several recent publications; let us refer to de Giorgi-Spagnolo [1], I. Babuska [1] [2], Bakhbalov [11, BensoussanLions-Papanicolaou [2] and to the bibliography therein. The state equation We assume that the state

(3.3)

in

+ + At)yc = f

(3.4)

ay = v avAt

(3.5)

ye,t=0 = y0

on

z

on

,

2


y&(v)

is given by

Q = 2x]O,T[


(3.6)

Let

Je(v) = jE lye(v)-zdl2 dE + N jE v2dz, N>O,zde L2(s)

Uad

be a closed convex subset of

L2(E)

51

.

.

By using Chapter 1, we know that there exists a unique optimal control

ue

, solution of

Je(ue) = inf

(3.7)

Je(v), ve Uad, ue Wad

The problem we want to study is the behavior of 3.2

as

e -+ 0

The homogeneized operator

Let us consider first the case when

then that, .when

(3.8)

ay

= v

f

is fixed.

One proves

in

on

E

,

Q

,

A

ylt=O = yO A

v

0 ,

e

at+Ay=

and where

u

on

SZ

,

is given by the following construction.

One defines firstly the operator

(3.9)

for every

Al = - Day- (aij(y) aa)

j

one defines

constant, of

Al(X3-yj) = 0 (3.10)

X3 Y-periodic

and one defines next

XJ

on

Y

;

as the unique solution, up to an additive

J. L. LIONS

52

aij

al

(Xj-yj, X3-yj),

JYJ = measure of

Y

(3.11) a a-'-y- dy

ayi

Then n 2

A

(3.12)

i j=1

aij axa axj

which defines an elliptic operator with constant coefficients; called the homogenized operator associated to 3.3

be defined by (3.8); we define

J(v) = fE Iy(v)-zdl2 dz + N fE v2 dz

and let

u

be the unique solution of J(u) = inf

(3.14)

J(v), veUad' ue Uad

We have:

(3.15)

ue -+ u

in L2(E)

as

a-0.

Proof:

Let us set (3.16)

Since

(3.17)

is

A convergence theorem

Let us consider the "homogeneized control problem": let

(3.13)

A

A`

ye(ue) = ye, y(u) = y

J (v) >_ N fZ v2dz

Plle L2(E)

we have

JE Iy-zdl

2

dE + N JE (u)2dz = X

E-+0 But for every

v e Uad' we know that (cf. (3.8)) y&(v) . y(v) ay

L2(0,T;H1(s)) weakly and also that

(tv)

a

- at y(v)

weakly; therefore (3.24)

ye(v)l,: - y(v)l,

so that

(3.25)

JE(v) - J(v)

in

L2(E)

in

strongly

in

L2(0,T;H-1(2))

54

J. L. LIONS

Then the inequality

Je(ue)

X < J(v), vs

(3.26)

Je(v)Vve Uad

gives

Uad

But one can show that y

(3.27)

so that

X = J(u), hence (3.26) proves that Since

(3.28) Since

lim sup Je(ue)

J6(ue) -+ J(u) jE dye-zdl

2

u = u

J(v) Vv, we have

.

dE - fE Iy-zd(2 dE

(cf. (3.23)) it follows from

(3.28) that

N fE u2 dz - N jE U2 dE

(3.29)

Since

u

ue-+u

in

-t u

in

L2(E)

weakly, it follows from (3.29) that

L2(E)

strongly.

Remark 3.2

Let us consider the optimality system:

ate +As ye = f,- ate + (As)

a

a

ave As

pe = 0

= us,

= ye- zd

av

on

(Ac)*

y6(O) = y 0, p E{ T) = 0, o n S t together with

jE (p+Nu (3.31)

I

ue s Uad

(v-ue) dE

0 Vve Uad

in


Then, as

e - 0

55

,

L2(O,T;HO(8)) weakly,

in

(3.32)

in L2(O,T;Ho(Q)) weakly, (3.33)

where

ue

u

{y,p,u}

in

L2(z)

,

is.the solution of the "homogeneized optimality system"

Q

(3.34)

avA

= u, -'3p

avA*

-

y - zd

y(O) = y0, p(T) = 0

on

on

.

z

2

with Jfz (p+Nu)(v-u) dz >_ 0 Vve Uad' (3.35)

ue Uad

Remark 3.3

In the "no constraint" case, (3.31) and (3.35) reduce to

pe + Nue

0, p + Nu = 0

on

z

The optimality system can then be "uncoupled" by the use of a non linear partial differential equation of the Riccati type.

The

above result leads in this case to an homogeneization result for these non-linear evolution equations.

Chapter 4

Systems Which Are Not of the Linear Quadratic Type 1.

State given by eigenvalues or eigenfuncitons. 1.1

Setting of the problem.

Let

2

this is not indispensable) boundary,

Let functions

Rn , with a smooth (although


a;1

be given in

r ;

R is supposed to be connected.

, satisfying

St

ai3 = aji &L"(Q), i.J = 1,...,n (1.1)

E aid{x} i:j

a > 0

>_ a E

a.e. in

s2

.

Let us consider, as space of controls:* (1.2)

U = L-(SZ)

and let us consider (1.3)

Uad

such-that

Uad = bounded closed convex subset of

L'(Q)

.

We then consider the ei envalue problem:

JAY+vYAY

in S2,

(1.4)

-Y = 0 -on r ; it is known (Chicco (1]) that the smallest eigenvalue in (1.4) is simple and that in the corresponding one-dimensional eigen-space there is an eigenfunciton

0

We therefore define the state of our system by (1.5)

.where

(y(v), a(v)}

X(v) = smallest (or first) eigenvalue in (1.4), and Ay(v) + vy(v) = a(v) y(v)

(1.6)

y(v) >_ 0

in

Sz

Iy(v)l = 1 (1-1 = L2norm)

57

in

s?

, y(v) = 0

on

r

58

J. L. LIONS

The cost function is given by (1.7)

J(v) = I.9 ly(v)-zdl2 dx

and the optimization problem we consider consists in finding (1.8)

inf J(v), ve Uad '

1.2

Optimality conditions.

It is a simple matter to see that

v - {y(v), x(v) } is continuous from

U

weak star

(1.9)

into H1(2)

weakly x R

.

Indeed

a(0) + I2vo (1.10)

x(v) = inf

2

dx

2 Im

where

aij(x)

a-

a

Therefore if (1.10) that

x(vn)

vn

v

in

dx

L°°(s) weak star, it follows from

is bounded, hence y(vn)

is bounded in

H,(Q)

can then extract a subsequence, still denoted by y(vn), x(vn)

that y(vn) - y in L2(2)

weakly and x(vn) .+ x .

strongly, and we have

Ay+vy=xy,y=0 on r,

Y'0, IYI=1 so that

y = y(v), x = x(v)

But

; we

such

y(vn) -P y in

59


it immediately follows from (1.9) that

r there

exists

ue Uad (not necessarily unique) such that

(1.11)

J(v) = inf J(v), vc Uad

We are now looking for optimality conditions. course to study the differentiability of make first a formal computation.

(1.12)

The main question is of

v - {y(v), x(v)}

.

We set

y(v0 + v)Ie=o = y, d (v0 + rv)1t=0

assuming, for the time being, these quantities to exist.

(1.6) v by v0+ v and taking the Ay + v 0

S

Replacing in

derivative at the origin, we find

y + v y(v0) = x(v0) Y + .Y(v0)

i.e.

(1.13)

AY + v0Y - x(v0)Y = -vy(v0) + xy(v0)

Of course (1.14)

Since (1.15)

on

,y = 0

Iy(v)I =

1

r

we have

(y, y(v0)) = 0 .

Formula (1.10) gives (1.16)

X(v) = a(y(v)) + f2 v y(v)2 dx

hence (1.17)

Let us

i = 2a(y(v0),.Y) + 2 f. v0 y y(v0)dx + f. v y(v0)2 dx

J. L. LIONS

60

But from the first equation (1.6) with the scalar product with

v = v0 we deduce, by taking

y

a(y(v0).y + I2 v0y(v0)Y dx = a(v0) I9 y(vo)y dx =

= (by (1.15)) = 0 so that (1.17) gives (1.18)

a = f9 v y(v0)2 dx

The derivative

.

6,a} is given by (1.13) (1.14) (1.15) (1.18)

Remark 1.1 Since

a(v0)

is an eigenvalue of

A + v 01

solution iff (-vy(v0) + a y(v0), y(v0)) = 0

(1.13) admits a

,

which is (1.18).

We can now justify the above calculation: {y(v), k(v)}

(1.19)

with values in

is Frechet differentiable in

L"(9)

D(A)x R

V

where (1.20)

D(A) = {oI 4e HO(ST), Ale L2(2))

This is an application of the implicit function theorem (cf. Mignot Cl)); we consider the mapping

(1.21)

1b,a,V _._L.

+ v4 - 4

D(A) x Rx U - L2(2)

.

This mapping, which is a 2d degree polynomial, derivative of F with respect to o,A at

(1.22)

q,?

(A

is C"

oo, a0, v0

The partial

.

is given by

+v0-X0)O -40

We consider S1

= unit sphere of

(D(A)nS1)x Rx U

.

L2(SZ)

If we take in (1.22)

and we restrict

F

to


00 = y(v0), x0 = x(v0)

(1.23)

then (1.22) is an isomorphism; therefore by applying the implicit.

function theorem, there exists a neighborhood in (D(A)f)S1)x Rx U

and there exists a

of y(v0),",v,

yxAxU

v,

function

C"

v - {K1(v), K2(v)} (1.24)

U -ii Y x A such that F(K1(v), K2(v), v) = 0, ve U , (1.25)

K1(v0) = y(v0), K2(v0) = x(v0)

We have (y(v0),x(v0,v0) y + ax (y(v0),x(v0),v0)i.

+ av (y(v0),x(v0), v0) = 0

which gives (1.13), hence (1.18) follows and (1.16) (1.15) are immediate.

N u

is

instead of

v0

We are now ready to write the optimality conditions: if an optimal control then necessarily (1.26)

(J'(u), v-u) >_ 0 We Uad

We introduce

with

v-u

instead of

v

,

anti

i.e.

Ay + uy - x(u)

'

= - (v-u) y(u) + x y(u)

(y, y(u)) = 0 , (1.27)

j2 (v-u) y(u)2 dx

y=0 in

r.

u

62

J. L. LIONS

Then (1.26) becomes (after dividing by 2), if y(u) = y (1.28)

f (Y-zd) ., dx >_ 0

Vve Uad

In order to transform (1.28) we introduce an adjoint state {p,µ} such that

Ap + u p - X(u)p = y-zd + µy (1.29)

p = 0 on r

;

(1.29) admits a solution iff (1.30)

(1+0 1Yl2 = (Y,zd).

We uniquely define (1.31)

by adding the condition

p

(p,y) = 0

Then taking the scalar product of (1.29) with

j'

, and since

(y,y) = 0,

we obtain

IQ (Y-zd) Y dx = Ap+up-a(u)p,Y) _

_ (p,AY+uY-a(u).Y) _ (p,(v-u)Y) + i(p,y) _ - (p,(v-u)Y)

so that we finally obtain the optimality system: in order for

u

an optimal control it is necessary that it satisfies the following system, where

y(u) = y

Ay + uy = X(u)Y, y '- d, IYI = 1 (1.32)

Ap + up - x(u)p = Y(Y,zd) - zd, (p,Y) = 0,

y, p = 0 and

on

r

to be


;fQ py (v-u) dx >_ 0

63

Vve Uad'

(1.33)

ue Uad

I

'

Let us also remark that the system (1.32) (1.33) admits a solution. 1.3

An example.

The following result is due to Van de Wiele [1].

We consider the

case: (1.34)

Uad= {vI k0_ 0

on

521,

ps0

on

520

p=0

on 2

(go U 21)

We are going to conclude from this result that

f if (1.37)

zd is not an eigenfunction for A+uI, and if u is any

optimal control, then necessarily ess sup u = k1, ess inf u = k0 .

(1) One can define more precisely these sets up to a set of capacity 0

J. L. LIONS

Suppose on the contrary that, for instance, ess sup u < kl Then one can find

such that

k0 < u + k < kl

,?.38)

E.,t

k > 0

y(u+k) = y(u), X(u+k) = x(u)+k

and

u+k

is again an optimal

control; we have therefore similar conditions to (1.36), but now, by virtue of (1.38), the analogs of ;(,,+k) = 0

-:1uded.

in

and

20

s21

are empty and therefore

i.e. (cf. (1.32) y(y,zd) = zd , a case which is

2 ,

Therefore ess sup u = kl

.

Another example of a system whose state is given by eigenvalues or eigenfunctions. 2.1

Orientation

We give now another example, arising in the operation of a reactor.

For a more complete study of the example to follow,

together with numerical computations, we refer to F. Mignot, C. Saguez and Van de Wiele [1]. 2.2


The operator.A is given as in Section 1.

v&L-(52), 0 < k0 < v(x) < kl a.e.

Uad = {vI

(2.1)

The state

{y(v), X(v))

is defined by

Ay(v) = a(v) v y(v)

in

2

(2.2)

y(v) = 0

on

r

X(v) = smallest eigenvalue, (2.3)

Y(v) and

y(v)

(2.4)

0

in

2 ,

is normalized by (y(v),g) = 1, g given in L2(St)

We set (2.5)

My(v) _

We consider

!g y(v) dx

.

in

i2)


65

and we define the cost function by (2.6)

J(v) = f2ly(v) - My(v)1

2

dx

We are looking for (2.7)

inf J(v), vc Uad

.

Remark 2.1

In (2.4) one can take more generally (2.8)

geH-1(2)

In particular if the dimension equals 1, we can take

g = z Dirac measures (cf. Saguez [1])

(2.9)

Remark 2.2

The above problem is a very simplified version of the operation of a nuclear plant where

y(v)

corresponds to the flux of neutrons

and where the goal is to obtain as smooth a flux as possible, which explains why the cost function is given by (2.6). 2.3

Optimality conditions

As in Section 1 we have existence of an optimal control, say- u, in general not unique.

We prove, by a similar argument to the one in Section 1, that-

is Frechet differentiable from Vad

v - y(v), a(v) set (2.10)

y(u) = y, a(u) =

I

y = °

d

we obtain from (2.2)

A ,

D(A) x R .

If we

J. L. LIONS

66

(A-xu) y = (au+x(v-u))Y,

r,

y=0 on (2.12)

(Y,g) = 0, i j) uy2 dx + x j., (v-u) y2 dx = 0

The optimality.condition is (2.13)

(y-My, Y-M(Y)) > 0

But

(y-My, My) _ MY-My),y) = 0

Vve Uad

so that (2.13) reduces to (y-My,Y)

(2,.14)

0

Vve Uad

We define the adjoint state

{p,µ}

(A-xu)p = y - My + pg,

, by

p = 0

on

r

,

(2.15)

(p,g) = 0

where

µ

is such that (2.15) admits a solution, i.e. (y-My,y) + F+(g,y) = 0

(2.16)

i.e.

µ = - (y-MY,Y)

Taking the scalar product of (2.15) with that

,y

and using the fact

(g,,') = 0 , we have

(y-M(y,Y) = ((A-xu)P,Y) = P,(A-xu)Y)

= ((iu+x(v-u))Y,P) replacing

x

by its value deduced from the last equation in (2.12), we

finally obtain j.9 LP -

y,uy Y] y (v-u) dx > 0

(2.17)

ue Uad

Vve Uad,


Therefore, if

67

is an optimal control, then one has

u

(A-),u)y = 0,

(A-au)P = y - My - (Y-My,Y)9, (2.18)

(9,Y) = 1, (9,P) = 0,

y=p=o and (2.17).

on

r

#

We go one step further, by using the structure of (2.1).

We introduce, as in Section, 1.3,

si =

( Ix (

u(x) = ki }, i = 0, 1 ,

St\(QoUS2I}

and we observe that (2.17) is equivalent to

iMY Y) s 0 y(p - Y,u

on

Q,,

0

on

5Z0,

Y(P - y;uy Y)

Y(P- Y,uy y) =0 on 2\(Q But since

(2.19)

y > 0

a.e. this is equivalent to

p - y,uy u y 1

2 .

so that

ou

is again an optimal

control and therefore one has the analog of (2.19) but this time with 20

and

empty; i.e.

2l

(2.22)

y

p -

0

a.e.

in

y.Uy

Q

From the first two equations in (2.18), we deduce from (2.22) that (2.23)

y - M(y) = (y-My,y)g

a.e.

in

52

hence the result follows, since (2.23) is impossible under the conditions stated on

g

in (2.20).

Remark 2.3

All what has been said in Sections 1 and 2 readily extend to other boundary conditions of the self-adjoint type. 3.

Control in the coefficients 3.1

General remarks

We suppose that the state of the system is given by

(3.1)

x(3.2)

where

(3.3)

- E aai (v(x) aY-)- f in 2, feL2(2), y = 0

on

r

ve Uad

Uad =

vsL (a), 0 < k0 0 (since vn, v c Dad) ; it follows from (3.13) that

J(vn) >_ J(v) - f2 (vn-v) Igrad y(v)l 2 dx

and since

vn-v - 0

in

L"(SZ) weak star and since

fixed L1 function, fS2 (vn-v)Igrad y(v)l 2dx -. 0

lim inf J(vn)

(3.16)

i.e. (3.9).

grad y(v)l2

is a

and (3.15) implies

J(v)

#

It immediately follows from (3.9) that (3.17)

problem (3.7) admits a solution.

#

Remark 3.4

We refer to Cea-Malanowski, loc. cit, for further study of problem (3.7), in particular for numerical algorithms. Remark 3.5

The existence of an optimal solution in problem (3.8) seems to be open; the proof presented in Klosowitz-Lurie loc. cit. does not seem to be complete, but this paper contains very interesting remarks on the necessary conditions satisfied by an optimal control, assumi.'tg it exists.

#

72

J. L. LIONS

Remark 3.6 cf. also Barnes [1] (these Proceedings). 4.

A problem where the state is given by an eigenvalue with control in the highest order coefficients.

Setting of the problem

4.1 In

(4.1)

L«(2)

we consider the open set

U = {vJ veL-(2), v >_ c(v) > 0

depends on For every (4.2)

Let

v e U Av

k

defined by

a.e. in

2 , where

c(v)

v}

we define the elliptic operator

_ - E az

be given

U

> 0

.

Mx) a

)

.

We define the state

y(v)

as the first

eigenfunciton of the problem Av y(v) = x(v) 1v+k) y(v)

in

2 ,

(4.3)

y(v) = 0

where

on

r ,

(we can normalize y

ly(v)l = 1)

by

x(v) = smallest eigenvalue, i.e.

f2vIgrad012 dx (4.4)

0(v) =

inf f2(v+k)

qseH1(2)

2

dx

We consider the cost function (4.5)

J(v) = f. v dx

and we want to minimize constraint (4.6)

x(v) = x(1)

J(v)

over the set of v's in

U

subject to the


73

Remark 4.1

This problem has been considered in Armand [1], Jouron [1].

In

the applic4tion to structural mechanics, n=2 , v corresponds to the width of the structure, and we want to minimize the weight for a first eigenvalue fixed, equal to the eigenvalues of the structure with uniform width equal to 1

.

Remark 4.2

One will find in Jouron, loc. cit, the study of the analogous problem under the added constraint v(x) >_ c > 0, c fixed.

(4.7)

4.2

Optimality conditions

We see, as in Section 1, that in the open set

U

the functions

is Frechet differentiable with values in

v - (y(v), ),(v)}

Hl(2) x R

If we set

JY

dr.

(4.8)

a =

1Y(v) = y, X(u) = A (u arbitrarily fixed for the time being), we obtain:

Au y + Av y = A(u+k) y+ A v y+ i(u+k) y i.e.

(Au - X(u+k))y = av y - Avy + i(u+k)y (4.9)

y=0 on t This is possible iff the right hand side is orthogonal in

L2(2)

to

hence (4.10)

a f

(u+k)y2 dx = !Q v[Igrad

yj 2

If we assume that there exists the set (4.6), then there exists that

e R

- a y2] dx

u c U

which minimizes (4.5) on

(Lagrange multiplier) such

y,

74

J. L. LIONS

(4.11)

(J'(u),v) + r, a = 0 Vv

i.e., using (4.10):

(4.12)

1

=

1

[Igrad yj

2

- X y2] = 0

fQ(u+k)y dx in (4.12) (4.13)

a=a(u)=a(1) (grad

yI2 -

=

Since

,

so that (4.12) can be written

X(1)Iy(2 = constant = cl

.

fQu)grad yJ2 dx

x(l)

we easily find that

fQ(u+k) y2 dx

(4.14)

N

cl = 12uldx > 0

2

We are going to check that, reciprocally;

if

u e U , with

y = y(u)

is such that

.(u)

satisfies

(4.15)

1 Igrad yJ2 - a(l)lyJ2 = cl = positive constant then

u

is an optimal control

Proof:

Let us multiply the equality in (6.15) by (v-u) and integrate over

Q ; we obtain cl[J(v)-J(u)] = JQvlgrad yJ2 dx - a(1) f2(v+k)lyl2 dx - [fgu Igrad yI2 dx - a(1)fQ (u+k)

IyI2 dxj

= IQvlgrad yI2 dx - a(1) f2(v+k) Iy12 dx >_ 0 (by (4.4))

5.

Control of free surfaces. 5.1

Variational Inequalities and free surfaces

Let

Q


a(o,*) be given on

Rn

HQ(Q) (to fix ideas) by

and let a bilinear form


a(q,*) = E f a..(x)

as dx +

ax

f2a

i

(5.1)

a0, aiJ eL"(Q), ai eL"(g)

75

a

'

.

We assume that

a(A,4)

(5.2)

a > 0, oc

aI1kl12,

Hi

0(S2)

where (5.3)

Let

IIqs11

K

in

= norm of k

H1(S2)

be given such that

(5.4)

K

is a (non-empty) closed convex subset of

Then it is known (cf. Lions-Stampacchia [1]) that if H-I(s2)

, there exists a unique

y

f

is given in

such that

(5.5)

a(Y,0-Y) > (f ,4-Y)

VocK ;

(5.5) is what is called a Variational. Inequality (V.I.). Remark 5.1 If we get

(5.6)

y = y(f)

IIy(fl)-y(f2)II

, we have

c

11f1-f211 -1 H

(52)

Remark 5.2

In the particular case when

(5.7)

a,

is symmetric:

V4i,4reHl(2)

then finding (5.8)

y

satisfying (5.5) is equivalent to minimizing

a(,k,o) - (f,1)

over

K

then the existence and uniqueness of y

in (5.5) is immediate.

dx:

J. L. LIONS

76

Example 5.1

Let us suppose that g

1K

a.e. in

21 , g given such that

(5.9) K

is not empty

Then one can, at least formally, interpret (5.5) as follows; if we set in general (5.10)

then

Ao _ - z aai

y

(aid

as )

+ E ai

ax

- + a0 0

should satisfy

AY-f>0 , y - g

(5.11)

0

(Ay - f) (y-g) - 0

in

Q

with

y = 0

(5.12)

on

r

We can think of. this problem as "a Dirichlet problem with an

obstacle", the "obstacle" being represented by

g

.

The contact region is the set where y(x) - g(x) = 0, xes

(5;13)

;

outside the contact region we have the usual equation Ay = f

(5.14)

where

represents, for instance, the forces.

f

The boundary of the contact region is a free surface. one has

y = g

and

axi

Formally

= N- on this surface. axi

Remark 5.3

For the study of the regularity of the free surface, we refer to Kinderlehrer [1] and to the bibliography therein.


77

Remark 5.4

For a systematic approach to the transformation of free boundary problems into Y.I. of stationary or of evolution type, we refer to C. Baiocchi [1] and to the bibliography therein. Remark 5.5

Actually it has been observed by Baiocchi [2] [3] that one can transform the boundary problems arising in infiltration theory into uq

asi Variational Inequalities (a notion introduced in Bensoussan-Lions

[1] [2] for the solution of impulse control problems).

There are many interesting papers solving free boundary problems by these techniques; cf. Brezis-Stampacchia [1], Duvaut [1], Friedman [1], Torelli [1], Conmincioli [1] and the bibliographies of these works. 5.2

Optimal control of Variational Inequalities

We define the state

of our system as the solution of the

y(v)

V.I. (with the notions of Section 5.1): y(v)eK, (5.15)

a(y(v), d-y(v)) '- (f+v, o-y(v))

VoeV

where (5.16)

v = control function.

ve U = L2(2),

The cost function is given by (5.17)

(where

2

J(v) = ly(v)-zdl

Jm = norm of

o

+

N1vi2

in

L2(2)).

The optimization problem is then: (5.18)

inf J(v), ve Uad = closed convex subset of

U

It is a simple matter to check that (5.19)

there exists

us Uad

such that

J(u) = inf J(v)

J. L. LIONS

78

Remark 5.6

For cases where we have uniqueness of the solution of problems of this type, cf. Lions [6].

#

Remark 5.7

One can think of prob;em (5.18) as an optimal control related to the control of free surfaces.

would be to try to find surface (in case

K

In this respect a more realistic problem minimizing the "distance" of the free

ve Uad

is given by (5.9)); cf. Example 5.1, Section 5.1)' This type of question is still largely open.

to a given surface. cf. also Section 6.

We assume from now on that

K

is given by (5.9).

It follows

from (5.6) that 5.20)

IIY(v1) - Y(v2)Il

c

Iv1-v21

so that, by a result of N. Aronszajn [1] and F. Mignot [2], the function

is "almost everywhere" differentiable (an extension

y(v)

v

of a result of Rademacher, established for

Rn ).

We set formally (5.21)

and we set

Y =

Y(u+(v-u))1t=0

y(v) = y ; a necessary condition (but this is formal since

we do not know if

u

is a point where

y

is differentiable; for

precise statements, cf. F. Mignot [2)) of optimality is (5.22)

(y-zd,y) + N(u,v-u) ? 0

The main point is now to see what

vve Uad

jy

satisfies:

f Ay - (f+u) >_ 0, (5.23)

y - g >_ 0, l (Ay - (f+u)) (Y-g) = 0

.

looks like.

The optimal state

y


79

Let us introduce

Z = set of x's in 4 such that (5.24)

y(x) - g(x) = 0

(Z is defined up to a set of measure 0

Then one can show that, at least "essentially":

rq=0 (5.25)

on 'Z

Aq = v-u

y=0

on

,

on 2\Z r

.

This leads to the introduction of the adjoint state by

p=0 (5.26)

on

Z

,

A* p = y - zd

p-=0

on

On

4 \Z

r.

Then

(Y-zd, y) _ (p, v-u)

so that (5.22) becomes (5.27)

(p+Nu, v-u)

0

the optimality system is'(formally) given by (5.23) (5.26)

Conclusion:

N

(5.27).

Example 5.2 Let ug assume that (5.28)

Uad = U .

Then (5.27) reduces to (5.29)

vve Uad, ue Uad

p + Nu = 0

80

J. L. LIONS

so that the optimality system becomes:

Ay + N p -. f >_ 0,

y-9'-0, (Ay +

p - f) (y-g) = 0

2

in

(5.30)

= 0

on

Z

(defined in (5.24))

A*p=y - zd

R\Z,

on

We introduce a bilinear

Let us give another form to (5.30). form on

0 = H0(2) - HO(Q)

by

A(y,p;,O,*) = a(y,o) + N a*(p,*) + . (p,,) - -

(5.31)

(y,4y)

where

(5.32)

a* (0,+y) =

We observe that

A(y,p;y,p) = a(y,y) +

..(5.33)

a*(p,p)

c[IIYII2 + IIPII2]

N

Given (5.34)

4

in

H1(2)

we set *(x) - g(x) = 0

Z(4s-g) = set of x's in 2 such that

Then (5.30) can be formulated as: (.

A(y,p;o-y,*-p)

z

(5.35)

d0,4 c

y=0

such that

Z(y -g)

on

p41

(5.36)

y,pEk, y ? g,

0

on

Z(y-g)

This is a quasi-variational inequality.

#

,


5.3

81

Open questions

Due to Remark 5.5, it could be of some interest to study

5.3.1

the optimal control of systems governed by quasi-variational inequalities.

Even after the interesting results of Mignot [2] for the

5.3.2

optimal control of stationary V.I., many questions remain to be solved for the control of Y.I. of evolution. Let us give now an interpretation (cf. Bensoussan-Lions

5.3.3

[2], [3]) of

y(v)

when

K = {k14 < 0 on

(5.37)

22}

and, to simplify the exposition,

fn grad 0 grad ,ydx + E f2 g,(x)

a(m,*) _

dx

xj

(5.38)

+f2addx, where the gj's are , say, in C1(sy) (in order to avoid here any technical Then

difficulty).

, the solution of the corresponding V.I.

y(x;v)

(5.15), can be given the following interpretation, as the optimal cost of a stopping time problem.

We define the state of a system, say

zx(t)

, as the solution of

the stochastic differential equation: dzx(t) = g(zx(t))dt + dw(t), (5.39)

zx(0) = x, x652 where

g(x) = {gj(x)}, and where w(t) is a normal Wiener process in Rn

In (5.39) we restrict Let

A

t

to be a.s. _ 0, (6.1)

y(v) - g '- 0,

I where

f , g

(Ay(v) - f) (y(v) - g) = 0 are given in

operator given in

2(0)

;

2(0)

and

in (6.1)

conditions that we do not specify. (cf. Section 5.1), denoted by

S(v)

in

A

y(v)

s(v)

is a second order elliptic Is subject to some boundary

This V.I. defines a free surface .

The general questions is: what are the surfaces can approximate by allowing and

r(1) ?

r(v)

S(v)

to be "any" surface between

that one r(0)

(Notice the analogy between this problem and a problem of

controllability).

Chapter 5

Remarks on the Numerical Approximation of Problems of Optimal Control 1.

General remarks.

Methods for solving numerically problems of optimal control of distributed systems depend on three major possible choices: (i)

choice of the discretization of the state equation (and

the adjoint state equation), both in linear and non-linear systems; (ii)

choice of the method to take into account the constraints;

(iii) Choice of the optimization algorithm. Remark 1.1

If the state is given (as in Chapter 4, Section 1) by the first

eigenvalue of course (i) should be replaced by the choice of a method to approximate this first eigenvalue.

#

The two main choices for (i) are of course (il)

finite differences;

(i2)

finite elements.

The main trend is now for (i2) and we present below in Section 2 a mixed finite element method which can be used in optimal control.

There are many ways to take into account the constraints, in particular, (iil)

by duality or Lagrange multipliers;

(ii2)

by penalty methods.

Remark 1.2

An interesting method (cf. Glowinski-Marocco [1]) consists in using simultaneously Lagrange multipliers and penalty arguments. Remark 1.3

One can also consider the state equation, or part of it (such as the boundary conditions) as constraints and use a penalty term for them (cf. Lions [1], Balakrishnan [1], Yvon (3]). The algorithms used so far for (iii) are: (iiil)

gradient methods in particular in connection with (il);

85

J. L. LIONS

86

(iii2)

conjugate gradient methods in particular in connection

with (i2); (iii3)

algorithms for finding saddle points such as the Uzawa

algorithm.

Remark 1.4

All this i; also related to the numerical solution of Variational Inequalities for which we refer to Glowinski, Lions,

Trfmolieres in. Mixed finite elements and optimal control.

2.

2.1

Mixed variational problems.

We first recall a result of Brezzi [1], which extends a result of Babuska [3].

(cf. also Aziz-Babufka [1].)

Let

be real

4,1, 412

Hilbert spaces, provided with the scalar product denoted by ( (and the corresponding norm being denoted by ll and

b

Ili

,

i=1,2)

.

,

)i

Let

a

be given bilinear forms:

(2.1)

is continuous on

41''1 - a(41,*1)

t1 "

1 b(41'''2) is continuous on 4l x 42

(2.2)

We shall assume throughout this section that the following hypothesis hold true: we define

B e L'(41;4'2)

by

(2.3)

= b(41,,2)

we assume (2.4)

a(41,41)

0 4 1e41

(2.5)

a(41,41)

c,111.1111

(2.6)

sup 01

lb(41'*2) Ih41 1

1

, a>O, V41e

KFr B

--c11*2112,c>0


87

Remark'2.1

If we introduce

(2.6)'

B*cL(02;01) , then (2.6) is equivalent to

IIB*4r2114,1> c 11*2112vy2 E 42

.

We now set (2.7)

n(o;tr) = a(01.*1) + b(+r1.02) - b(61.$2)

4) _ t1 x 4,2

where

on

4' x 4,

an

a , we look for

.

Problem: given a continuous linear form

L(*)

e' such that

(2.8)

n(,O;4) = L(4V) V*&$

This is what we call a mixed variational problem. refer to Brezzi, loc cit. and to Bercovier [1].

For example, we The result of Brezzi

is now:

under the hypothesis (2.4) (2.5) (2.6) problem (2.8) admits a unique solution and

(2.9)

11,0114)

_ allolllI +

E1102112

and therefore there exists unique

(2.40)

4,

e,6

such that

t1 (4E;y) = L(ay) V*ei .

One has then (cf. Eercovier, loc. cit.)

(2.41)

III-4Elilt < C e

1IL114>

Remark 2.2

Let us define the adjoint form

(2.42)

by

tt*(O;y) = n(`V;O)

If we define

(2.43)

tt*

a*

by

a*(01,*1) = a(*1,,O1)

then

(2.44)'

tt*(4;*) = a*(o1,*1) - b(p1,m2) + b(41,*2)

This amounts to replacing

a

not affect (2.4) (2.5) (2.6).

by

a*

and

b

by

-b .

These changes do

We have therefore similar results to the


93

above ones for the adjoint mixed variational problem. Remark 2.3

Usual variational elliptic problems can be formulated in the preceding setting (cf. Bercovier, loc. cit.); then the approximation results (2.20) or (2.41) lead in a natural way to mixed finite element methods.

Optimal control of mixed variational systems.

2.3

Orientation We now introduce the standard problems of optimal control for elliptic systems in the mixed variational formulation. Let operators

and

U

and

K

(2.45)

K e L(U;4')

(2.46)

C & L(4';H)

Let

be real Hilbert spaces; we are given two

H C

:

be given by (2.7) and we assume that (2.4) (2.5) (2.6) hold

n

y(v) e p

Then there exists a unique element

true.

(2.47)

n(y(v);'V) _ V*e(P

such that

.

This is the state of our system.


(2.48)

J(v) = IICy(v)-zdIIH + Let

Uad

NIIvIIU ,

N>0,

Zd&H

be a (non-empty) closed convex subset of

U

Ype optimization problem we want to consider is now inf

(2.49)

Since U -, 4,

(2.50)

v - y(v) ,

J(v), ve Uad

is an affine continuous (cf. (2.41)) mapping from

(2.49) admits a unique solution y(u) = y

,

u

; if we set

J. L. LIONS

94

it is characterized by (CY-zd, C(y(v)-y))H + N(u,v-u)U > 0 Vve Uad (2.51)

ue Uad

The adjoint state Using Remark 2.2, one sees that there exists a unique element

pe4

such that

(2.52)

n*(P:V') _ (Cy-zd,C*)H V>res

we call

p

the adjoint state.

Transformation of (2.51). By taking

i' = y(v)-y

in (2.52) we obtain

(Cy-zd, C(Y(v)-Y))H = x*(P; Y(v)-Y) = n(Y(v)-Y;P) _ (2.53) =

We define (2.54)

K*

K(v-u),p>

.

by

(K* p, v) U = _ 0

Vve Uad

(2.55)

ue Uad

The optimality system is finally:

0

But (2.66) is equivalent to

IIC(Y-y)IIH

(2.71) (Cy-zd,

C(y(u)-y)) + (Cy-zd, C(Y(u)-Y))

Using (2.59) we have

IIYO)4II0 = 11y0)46)114'

CpI iJIN.

6(u)-Y114' = 11;(u)-y(u)114'

CPIiuIIU

so that (2.67) implies (2.72)

We obtain

Iju-ull

cp(IiuIIu + 011U)

But if we choose a fixed

NII;II

v0E Uad

i(u) < i(v0)

so that (2.68) implies (2.65).

we have

constant

97

98

J. L. LIONS

Remark 2.8

We can extend all ti;i i tr;eoor, to the case of evolution equations. Remark 2.9 For some extens;cr.

o n.:c-1irear problems, we refer to

Bercovier, loc. cit. Remark 2.10

By using the methods of finite elements for standard elliptic problems (as in Aziz ed: [1], 8abus`ka [1], Brezzi [1], Ciarlet-

Raviart [1], Raviart-Thomas [1], Oden [1]) and the above remarks, one obtains in a systematic manner mixed finite element methods for the optimality systems; cf. Bercovier [1]. Remark 2.11

For other approaches, cf. A. Bossavit [1], R. S. Falk [1]. Remark 2.12

We also point out the method of Glowinski-Pironneau [1] who transform non-linear problems in P.D.E. into problems of optimal control, this transformation being very useful from the numerical viewpoint.

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83. 84. 85.

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J. MOSSINO [1] An application of duality to distributed optimal control problems...J.M.A.A. (1975). 50, p. 223-242. [2] A numerical approach for optimal control problems...Calcolo (1976). F. MURAT [1] Un contre exemple pour le probleme du contrSle dans lea coefficients. C.R.A.S. 273 (1971), 708-711. [2] Contre exemplea pour divers problPmes ou le contr9le intervient dans les coefficients. Annali M. P. ed. Appl. 1976. F. MURAT and J. SIMON [1] To appear. R. H. NILSON and Y. G. TSUEI [1] Free boundary problem of ECM by alternating field technique on inverted plane. Computer Methods in Applied Mech. and Eng. 6 (1975), 265-282. J. T. ODEN [1] Generalized conjugate functions for mixed finite element approximations..., in The Mathematical Foundations of the Finite Element Method, A. K. Aziz, ed., 629-670, Acad. Press, New York, 1973. 0. PIRONNEAU [1] Sur les problPmes d'optimisation de structure en M6canique des fluides. Thesis, Paris, 1976. [2] These proceedings. M. P. POLIS and R. E. GOODSON [1] Proc. I.E.E.E., 64(1976), 45-61. P. A. RAVIART and J. M. THOMAS [1] Mixed finite elements for 2nd order elliptic problems. Conf. Rome, 1975. W. H. RAY and D. G. LAINIOTIS, ed. [1] Identification, Estimation and Control of Distributed Parameter Systems. R. T. ROCKAFELLAR 1 Conjugate duality and optimization. Reg. Conf. Series in Applied Math. SIAM. 16, 1974. D. L. RUSSELL [1] These proceedings. [2] Control theory of hyperbolic equations related to certain questions in harmonic analysis and spectral theory. J.M.A.A. 40 (1972), 336-368. C. SAGUEZ [1] Integer programming applied to optimal control. Int. Conf. Op. Research, Eger. Hungary, August 1974. J. SAINT JEAN PAULIN [1] Contr8le en cascade daps ua problZme de transmission. To appear. Y. SAKAWA and T. MATSUSHITA (1) Feedback stabilization of a class of distributed systems and construction of a state estimator. IEEE Transactions on Automatic Control, AC-20, 1975, 748-753. J. SUNG and C. Y. YU [1] On the theory of distributed parameter systems with ordinary feedback control. Scientia Sinica, SVIII, (1975), 281-310.


87.

88.

89. 90. 91.

92.

103

L. TARTAR [1] Sur 1'6tude directe d'equations non lineaires intervenant en th6orie du contr8le optimal. J. Funct. Analysis 17 (1974),1-47. [2] To appear. A. N. TIKHONOV [1] The regularization of incorrectly posed problems. Doklady Akad. Nauk SSSR,153 (1963), 51-52, (Soviet Math. 4, 1963, 1624-1625). G. TORELLI [1] On a free boundary value problem connected with a nonsteady filtration phenomenon. To appear. A. VALLEE [1] Un problpme de contr8le optimum dans certains problcmes d'evolution. Ann. Sc. Norm Sup. Pisa, 20 (1966), 25-30. J. P. VAN DE WIELE [1] REsolution numerique d'un probl6me de contr8le optimal de valeurs propres et vecteurs propres. Thesis 3rd Cycle. Paris 1976. R. B. VINTER [1) Optimal control of non-symmetric hyperbolic systems in n-variables on the half space. Imperial College Rep. 1974.

93. 94. 95. 96.

97.

R. B. VINTER and T. L. JOHNSON [1] Optimal control of non-symmetric hyperbolic systems in n variables on the half-space. To appear. P. K. C. WANG [1]. J. L. A. YEBRA (1). To appear. J. P. YVON (1] Some optimal control problems for distributed systems and their numerical solutions. [2] Contr8le optimal d'un probleme de fusion. Calcolo. [3] Etude de la methode de boucle ouverte adaptee pour le contr8le de systbmes distribu6s. Lecture Notes in Economics and Math. Systems, 107, (1974), 427-439. [4] Optimal control of systems governed by V.I. Lecture Notes in Computer Sciences, Springer, 3 (1973), 265-275. J. P. ZOLESIO (1) Univ. of Nice Report, 1976.

We also refer to: Report of the Laboratoire d'Automatique, E.N.S. Mecanique, Nantes: Calculateur hybride et Syst1'mes a parambtres r6partis, 1975

"STOCHASTIC FILTERING AND CONTROL OF LINEAR SYSTEMS: A GENERAL THEORY" A. V. Balakrishnan* A large class of filtering and control problems for linear systems can be described as follows. y(t)

(say, an

0 < t < T < m

m x .

1

We have an observed (stochastic) process

vector),

t

representing continuous time,

This process has the structure:

y(t) = v(t) + n0(t) where

n0(t)

is the unavoidable measurement error modelled as a white

Gaussian noise process of known spectral density matrix, taken as the Identity matrix for simplicity of notation. composed of two parts: random 'disturbance'

The output

the response to the control input nL(t)

v(t) u(t)

is

and a

(sometimes referred to as 'load distur-

bance' or 'stale noise') also modelled as stationary Gaussian; we also assume the system responding to the control is linear and time-invariant so that we have: t

B(t-s) u(s)ds + nL(t)

v(t) = J

O

where

is always assumed to be locally square integrable, and

where

is a 'rectangular' matrix function and

IIB(t)II2dt < -

.

FO

* Research supported in part under grant no. 73-2492, Applied Mathematics Division, AFOSR, USAF

105

A. V. BALAKRISHNAN

106

We assume further more that the random disturbance is `physically realizable' so that we can exploit the representation: t

nL(t) = I0 F(t-p) N(p) do

where

is a rectangular matrix such that

F(p)

f o IIF(s)II2ds < where, in the usual notation, AA*

IIAII2 = Tr.

We assume that the process noise process

n0(t)

nL(t)

is independent of the observation

.

It is more convenient now to rewrite the total representation as: y(t,w) = v(t,w) + Gw(t)

t

t

v(t,w) = J0 B(t-s) u(s)ds + J0 F(t-s) w(s)ds

where GG* =

I

.F(t)G* = 0 w(-)

is white noise process in the. appropriate product Euclidean space,

and

IIF(t)II2dt

0)

This work was supported in part by the National Science Foundation under grant MPS71-02656 A04.

151

H. O. FATTORINI

152

and the same boundary condition as for (1.2)

u(x,y,t) = 0

t < 0

((x,y) E r, t > 0)

.

We assume that the magnitude of the force is restricted by the constraint (1.3)

(C

ff

If(x,y,t)l2 dx dy < C

(t > 0)

a positive constant fixed in advance) while its objective is that

of bringing the energy of the membrane

E(t)

to zero in a time

7

7

2

{

T > 0

U) at

2

aU

+

c2

[ ax)

as short'as possible.

21

2

+

ay

dx dy

J

In other words, we

want to bring the membrane to a standstill as soon as practicable within the limitations imposed on the use of force by the constraint (1.3).

applied

here E(t)

=0

Figure 1

153

THE TIME OPTIMAL PROBLEM

Three questions arise naturally in connection with this problem. (a)

Is it at all possible to reduce the energy by means of a force

T

a finite time

E(t)

to zero in

subject to the constraint

f

(1.3)? (b)

exist a (c)

Assuming the answer to (a) is in the affirmative, does there f0 If

that does the transfer in minimum time? exists, is it unique?

f0

What additional properties

(say, smoothness) does it have?

Problem (a) is a typical controllability problem, and we show Section 3 that it has a solution.

This is scarcely surprising in view

of the extremely lavish class of controls at our disposition.

(A more

realistic situation would be that in which we can only use a finite number of control parameters, for instance

m f(x,Y,t) =

where the functions fl, ..., fm

E k=l

fk(t)bk(x,Y)

b1, ..., bm

are fixed in advance and we can vary

subject to constraints of the form

Ifk(t)1 < C

(1

0).

Problem (b) refers to the existence of optimal controls and it is well known that, at least in the linear case considered here, its ....solution follows from the solution to (a) via a simple weak compactness

argument as in [2], [3], [1].

This is done in Section 4.

We examine in Section 5 problem (c).

There we prove an analog of

the celebrated PONTRYAGIN maximum principle in the form obtained by

BELLMAN, GLICKSBERG and GROSS for linear systems in [2] and generalized

H. 0. FATTORINI

154

to infinite-dimensional situations by BALAKRISHNAN [1] and the author The basic technique here is that of "separating hyperplanes" used

[6].

in [2] for the solution of a similar control problem in finite dimensional space.

It turns out that treating the present problem directly would

involve us with some of its special features (say, finite velocity of propagation of disturbances) that play no significant role on it.

It

is then convenient to cast it into the formalism of second order differential equations in Hilbert spaces.

This is done in Section 2

and the results obtained in the following sections are then seen to be applicable to many different situations.

We examine in Section 6 some variants of the original problem obtained by replacing the constraint (1.3) on the control (which is not necessarily the only physically significant one) by other types and we show that versions of the maximum principle also hold in these cases. Second-order equations in Hilbert space.

2.

We begin our quest for

generality by considering the problem in Section 1 number of dimensions. space

RP(p >_ 1)

in an arbitrary

be a bounded domain in Euclidean

Let then s

with sufficiently smooth boundary

r

and consider

the operator

(2.1)

aku

(Au)(x) = c'au(x) = c2 k=l Z

is defined in the customary way as the

(ak = a/axk).

The domain of

set of all

in the Sobolev space

u

A

that satisfy the Dirichlet

H1(E)

boundary condition (2.2)

u(x) = 0

(x ( r)

and such that. ou , understood in the sense of distributions, belongs to

L2(c)

.

(Recall that

distributional derivatives

consists of all

H1(E))

alu, ..., a u p

belong to

P

Pull 2l H

=

(lull2L2

(c)

laku1122 k=l

L (E)

)

u E L2(g)

whose

L2(g)

with norm


It is well known ([7]) that operator in

L2(S)

is a negative definite self adjoint

A

u(x,t)

If

.

155

is a solution of the inhomogeneous

wave equation 2

(2.3)

a u =

c2ou + f

(x

0)

that satisfies the boundary condition u(x,t) = 0

(2.4)

and we denote by

(x E r,t _ 0)

u(t)

the function in

t -_ 0

with values in

L2(g)

given by

u(t)(x) = u(x,t) and define

similarly, then

f(t)

is (at least formally) a

u(-)

solution of the abstract differential equation u"(t) = Au(t) + f(t)

(2.5)

(t > 0)

.

We are then naturally led to the following abstract formulation of the time-optimal problem considered in Section l: Let space and

A

(Au,u)

for some

w > 0

0)

.

the unique self adjoint, positive

Then it is not difficult to deduce from

standard functional calculus arguments that if

(2.13)

u0 E D(A), ul E K

,

u(t) = C(t)u0 + S(t)uI

is a solution of (2.10) with initial data (2.8) and that, moreover, it is the unique such solution.

As for the nonhomogeneous equation (2.5),


if

is, say, continuously differentiable in

f

t > 0

157

the (only)

solution of (2.5) with null initial data is given by the familiar formula

t

(2.14)

S(t-s)f(s)ds

u(t) = 0

is of

(the solution with arbitrary initial data

u0 E D(A), u1 E K

course obtained adding (2.13) to (2.14)).

However, the nature of our

control problem is such that the definition of solution introduced

above is too restrictive (for instance, we will be fprced to consider controls

f

that are much less than continuously differentiable).

is continuous (as a H-valued

C(t)u

functional calculus that

t

function) for any

and continuously differentiable for

u E H

with

(C(t)u)' = AS(t)u ; note that

into

D(A))

S(t)u

t

tive

and

AS(t)u

S(t)

maps

is continuous for any

.

into

H

u ( K).

is continuously differentiable for any

(S(t)u)' = C(t)u

In

It is again a consequence of the

view of this, we proceed as follows.

u E H

K

u

E K

(thus

K

Also,

with deriva-

Making use of all these facts we extend the

previous notion of solution in a way customary in control theory, namely we define rt

S(t-s)f(s)ds

u(t) = C(t)u0 + S(t)uI +

(2.15)

1

0

to be the (weak) solution of (2.5), (2.8) whenever f

u0 E K, ul E H

and

is a strongly measurable, locally integrable function with values in

H. (5)

It is not difficult to see, on the basis of the previous

observations, that

is continuously differentiable (with

derivative t

(2.16)

u'(t) = AS(t)u0 + C(t)ul + J

C(t-s)f(s)ds

0

and that the initial conditions (2.8) are satisfied.

It is not in

H. O. FATTORINI

158

general true that

u

can be differentiated further, so that it may not

be a solution of (2.5) in the original sense. 2.1 Remark.

In the case where

A

is defined by (2.1), (2.2) the

functional calculus definitions of

(-A)1"2, C(t), S(t)

can be

explicited as follows.

(0 < a0 < a1 5 ...)

be the

eigenvalues of functions.

A ,

Let

{cpn}

Then 1

(2.17)

(-an}

a corresponding orthonormal set of eigen-

1

(-A)2u

=

n(u,con)wn

E k=l

(-A)1"2

the domain of

consisting of all

u

E

E

such that the series

on the right-hand side of (2.17) converges, or, equivalently, such that

We also have

E> 0

,

the solution of (2.5) with preassigned

u0 , u'(0) = u0

other words, such that

satisfies

u(T) = u'(T) = 0

;

in

159


(3.2)

J S(T-t)f(t)dt = - C(T)u0 - S(T)u1

(3.3)

Jo C(T-t)f(t)dt = - AS(T)u0 - C(T)u1

Existence of a solution to (3.2), (3.3) for

T

large enough will

follow from some simple manipulations with

C(-)

by introducing some useful notations.

K = K x H

Let

and

S(-)

.

We begin

endowed with the

norm 2

2

II(u,v)IIK = IIuIIK +

where the norm in immediate that

K

K

IIvIIH

is defined by

IIuji

is a Hilbert space.

=

II(-A)1/2uIIH

Elements of K

.

will be denoted

by row vectors or column vectors as convenience indicates.

S(t)

the operator from

H

into K

It is We denote by

defined by

S(t)u =

and observe that, in this notation, the two equations (3.2), (3.3) can be condensed into the single equation C(T)u0 + S(T)u1

f S(T-t)f(t)dt = -

(3.4)

AS(T)u

Let now 0

cp,,y

t 5 T

E D(A)

both

differentiable and S'(O)u = u

1

such that

p(T) = -1

,*(0) - 0 yr(T) = 0 u

+ C(T)u

be twice continuously differentiable scalar functions in

p(0) = 0

If

0

;

t

S(t)u

(p'(0) = 0 gyp' (T) - 0

*'(0) = 0 *'(T) _ -1 and

t - C(t)u

.

are twice continuously

S'(t)u = C(t)u, S"(t)u = AS(t)u, S(0)u = 0,

C'(t)u = AS(t)u, C"(t)u = AC(t)u, C(0)u = u, C'(O)u = 0

H. O. FATTORINI

160

(see the comments preceding (2.15). that, if

u, v E D(A)

Then integration by parts shows

and

f(t) = cp(t)Au - cp"(t)u + *(t)Av - it"(t)v we have

(3.5)

IT S(T-t)f(t)dt

=

(:)

and it is easy to see that we can.choose gyp, y in such a way that

(3.6) M(T)

lif(t)II

0

which does not depend on and

We perform now some computations with follows directly from its definition that

S(.)

It

.

satisfies the "cosine

functional equation"

(3.7) for all C

C(C)C(TI) = 2' C(C+TI) + - C(C-TI) ,

Tl

.

to

,

rl

and integrating with respect to

1:'

in

that

0 < ' 1 - IIC(t)uli

=.q =

t/2

we


which makes it clear that if words, that measure of

(3.16) for all

e(u) e(u)

and

E E

.

then

t/2 fe(u)

are disjoint.

'lf e(u)

cannot exceed

fo IIC(t)u11 2 dt u

E e(u)

t

or, in other

This means that the

2T/3 ; hence

Ilull2

27

(This argument is due to GIUSTI.)

Define now

N(T)u = fT C(t)2u dt

Clearly

N(T)

is a self adjoint operator and we can write (3.15) as

follows:

(N(T)u,u) ' which shows that

27 Ilull2

N(T)

is invertible and that

IN(T)-lll `5 T' We examine now (3.14) again in the light of the preceding comments on N

.

Write

f1 = f2 + f3

f2(t) =

where

C(T-t)N(T)-1S(T)ul

T

Then it is clear that fp C(T-s)f2(s) ds = T S(T)ul

Call now

v(T) = f 0 S(T-t)f2(t) dt

preceding (3.5) construct an f3

and, making use of the comments

such that S(T)u0 + Tv(T)

fn S(T-t)f3(t) dt = - T 0

To prove that this is possible, and that small norm for

T

f2

will have sufficiently

large enough we only have to show that

163

S(T)u0 -

H. 0. FATTORINI

164

Tv(T)

and that

E D(A)

The statement for

IIA(S(T)u0 - Tv(T))II

S(T)u0

preceding observations; as for help of (3.9).

v0 = VI = 0

remains bounded as

T

is a direct consequence of (3.15) and Tv(T)

it can be easily proved with the

This ends the proof of Theorem 3.1 for the case The general case can be easily deduced from the one just

.

solved using the invariance of equation (2.5) with respect to time reversal.

u0, v0 E K, ul, vl E H

In fact, let

that there exists a solution

.

Take

T

so large

(resp. f2) of the controllability

fl

problem with (u0,u1) (resp. (v0,vl)) as initial data and zero final data in

0

t_ T with

Ilfl (t)II

(rasp. IIf2(t)II < T)

...

,

Tn

in time (u0,u1)

TO

T

.

to

that

Choose now a sequence

{fn}

Tn

with

(v0,v1)

in time


and consider

{fn} as elements of the space

Chapter III) extending {fn}

Since the sequence

is uniformly bounded in

exists a subsequence (which we still denote to an

;

H) (see [8],

fn = 0 L2(0,T1

there.

there

H)

;

{fn}) that converges weakly

which, as easily seen, must vanish in ta TO

f0

satisfy (3.1) almost everywhere. to

L2(0,T1

by setting

(Tn,T1)

to

fn

165

f0

The fact that

and must

drives

(u0,u1)

follows from taking limits in the sequence of equalities

(v0,v1)

C(Tn)u0 + S(Tn)ul - v,

-

IT

n S(T -t)f (t) dt = 0

n

AS(Tn)u0 + C(Tn)u1 -

°

v1

which can be easily justified on the basis of the weak convergence of {fn}

(see [3] for further details)). Let

The maximum principle.

5.

(u0,u1)

and

f0

be a control joining two points TO

in minimum time

(v0,v1)

isochronal set (of f0) to be the set of all

=

f0 0

v

(u,v) E H x H

,

the

of the form

S(T0-s)f(s) ds

f

(that is, for some strongly measurable

f

for some admissible control f

S2(=2(T0))

T

(U) (5.1)

and define

that satisfies IJf(t)II

a:e. in

_ 0)

.

We assume in the sequel (as we plainly may) that

is convex(6).

that

2

from the definition of

2 c K

.

C = 1

.

It is clear

It is also immediate that

S2

Two crucial properties of the isochronal set are:

(1) The interior of

2

(in

K)

is non void.

(ii) (w0,w1)=(v0,v1)-(C(t)u0+S(t)u1,AS(t)u0+C(t)ul) is a boundary

point of

9

.

The proof of (I) follows essentially form that of Theorem 3.1. (u,u') E K .

Let

By "running backwards" equation (2.10) we can assume that

(u,u') = (u(T0),u'(T0))

for a solution

(u(0),u'(0)) E K ; precisely,

of (2.10) with

H. O. FATTORINI

166

C(TO)u(0) + S(TO)u'(0)

u

(5.2)

AS(TO)u(0) + C(TO)u'(0)

u'

where u(0)

C(TO)u - S(TO)u'

u'(0)

-AS(TO)u + C(TO)u'

(5.3)

(the justification of (5.2) and (5.3) is an easy consequence of formulas (3.7), (3.8) and (3.9)). find a control

According to Theorem 3.1 we can now

such that

f

TO

S(TO-t)f(t) dt =

f

10

(:)

with

!If(t)II < MII(u(0), u' (0))IIK M

a constant independent of

(0 5 t s TO)

(u(0),u'(0))

.

,

But, on the other hand, it

follows from (5.3) that

II(u(0),u' (0)HK = II(u,u' )IIK so that if

is sufficiently small the control

II(u,u')IIK

admissible.

f

will be

This shnws that the origin is an interior point of

The proof of (i!) follows from (i). not a boundary point of

SZ

.

Sd

In fact, assume (w0,w1)

is

Taking into account that the function

t -+ C(t)u0 + S(t)u1

is continuous in

is continuous in

it is not difficult to deduce the existence of a

T1 < TO

and a

H

v1

t -+ AS(t)u0 + C(t)u1

Ct)u0 + S(t)u1 E 9

-

r

and that

such that

r < 1

vO (5.4)

K

AS)uO + C(t)u1

(Tl

t h We shall assume this to be the case. This amounts to .

the assumption that the optimal column nowhere achieves the maximum allowable thickness determining

b

We leave to the reader the problem of

.

in the case where the maximum allowable thickness is

p*

achieved by the optimal column. In the case we are considering the point along the curve (3.5) on the interval (3.4) on the interval Fig.

1.

Clearly,

[s.-xl,,¢]

y'(z/2) = 0

,

(y(x),y'(x))

as is indicated by the arrows in Moreover, since eigenfunctions are

.

unique only up to a scalar multiple we may assume that scaled so that

y(2/2) =

1

.

Solving this equation for

y

has been

Equation (3.5) must then be given by

y'2 + 4an3/4y112 = 4X9

(3.6)

moves

(xl,x.-x1) , and along the curve

3/4

y'(xl)

.

and substituting into (3.4) gives

the equation

for

3anH-1/3

Y'2 + AHy2 = 4an3/4 _

(3.7)

(y(x),y'(x)),

0 < x < xl

.

This equation can be solved for

on this interval the solution shows that

terms of

x

(3.8)

xl -

arc sin 1XH

4n

3/4H _

nH-4/3 -3nH

y

in

MAX-MIN PROBLEMS IN OPTIMAL DESIGN STUDIES

191

I u,vi

,

`

s , y.

AH y

=4111

3A H

(Y(11. '4111

I

Figure 1. The path of the solution is indicated by arrows.

In order to obtain a second expression for function

e(x)

we introduce a new

xl

defined by the requirement xl < x < k/2

y(x) = sin4e(x),

.

Substituting this into (3.6) gives the differential equation

i.= 3de sino

an3/4 ,

e(x1) = arc sin

for

(3.9)

e

.

xl

< x < z/2

(nl/8H-l/6) ,

o(t/2) = ,r/2

This implies that

x1 = s,/2 -

2

an3/4

,r/2

2

sin ode

arc sin (nl/8H-1/6)

192

EARL R. BARNES

n1/8H-1/6

Z =

Let

(3.8) and (3.9) then imply the equation

Z

and

Z

a

3

sin ede

J

443Z

for the unknowns

("/2

2

2

arc sin

H

(3.10)

.

= 112

aH arc sin Z

ZZ

A second equation for these unknowns is

.

implied by the condition f£/2 0

We have x

rt/2 p-1/3(x)dx

= f

0

I H-1/3dx + 112 n-1/4sin2edx

0

H-1/3

A

+ 2Z 3 J

F72for

Z

.

5

sin ede = V/2 Z

n/2

5

sin ede arc s in

Z

arc sin

r/2

+

44

arc sin -Z2+ 2Z_15 f (3.11)

2H-1/3

Z2

arc sin

YrT

xl

Z Z

=

V/at

3 sin ede

Tr/2 arc c sin Z

It is easily shown that this equation has a unique solution

in the range

0 < Z
h , then the optimal column never achieves the maximum allowable thickness. case the appropriate value for

Given these values of

n

and

a a

can be obtained from equation (3.10). , the cross sections

A(x)

strongest thin-walled tubular column are obtained as follows. solve the initial value problem

Y"+as(Y)Y=0, 0<x 0 and all nontrivial A = 0

the equation has a

Our main purpose in this section is B

that will guarantee the bounded-

We begin with an example to motivate these

considerations.

Consider a thin uniform rod of length L verse vibrations due to periodic and forces Let the line segment its undeflected state.

0
n .

a simple phase plane analysis similar to that

used in Section 2 shows the existence of a point

0 < xl

< t/2

such

that

sin a gh x

p <x <x l

VX_ ON

T r

,

xl

<x

12 .

R

p*(x)dx = M

The condition

implies that

f0

a1M FA-07h

x 1

The condition

y*'(xl) = 0

holds for

y*'(x) = 0 implies that

xj = al(p*)

xl < x
H , a direct

EARL R. BARNES

198

substitution

of (4.13) into (4.8) and solving the resulting equation,

yields the equation

A- gh cot ash xl = T tan Vx_- jH (t/2-x1) for a = al(p*) , where RH-M 2 H-h

_ xl

To find

an

for

n >

divide the interval

1

[0,1]

into

n

equal parts and consider the boundary-value problem y" + (a-8p(x))y = 0

,

0 h

and

< H

,

how

should it be tapered in order to maximize the rate of heat dissipation to the surrounding medium.

The answer was conjectured by Schmidt [18]

in 1926 for fins with no minimum or maximum thickness constraint imposed on them.

He proposed that the optimum fin should taper, narrowing in

the direction of heat flow, in such a way that the gradient of the temperature in the fin is constant. by Duffin [19] in 1959.

This conjecture was proved rigorously

Since that time a number of papers have

appeared treating various aspects of the optimal design problem. list [20], [21], [22], [23], to name a few.

We

In [23] the present author

obtained the optimum taper of a rectangular fin subject to thickness constraints.

We shall now show how to obtain analogous results for

annular fins.

Figure 2.

Annular Fin on a Cylinder

MAX-MIN PROBLEMS IN OPTIMAL DESIGN STUDIES x, y, z

In

plane.

201

space we take the fin to be parallel to the x,y

We assume the fin is sufficiently thin that there is no appre-

ciable change in its temperature in the

z

direction.

If the tempera-

ture of the surrounding medium is taken to be zero, and if we assume that Newton's thermal conductivity is unity, then the flow of heat in the fin is governed by the steady state heat equation (5.1)

(p 2X )

ax

where

+ ay (p 2y) - q u = 0, (x,y)

ES

is the annular region of the fin in the

S

u = u(x,y)

is the temperature in the fin,

of the fin, and

q > 0

y

constant.

x,y

p = p(x,y)

plane.

is the thickness

is the cooling coefficient, here assumed

We shall assume that the outer edge of the fin is insulated

so that the appropriate boundary conditions are u = T (= steady state temperature)on

rl

(5.2)

where

of

S

and

rl .

v

r2

are, respectively, the inner and outer boundaries

is a unit outward normal to the boundary of

S

.

Newton's law of cooling implies that the heat dissipated per unit time by the fin is given by (5.3)

ff.q u dxdy S

The weight of the fin, which we assume to be fixed, is proportional to (5.4)

f p(x,y)dxdy = M

.

The thickness of the fin satisfies the constraints (5.5)

h < p(x,y) < H

for some positive numbers

and

h

H

satisfying

h f f dx dy < M< H f f dx dy S

S

Our problem is to determine

p

and

u

satisfying conditions

(5.1), (5.2), (5.4), and (5.5), in such a way that the integral (5.3) is maximized.

Ouffin and McLain [24] have given a max-min formulation

EARL R. BARNES

202

of this problem. (5.1) by

In order to obtain this formulation multiply equation

u

and integrate, using Green's theorem, to obtain

I

LP (aX)

2

2

+qu2] dxdy.

+ p (ay)

JJ [qu

ax (p ax)

ay (p

ax)]udxdy

S

(5.6) r

+ j

=T1

rl

+

up a do

I

lr2

p

rl

do

up av

avdo

On the other hand, by simply integrating equation (5.1) over the region S

, again making use of Green's theorem, we obtain f1 qudxdy = jI [ax (p ax) + ay (p ay)] dxdy S

S

(5.7)

=J

paw do

rl

Combining this with (5.6) we obtain 2

(5.8)

ff

qudxdy = 1J

T

S

(p ax)

2

+ p (ay)

+ qu2] dxdy

S

The differential equation (5.1), together with the boundary condition

u = T

on

is just the Euler equation for minimizing the

rl

integral on the right in (5.8).

We can therefore write 2

(5.9)

min ff Lp (ax)

If qudxdy = S

T

v

2

+ P (ay)

+ qv2] dxdy

S

where the minimization is taken subject to

v = T

on

rl

.

The problem

of tapering the fin to maximize the rate of heat dissipation is therefore equivalent to the max-min problem

MAX-MIN PROBLEMS IN OPTIMAL DESIGN STUDIES

2

(5.10)

max min p

2

+ p(ay)

JJ [p(aX)

u

203

+ qua] dxdy

S

where the minimization is taken over functions rl

satisfying

u

and the maximization is taken over functions

p

u = T

on

satisfying (5.4)

and (5.5).

The problem we have formulated is valid for fins on cylinders of arbitrary cross-sectional type.

However, we shall now restrict our

attention to circular cylinders of radius

R

In this case it is clear that the functions

and fins of length p'

and

R

which solve

u

(5.10) must depend only on the distance to the center of the cylinder. Let the center line of the cylinder be along the Fig. 2.

p(x,y)

Introduce the variables .

r =

+y

z

axis as in

(r) = u(x,y), p(r) =

In terms of these variables the problem (5.10) becomes

R +R

(5.11)

max min J p

[rp(r)o'2(r) + gro2(r)]dr R

m

where the minimization is taken over absolutely continuous functions

4,

satisfying (5.12)

(R) = T

and the maximization is taken over piecewise continuously differentiable functions

p(r)

satisfying

R+R

(5.13)

rp(r)dr = fR

h < p (r) < H

.

,

EARL R. BARNES

204

The boundary-value problem (5.1), (5.2) transforms into

dr (rp(r) dr) - grm(r) = 0 (5.14)

0(R) = T, 4'(R+R) = 0

.

The technique used to prove condition (3.14) in [23] can be used to prove the following theorem. Theorem 5.1.

Let

The pair

(5.14).

there exists a constant (5.15)

max h < p < H

for each

r

(r)-n] =

.

p*(r)[m*'2

be a solution of (5.11).

condition (5.15) implies that close to R+q

such that

n > 0 p[OI2

be functions satisfying (5.13) and

q*

is a solution of (5.11) if and only if

(r)-n]

[R, R+z]

in

(p*,m*)

Let

and

p*

(p*,p*)

p*(r) = h

For these values of

r

Since

*'(R+¢) = 0

for values of

r

sufficiently

, equation (5.14) is of Bessel

It therefore seems unreasonable to attempt an analytic solution

type.

of (5.11).

Instead we shall give an iterative procedure which can be

used to obtain numerical solutions.

h < p(r) < H

are no constraints of the form satisfies

0'2(r) =

n,

First we remark that in case there

R < r < R+.t

p, the optimal

on

o

Substituting this into (5.14)

.

gives a simple differential equation from which

p

can be determined.

This analysis is carried out in [19]. To facilitate the discussion of a numerical solution of (5.11) we introduce some notation.

All functions involved will be considered as

iembers of

L2[R, R+A], the space of square integrable functions on

[R, R+t]

We shall use the symbol

.

fR0o(r)p(r)dr

of two functions in

will be denoted as usual, by functional

g

11

g(p) = min

to denote the inner product

L2[R, R+,t]

The norm in this space

For convenience, we define a

.

on the class of nonnegative r

(5.16)

11

0-p

p's

by

R+z

[rp(r)o'2(r) + grm2(r)]dr

1

R

where the minimization is taken over functions satisfying (R) = T Problem (5.11) is then to maximize

g

subject to (5.13).

MAX-MIN PROBLEMS IN OPTIMAL DESIGN STUDIES Let

(p2,02)

205

be pairs of functions satisfying (5.14).

Then by arguing as in the proof of Theorem 3.2 in [23], it can be shown that

(5.17)

1

.

rm'2(p2-p1)dr - 2 JO (p2-pl)2dr

9(p2) - g(pl) < I

ro'2(p2-pl)dr

0

where

IT (2R+L)L}2/h

K

.

This means that g is differentiable,

and its gradient is given by dg(p) =

where

r4'2

is the function which solves the minimization problem (5.16).

0

Let

be the function which maximizes

p*

sequence of functions converging to i)

ii)

Let If

pl

subject to (5.13). 'A

g

can be generated as follows.

p*

satisfying (5.13) be chosen arbitrarily.

p1,...,pk

have been chosen, take

pk+1

to be the solution

of R+A

maximize

[vg(pk)'(p-pk)

(

-

T (p-0k)

2]dr

R

subject to rR+R

rp(r)dr -

Z(5.18)

)R

n

h < p(r) < H pk+l

is the solution of a simple moment problem which can be easily

solved numerically.

It is clear from (5.17) that

We shall now show that the sequence converges to

g(pk+l) > g(pk)

g(pl), g(p2), ..., actually

g(p*)

By completing the square in the integrand in ii) above, we see that

pk+l

is the point in the convex constraint set defined by (5.18),

nearest the point

pk + K Vg(pk).

It follows that

EARL R. BARNES

206

(5.20)

(Pk + K vg(Pk) -

satisfying the constraints in (5.18), and in particular for

p

p = pk+1 For

k >

1

,

(5.17) implies that

"P W -PO

g(pk+l) - g(pk)

K(Pk +

vg(pk)-pk+l).(Pk+l-Pk)

K

+ "Pk+l-nk"

Ilpk+l-pkII

-

2 > 2 Upk+l-pkll2 Since

(by (5.20)).

[9(pk+l) - 9(pk)] _ 0 Let

6E

T

(with self explanatory notations)

VS'

E

e

be the left member of (4.7).

Suppose that a > 0, by (4.7)

6E = J [IOs-'kdl2-Ims'-mdf21dx + IQQIdJ s

it can be shown that 1I(oS-'k S2 -" 0

when

a -+ 0 ,

so that, by

L

differentiation and from the mean value theorem (since narrow strip around S ) 6E =

I25

1

w here

60 = 4

(4.8)

-4,S .

-d)6

S'-2S

+ o(a)

The P.D.E. in (4.4) in variational form is

(v4vw+4w)dx = 2S

+

S2

fwdx

I

2S

Yw ( H1(52s)

.

is a

218

O. PIRONNEAU

Therefore (4.9)

satisfies

6o,h

(vosvw+bow)dx = J xa(fw-Vovw-ow)dS

1

S

12S

Thus if we let

(4.10)

*

be the solution of

(v,yvw+yw)dx = f2s (0S-0d)wdx

Vw E Hi(2S)

J

2S P utting

in (4.9) and

w = ,y

f

w = 69s

aa(f*-v4v*-oir)dS

(,k S-0d)6odx =

2S

in (4.10) leads to

S

so that 6E = Is(1d2 + f-*-vy}dS + o(a)

(4.11)

A similar computation can be done for (4.5) if one notes that in

2

S

64,

(4.12)

where

oho = 0, 6415 = - ),a

ka(IAS-'bdI2

6E = 2 fs

*

(4.13)

is a solution of

- an an )dS

an

+ o(1t)

so that

+ o(X)

is the solution of

any =

0S-Od

in

9S, *Is = 0

.

It can be shown that (4.11) and (4.12) are valid for non positive

a' s . Therefore let us consider the-following algorithm for solving say (4.4) where

S = IS twice differentiable}

.

2t9

FREE BOUNDARY PROBLEMS AND OPTIMUM DESIGN PROBLEMS

Algorithm 1

Step 0

(4.14)

Choose

S

0 0i

set i= 0

E S S

Step 1

Compute

Step 2

Compute *i by solving (4.10) with S = Si, 0S = 0i

Step 3

Set

Step 4

Compute

= 0

by solving (4.8) with

i

S = Si

ai = - (0 1 -md)2 - fyi + 01*i + V¢iV4t Xi

solution of

min E(S'(),)) with S'(x)={xi+1(s)=xi(s)+xai(s)ni(s)IsE[0,1]}

and set

Si+1

=

S'(xi) ,

i

= i + I

and go back to 1

.

.

THEOREM 1

Any C2-accumulation point of a sequence {Si}i>0 generated by algorithm 1

(4.15)

is a stationary point of (4.3) and satisfies 10S-4dl2

+ (f-,k)* - 707* = 0

Proof: Algorithm 1 is a method of steepest descent for (4.4) locally in the space of admissible

a's such that

S' E S

.

The convergence

proof proceeds just as in the case of an ordinary gradient method. From (4.11) with

a

as in step 3

Ei+l - Ei = -xi

,

iV*i]2dSi+o(xi)

[IBS S

So that each iteration decreases quantity in (4.15) is zero. 4.3.

E

of a non trivial quantity until the

For a rigorous proof see Pironneau (1976).

Implementation of algorithm 1

Thus algorithm 1 proceeds like the method of relaxation except that its convergence can be proved, at the cost of integrating a second P.D.E., the adjoint system.

O. PIRONNEAU

There are cases where algorithm 1 can be implemented directly: i) when the numerical integration of the PDE's can be done with a very good precision (see figure 5); ii) when the cost function of the optimum design problem is very sensitive to the positions of the free boundary. In all other cases a simple discretization of the equations of continuous problem would fail to enable us to solve the optimum design problem with a good precision because the numerical noise of the discretization makes it impossible to find a positive

a

solution of (4.14).

Therefore let us derive a formula similar to (4.11) for the

discrete case. For a given

S

let

0

zh

be a triangulation of

2h , a polygonal

approximation of 2So n

i e. T

2h =

f

= triangle where

Th ,

_I

h

Th f1 Th = one side or one node,

denotes the length of the largest side, let

H = {wow linear on Th}, i=l,...,n ; w continuous

If

is,the function in

wJ

'H

which equals 1 at the node j and

zero at all other nodes, then {w3}mj=1

is a basis for H (m - number of

nodes).

A first order finite element discretization (4.8) is (r = number of interior nodes) r

(4.16)

(vOvw4+.,w1)dx = fQ

f2

fwJdx

E J=1

h

h

o"w.

J=l,...,r ; m =

Equation (4.16) is a linear system in

0J

which can be solved by a

relaxation method, for example. Similarly (4.10) is approximated by m (4.17)

.r

(v*vw +*w )dx = t

21h

where

is the solution of (4.16)

('"d)wJdx, J=1,...,m ;

i< =

E *Jwj j=l


Let

S'(a)

be the boundary obtained from

S

221

by moving the nodes

0

along the lines of discretization "perpendicular" to

(see figure 6)

2h

then the discrete analogue of (4.11) can be obtained in a similar way: (1d-0d)2

(4.18)

1 6Eh = '2' -Q

+ f*-4*-VOV*)dx + oh(x)

h-2 h

In establishing (4.18) we have used the fact that continuously upon the coordinates of the nodes.

and

k

9'

depend

Now from the mean

value formula for integrals

(4.19)

oh(o)

1 6Eh = IS 0

Thus the discrete version of algorithm 1 is Algorithm 2: We assume that the user has an automatic triangulation subroutines which generate the interior nodes from the boundary nodes. For simplicity we assume that the first point of Step 0 - Choose

0

, set i=0 , choose

Step 1 - Compute

qs

from (4.16) with

Step 2 - Compute

,,

from (4.17) with

Step 3 - Let

z

tih

sh sth

remains fixed.

choose 0 E (0,1)

Qh Qh

sJ

is the curvilinear abcissa of the node

determined as follows: let

ai

J

and

al

where

6Eh3 = ISi (I-d12 +

a1J(s)

are

be the sines of the angle of the

discretization lines, let

(4.21)

S , set

be the number of boundary nodes on

s-s. i it i i a (s) - aJ + sJ+ j (aJ+l-aj) for s E (si ,sj+1), J=1,...,e

(4.20)

where

S

So

is given by (4.20) with

A a`) a'J(s) aids

as as

ak = 6Jk ,

k=l,...,e

0. PIRONNEAU

222

then take a

(4.22)

aEhj/(sj-sj-1)

Step 4 - Obtain. S(a)

j=2,...,m ; a = 0

by moving the nodes of

discretization lines of a quantity x aj

on the

Si

and compute the smallest

integer k such that (with self explanatory notations)

Eh(S(a)) - Eh(Si) f - pa E

(4.23)

a = ()k

j=1

and such that Si

,

to the next discretization line "parallele" to Step 5 - Set

Si+l

of the distance of

is not nearer than, say

S(2_k)

Si+1 = S(2-k)

Si

compute a 'new triangulation around

from its nodes computed in step 4, set

i = i+l

and go back to

step 1.

Conjecture - If the automatic triangulation subroutine is such that the position of the interior nodes depend continuously upon the position of the boundary nodes (and it is the case-of some such subroutines, for a given t

)

then the sequence {Eh(Si))i>O

will be a

decreasing sequence and {Si) will converge to an accumulation point of the discrete analogue of (4.4).

Justification - First of all (4.23) has always a non zero solution because t

i2

jZl (aP (s-sj-1) is the slope of x -o. Eh(S(a))

Therefore algorithm

at A = 0 (see (4.19)(4.21)(4.22)).

2 is the method of steepest descent applied to

(4.4) with 2 - 2h .and (4.16) instead of the continuous PDE; with one difference however, the triangulation is rebuilt at each iteration (so that each triangulation is well suited to the new geometry). since the cost function changes with the discretization, with the new triangulation might be greater than (4.23).

Eh(Sj)

Therefore,

Eh(Si+l) ,

despite

It should be noted that it might be possible to prove the

convergence of algorithm 2 by using.a model similar with the one described by Klessig-Polak (1972), by asking to increase the number

223


of nodes when the cost increases.

However, the following argument seems

to indicate that the cost is unlikely to be increasing. It is theoretically possible to consider

Eh

as a function of the

coordinates of all the nodes (instead of the boundary nodes only).

And

indeed in doing so one is very close to the concept of mappings of 2S into

20 , studied in the previous section. O'Carroll et al (1976) have

tested this method, by the way.

Now the change of cost due to a shift

of a boundary node tangentially to

S

is certainly smaller than the

change of cost due to a normal shift.

In the first case one changes

the definition of the continuous problem also while in the second case one changes the discretization of the problem.

It seems also that the

change of cost due to a shift of the interior nodes would be of the same order, if not smaller, than for the above tangential shift. Similarly, an implementable algorithm can be obtained for problem To derive an analogue of (4.12) one needs to use the following

(4.5).

formula (Murat-Simon (1974)): if 4 satisfies

then when

=J

J2

VoVwdx

a

is replaced by

fwdx

vw E H01 (2) ,4 E H01 02)

52'

, 64 E H1(2)

satisfies

J2vs0vwdx = - j,xvw.(f;+ dive" -(e'+te')v4)dx+o(x) vw E

HQ(S2)

(x,y) + xe(x,y) _ (x+), el(x,y), y+x e2(x,y)) are the coordinates of the,

transform of (x,y) by the map

I+xe

that transforms

st

into

ei

Then the analogue of (4.19) is aEh = -2j 9xv$[fe+divev4-(6'+t$')v4]dx + rsxa145`#d12dS+oh(x)

where

is the solution in

r

of

H,(9)

f2 v*vwdx = f2(4S-4d)wdx

and Ho 2h The reader will easily construct an implementation of

These equations remain valid when into

K

.

Vw E Ho(st)

$2

algorithm 1 in this case, by letting.

triangles that touches S , where

e

is changed into

0=0

except in the strip of

is taken linear and in tune with a.

O. PIRONNEAU

224

5.

CONCLUSION

An engineer faced with a free boundary problem may be in one of the two positions: either he knows very well his problem intuitively and he is able to find a good estimate of the solution; or he does not know much about the answer and he is not willing to develop a special subroutine to find it.

Then, if he is in the first case he will be

probably better off by using the relaxation method; but if he is in the second position he would prefer to call an "automatic" subprogram. Such efficient automatic scheme is still to be found; the last section of this paper contains one that looks reasonably promising on account of the fact that free boundary and optimum design problems are numerically usually quite stiff.

ACKNOWLEDGMENT I wish to thank J. R. Bourgat, A. Dervieux and P. Morice for their helpful suggestions. REFERENCES 1. 2.

3. 4.

5.

Baiocchi C. (1975) - Cours au Collage de France. Begis D., Glowinski R. (1975) - Application de la m6thode des 616ments finis A l'approximation d'un problCme de domain optimal. Applied Math. & Optimization, Vol. 2, No 2. Benssouasan A., Lions J. L. (1973) - CRAS 276, pp. 1189-1193, Paris. Bourgat J. F., Duvaut G. (1975) - Calcul num6rique avec ou. sans sillage autour d'un profil bidimensionnel sym6trique et sans incidence. Rapport Laboria No 145. Bourgat J. F., Duvaut G. (1976) - R6solution num6rique d'un problCme de Stephan A 2 phases par une in6quation variationnelle (a paraitre).

6.

Bourot J. (19714) - CRAS A.278.455.

7.

C6a J. (1975) - Une m6thode nur6rique pour Is recherche dun Romaine optimal. Proceedingal.F.I.P. Congrps Nice. C6a J., Glovinaki R. (1972) -.M6thodes num6riques pour 1'6coulement laminaire d'un fluide rigide visco-plastique incompressible. Intern. J. of Computer, Math. B. Vol. 3, pp. 225-255. Chavent G. (1971) - Th6se de Doctorat, Paris. Chesnais D. (1975) - On the existence of a solution in a domain identification problem. J. of Math. Anal. and Appli. 52,2. Cryer C. W. (1970) - On the approximate solution of free boundary problems using finite differences. J. of the Association for Comp. Machinery, Vol. IT, N° 3, pp. 397-4n. Garabedien P. R. (1956) - The mathematical theory of three dimensional cavities and jets.Bull. Amer. Math. Soc. 62, pp. 219-

8.

9. 10.

11.

12.

235.


13.

14. 15. 16.

17. 18. 19.

20. 21. 22. 23. 24. 25. 26.

27.

225

Garabedian P. R. (1964) - Partial Differential Equations. Wiley, New York. Glowinski R., Lions J. L., Tremoliere R. (1976) - Analyse num6rique des in6quations quasi-variationnelles. Dunod (to be, published). Hadamard J. (1910) - Legons sur le calcul des variations. Gauthiers-Villars. Klessig R.,Polak E., (1970) - A method of feasible directions using function approximation with applications to min-max problems. E.R.L.-M287, University of,California, Berkeley. Lamb H. (1879) - Hydrodynamics. Cambridge University Press. Lions J. L. (1972) - Some aspects of the optimal control of distributed parameter systems. RESAM, 6 SIAM. Murat R., Simon J. (19T4) - Quelques r6sultats our le contr8le par un domain g6om6trique. University de PARIS VI, Rapport interne Lab. Analyse Num6rique, R° 74003. Morice P. (19714) - Une m6thode d'optimisation de forma de domain e. Proc. CongrLs IFIP-IRIA, Paris. Springer Verlag. O'Carroll M. J. and H. T. Harrison (1976) - Proc. ICCAD Conf. Rappalo, Italy. Pirdnneau 0. (19T3) - On optimum profiles in Stokes flow, J. Fluid Mech., Vol. 59, pp. 117-128. Pironneau 0. (1976) - These de Doctorat, Paris. Southwell R. V. (1946) - Relaxation methods in theoretical physics, Vol. 1, Clarendon Press, Oxford. Stanitz J. (1952) - Design of two-dimensional channels, NACA technical note 2595. Trefftz B. (1916) - Uber the Kontraktion kreisf6rmige FlUssigkeitsstrahlen. Z. Math. Phys. 614, pp. 34-61. Zienkievicz 0. C. (1971) - The finite elements in Engineering Science. McGraw-Hill, London.

226

O. PIRONNEAU

Figure 1

- (From Bourgat-Duvaut (1976)).

Fusion of an ice cube when the bottom plane is heated.

Even though

tlye free boundary is not well approximated, the isothermal lines are computed. accurately.


227

11

im

am

III

Figure 2

Illl 111111 111111 111111 111111

Ilj 11j11IfII

111111 1

111

11 II I

11111

I11III

/111111

Ill

IN IN

IIIIIIII

lull VIII

11111111

III

11111111

11111

11111111

VIII

111 II111111

VIII 1101

1011111

III

1111101

VIII

IIIIIIII

0111

I II 11111111

VIII

Figure 3 - (From Begis-Glowinski (1975)).

Computation of a free boundary (upper line) by the method of BegisGlowinski and the corresponding finite element triangulation in a good case and in a bad case.

228

0. PIRONNEAU

Figure 4 - (From Morice (1974)).

Computation of a free boundary by Morice's method and the corresponding triangulation.


Figure 5 - (From Bourot (1974)).

Computation of the minimum drag profile in Stokes flow by the method of Pironneau (1973).

Figure 6

Sl is obtained from So by movingrthe nodes of discretization of So of a suitable quantity: aaj . along the lines of discretization of

2so "perpendicular" to So

229

"SOME APPLICATIONS OF STATE ESTIMATION AND CONTROL THEORY TO DISTRIBUTED PARAMETER SYSTEMS" W. H. Ray

1.

Introduction In this chapter we shall discuss a large number of real and

potential applications of state estimation and control theory to distributed parameter systems.

Because of the inherent practical difficulties

of state measurement in distributed systems, the major problems in control implementation are often due to difficulties in obtaining reliable estimates of the state variables,

For this reason, the major emphasis

of this chapter shall be on distributed parameter state estimation.

The

reader is referred to other survey articles for an overview of parameter identification [1,2] and to the other chapters in this volume for a discussion of the control of distributed parameter systems.

A second

troublesome problem arising in applications is the development of efficient computational algorithms which will allow real time implemen. tation of these state estimation and feedback control schemes.

In what

follows, we shall give examples aF how these numerical problems can be solved in practice.

The next section gives an overview of practical considerations and the algorithms available for distributed parameter state estimation and presents a survey of reported applications.

Following this, a

detailed case study of the real time implementation of state estimation and feedback control to a laboratory model process will be discussed in some detail.

Finally, a general survey of remaining practical problems

will be presented along with some suggestions for future research

directions. II.

A Survey of State Estimation Algorithms and Applications The field of sequential state estimation has grown enormously in

scope and popularity since the original work of Kalman [3] and Kalman

231

232

W. H. RAY

and Bucy [4] in the early 1960's.

However, it was not until 1967 that

the first extensions of these ideas to distributed parameter systems began to appear.

The earliest paper seems to be due to Kwakernaak [5],

and was quickly followed by the work of Falb [6], Tzafestas and Nightingale [7-10], Balakrishnan and Lions [11], Thau [12], and Seinfeld [13], all of whom apparently developed their results independently and nearly simultaneously.

Since these early papers, work has

progressed so that state estimators are now available for numerous classes of linear and non-linear distributed parameter systems. Unfortunately, the reported practical applications of distributed parameter estimation algorithms has badly lagged the theory.

This is

perhaps due to the relative complexity of the estimation equations when compared to the classical Kalman-Bucy filter.

Another factor is the

lack of meaningful contact between potential industrial users and research engineers familiar with the capabilities of these techniques. In this section we shall summarize the results available for distributed parameter estimation and describe recent computational and experimental example applications.

In this way it is hoped to provide

the potential user of these techniques with a guide to the theory as well as inspiration for further applications. This survey section, which has been updated from an earlier lecture [14] by the author, shall begin with a discussion of the theoretical results available for both linear and nonlinear systems and then review the example applications which have appeared to date. A.

Theoretical Results

There have been a large number of theoretical approaches taken in the derivation of distributed parameter state estimators.

The most

direct approach is to somehow approximately lump the distributed system and then apply the classical lumped parameter theoretical results [3,4]. One can certainly proceed in this way and obtain useful results (e.g., [15-20]), although in certain cases the distributed parameter formulation does seem to have advantages (e.g., when there is boundary noise).

Because there are so many ways of Pumping a distributed system

(e.g., finite difference, modal representation, splines, etc.) it is hard to compare exact distributed parameter results with lumped approximations, and thus the possible pitfalls of lumping have not been well

APPLICATIONS OF STATE ESTIMATION AND CONTROL THEORY

defined.

233

In the discussion of theoretical results which follows, we

shall deal mainly with results derived for the full distributed parameter system model; however, several cases of "lumping approaches" to filtering will be included in the applications section.

Even excluding lumping, there have been a wide range of approaches taken in arriving at theoretical results ranging from "formal" such as minimizing a given error functional to more "rigorous" approaches explicitly treating the evolution of the probability distribution of the state variables.

While the more rigorous

approaches yield more precise statistical information about the problem, they are limited in the classes of problems which may be treated.

The

formal appt«)ches yield less detailed statistical information, but have t.he,,ability to treat a wide range of both linear and non-linear problems.

Therefore, both methods of analysis have value and in fact are usually found to yield identical filter equations in situations where both techniques apply.

Because this survey section is emphasizing applications, it is not our intention to attempt an extensive review of the theoretical approaches for developing distributed parameter state estimators, but rather shall concentrate on the results which may be of direct interest to the practitioner.

For a fuller discussion of the theoretical ideas

see the review by Tzafestas [21]. 1.

Observability and Measurement Location Perhaps the first consideration in the application of a state

estimator is the question of observability and the optimal location of measurement devices.

Observability, which has been defined for lumped

parameter systems as the ability to determine the initial state based on the system model and measurements available, is a more difficult question for distributed parameter systems because of the strong influence of the location of the sensing devices.

Thus for a fixed

number of sensors, it is possible to choose locations for which (i) the system is not observable, or (ii) the system is.observable, or (iii) for which the system is both observable and the measurement locations yield the maximum amount of information in the presence of noise.

For these

reasons, the choice of type, number, and location of tensors is crucial

W. H. RAY

234

in the design of a distributed parameter state estimator.

Wang [22) seems to be the first to discuss observability for distributed parameter systems and proposed a definition based on the recovery of the. initial state through analogy to lumped parameter systems.

of

Goodson and Klein [23] define observability as the existence

a unique system state based on the model and the observation device

and have developed general criteria for observability for a number of linear partial differential equation systems.

For first order quasi-

linear systems, their criteria includes the requirement that each of the characteristic lines in the system intersect a sensing device.

Yu and

Seinfeld [24J have quantified this requirement and, by applying the known lumped parameter observability theorems along the characteristic lines of the system, have developed criteria specifying which combinations of states must be measured.

Their results take the

familiar form of determining the rank of an observability matrix. Recently Thowsen and Perkins [25] have extended these results to a slightly more general form of the equations, and Sendaula [26] has considered the related problem Of time delay systems. Goodson and Klein [23] also consider linear diffusion and wave equations and are able to develop sufficient conditions for observability based on the choice of sensor placement.

Cannon and Klein [27]

have numerically investigated the optimal placement of the measurement location for a simple linear example problem of heat conduction with specified surface temperatures.

Yu and Seinfeld [28] have developed

sufficient conditions for observability for linear systems amenable to a modal representation.

Their approach is to represent the system in

terms of N eigenfunctions and then apply the known observability conditions [3,4] for lumped parameter systems to the resulting ordinary differential coefficient equations.

The rank of the observability

matrix depends on the values of the eigenfunctions at the measurement locations.

These results were applied to both one and two dimensional

heat conduction and diffusion examples.

In addition, numerical studies

were performed to show the influence of the measurement location on the quality of the estimates.

In spite of these valuable contributions, at present there exists no'general theory of observability for either linear or


235

nonlinear distributed parameter systems; there exists only a few results for the particular linear cases studies.

Nevertheless, several recent

papers [29,30] would seem to lead toward such a theory.

Similarly no

general results are available for determining optimal sensor locations; all that one can do is numerically investigate each problem individually for a range of model parameters, noise statistics, etc.

Several workers

have developed systematic algorithms for carrying this out [31-33]. However, further theoretical results would be extremely useful here. 2.

State Estimators Available

State estimators, for smoothing, filtering, and prediction, have been developed for a wide range of distributed, parameter systems.

In

this section we shall discuss explicitly the filtering results; however, the related prediction and smoothing algorithms can be found in a straightforward manner once the filtering equations are known.

Just as

in the case of lumped parameter systems, linear distributed parameter systems allow the possibility of much more rigorous and precise results than-do nonlinear systems.

Not only do the stochastic system modelling

equations have precise meaning, but the estimate covariances arise naturally in the filter calculations and provide valuable statistical information about the quality of the estimates.

On the other hand

nonlinear state estimators, developed by more formal techniques, are much more general, greatly expand the range of practical problems which can be handled, and have shown to reduce to known, rigorously derived, linear filters when the nonlinear equations are made linear.

Thus the

practitioner is advised to take advantage of the rigorous linear filter, with its precise estimate statistics, where possible, and to make use of the more general nonlinear filtering results, when necessary. Because of the fact that the independently derived linear filters appear as special cases of the more general nonlinear results, in what follows we shall group the state estimation results under their general nonlinear form.

A summary of available theoretical results is presented

in Table 1.

The first class of distributed parameter systems to have filters developed were systems with time delays.

Kwakernaak's classic paper [5]

led the way and showed the structure of the distributed parameter

236

W. H. RAY

filtering problem.

Although a number of linear filtering results have

appeared since, [17,34-42], for the purpose of our discussion here these can be treated as special cases of the nonlinear filtering results of Yu et al. [43].

In this general formulation, the filtering problem for

time delays takes the form z = f(x(t), z(r1,t),

--z(r,,t)) +

(1) JI (z(r,t)dr + g(t)

zt(r,t) = -M(r,t),(r,t) + (2)

g(z(r,t),r,t) + t(r,t) N N .y

y(t) = h(x(t), z(r1*,t), --z( Y*,t),t) +

3) j

(z(r,t),r,t)dr + 1(t)

x(0) = xp

(4)

z(r,0)= z,(r)

(5)

z(O,t) = b(x(t))

(6)


237

Table I. Summary of Theoretical Results Topic

References

Systems with time

[5,17,34-43,89]

delays

Systems described by

Linear Systems [7-9,11,12,27,

second order partial

28,45-641

differential equations

Nonlinear Systems [10,13,15, 44,58,65-67]

Linear systems described

[6,9,47,48,54,55,57,58]

by integral or integrodiffer4ntial equations Systems described by

[68]

partial differential equations with moving boundaries

Separation principles

[8,35,38,52,57,69,70]

and stochastic feedback

control where Eqs. (1,2) represent the behavior of a coupled lumped and distributed parameter system defined for donain

r E [0,1]

conditions (4-6). defined by (3).

.

A(t)

and

z(r,t)

t > 0

and on the spatial

are state vectors with boundary

Observations consist of the output vector y(t) The quantities

and n(t)

represent

zero mean random processes with arbitrary statistical properties. Within this framework an extremely wide range of time delay problems

may be treated. 'A second class of distributed parameter systems for which a large number of results are available are second order partial differential equations systems. spatial variable is [44]

A very general form of this problem for one

W. H. RAY

238

zt(r,t) =

(r,t) (7)

t>0,0(zm*)J

while (32)

4,(t) = 2a2'E(0)R -1(t)'ET(0)

2a2,,8(1)R1-l(t),'ET(1)

D1(t) =

and

(33)

is the matrix of coefficient arising from

D(t)

R+(z,z',t) =

N

N

E

E

di j (*ci(z)cpj(z')

(34)

i=0 j=0

Due to the symmetry of the covariance equations only ODE's for the along with

N+l

N+l 2+ N+l

need be solved (and these can be done off-line)

Bij(t)

ODE's for the estimates, al(t).

The computational

effort may be summarized in tabular form as Table 3

Computational effort for filter implementation Number of

Estimate Eq'ns

Covariance

Total Eons to

Eigenfunctions

(on-line)

Eq'ns (off-line)

be solved

needed, N+l

(N+1)ODE's

N+1 2+ W '10E's

ODE's 5

2

2

3

3

3

6

9

4

4

10

14

5

5

15

20

6

21

27

6

W. H. RAY

254

In the present experimental study, it was found that

N+l = 3

was adequate to represent all the temperature profiles studied; thus a total of 9 ODE's had to be solved.

For the sake of convenience all

9 ODE's were solved on-line in real time and yet the computational effort required was found to be less than 0.5% of real time. The experimental results for this case study were very impressive (cf [78] for more complete details).

Figure 6 shows typical open loop

results when only a single temperature measurement at

z = 0

was

taken and a substantial random error added to the measurement before passing it to the filter.

In this case a random error was taken from

a Gaussian distribution haVing a standard deviation of 8°C.

As can be

seen, the initial condition given the filter was some 30°C in error, and yet the filter converges very quickly to within 1-2°C of the exact

profile. Typical closed loop results are shown in Figure 7.

In this case

the filter, having only one temperature measurement (at z=0) with 8°C std. deviation added measurement error, is coupled with a proportional + integral modal feedback controller making use of the certaintyequivalence principle.

In all cases the stochastic feedback controller

worked well with the estimates and exact profiles converging quickly and both reaching the desired set point.

The results in Figure 7 are

for an untuned filter and controller; thus one would expect that with tuning the rate of convergence of the stochastic controller could be improved still further.


Figure 6.

Open loop distributed parameter filter performance for the heated slab with one temperature sensor (z=0) and 8°C standard deviation added measurement errors.

255

100

Desired profile

'C

Filter Measured profile 80

60

T 40

20 t =0 sec.

01 0

Figure 7.

.2

.4

z

6

8

1.0

Closed loop distributed parameter filter and stochastic feedback controller performance for the heated slab having only one temperature sensor (z=O) and 8°C standard deviation added

measurement errors.


IY.

257

Concluding Remarks

From the discussion above it should be clear that there are a wide range of rigorous theoretical results available for optimal state estimation in linear distributed parameter systems.

At the same time,

a substantial number of approximate optimal filtering results leading to useful algorithms systems have appeared.

Nevertheless, there are

several areas in which further theoretical work would be of great help in applications.

Some examples which spring to mind would be conditions

for observability which extend to a wider range of systems and which are easy to apply. sensors.

This would greatly aid in the choice of location for

A second area of need is the development of an interaction/

decomposition theory for stochastic feedback control which extends to a wider range of linear and nonlinear systems and to a wider range of error distributions.

If such a theory were able to contain sensitivity

information with regard to sensor and controller placement, this would be extremely helpful in choosing the optimal sensor location and controller application points.

In the applications/implementation sphere, further research is, perhaps, even more urgently needed.

Our experimental study described

in the last section (which seems to be the only experimental work yet to appear) has done much to show that distributed parameter state estimation and feedback control algorithms can be implemented on minicomputers if one is careful in choosing the numerical algorithms. Thus one pressing need is for work developing and demonstrating efficient algorithms for real time implementation. proceeding In this direction.

Our own research is

We currently have under development

algorithms suitable for linear multi-dimensional systems as well as others for nonlinear systems in single or multiple space dimensions. All of these techniques are being applied to experimental model processes through the use of our PDP 11;40 minicomputer system and the results shall be reported elsewhere in due course.

As noted above, nearly all of the applications of distributed parameter state estimation and feedback control ,eported remain potential applications because of the lact of cor,v;ncing

that the wmplex equations arising in real life problems can be

ins

W. H. RAY

258

feasibly solved on-line with minicomputers.

It is our hope that the

encouraging experience from these first experimental results will inspire the potential industrial and governmental user to begin applying these techniques to the solution of relevant real world problems.

This

in turn should provide feedback to the theoretician in the form of new challenging problems and thus bring enhanced vitality to the entire field.

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3. 4. 5. 6. 7.

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"NUMERICAL SOLUTION OF THE TRANSONIC EQUATION BY THE FINITE ELEMENT METHOD VIA OPTIMAL CONTROL"

M. 0. Bristeau, R. Glowinski, 0. Pironneau

ABSTRACT It is shown that the transonic equation for compressible potential

flow is equivalent to an optimal control problem of a linear distributed parameter system.

This problem can be discretized by the finite element

method and solved by a conjugate gradient algorithm.

Thus a new class

of methods for solving the transonic equation is obtained.

It is

particularly well adapted to problems with complicated two or three dimensional geometries and shocks. 1.

2.

PLAN 2.

Introduction

3.


4.

Gelder's algorithm for subsonic flow

5.

Formulation via optimal control

6.

Discretization and numerical solution

7.

Numerical results

8.

Conclusion

INTRODUCTION

The transonic equation is a non linear partial differential equation which has an elliptic behavior in the subsonic regions of the flow and a hyperbolic behavior in the supersonic regions.

At the

interface the normal component of the speed of the flow can be discontinuous (shocks).

Some finite difference methods have been successfully

developed even for flows around simple 3-D objects (Jameson (1974), Garabedian-Korn (1971)).

However the method of finite differences is

265

M. 0. BRISTEAU etaL

266

not well suited to complicated geometries. An alternative approach using finite elements was studied by Gelder (1971), Norries and de Vries (1973), Periaux (1975) but their methods explode at supersonic speeds. Following Gelder's approach we shall replace the transonic equation by the minimization of a functional in an abstract space, a problem which can be solved by the methods of the theory of calculus of variations and optimal control theory. 3.

STATEMENT OF THE PROBLEM Stationary adiabatic monophasic compressible flows, in which the

effects of viscosity are neglected, are well described by the set of equations apu3 apu2 + a + a x2 x3

apul

0

(3.1)

0

a

xl

(1

2)

= 0

1

(3.2)

C*

u = vO (ui =

(3.3)

p

where and

y

(1971).

a axi

i - 1,2,3)

,

is the density, u is the speed of the fluid and where

po,C*

are constants (y=1.4 for di-atomic gas, see Landau-Lifschitz

We shall denote

1

k =

- a = 1/(Y-1). Therefore, if 1

2

C* the region occupied by the fluid, one must solve the nonlinear partial

differential equation: 0

(3.4)

with the boundary conditions (3.5)

(3.6)

OIr1 = Ol

an 'r2

g2

in

2

is

267

NUMERICAL SOLUTION OF THE TRANSONIC EQUATION

where

and

r1

r2

are parts of the boundary

rI U r2 = a2

assume that

rl n r2 = 0

and

.

ac

of

2

.

We shall

In addition, if there

are shocks (i.e. lines or surfaces where the tangential speed of the flow is continuous but the speed normal to these lines or surfaces is discontinuous) then, across the shock: (3.7)

(pu)+ = (pu)

(Rankine-Hugoniot condition)

(3.8)

un

(entropy condition)

un

where it is understood that the particles of the fluid move from - to +. Note the (3.4) multiplied by

w E

and integrated by parts,

C1(62)

leads to (3.9)

fS2(1-klvol2)avo vwdx = Ir (1-kIvfl2)Lg2wdr2 2

Vw ( C1(52) s.t. wlr1 = 0 ; 1r1 = 0

If the notion of derivative is extended and the space

replaced by

H1(s2) = {wEL2(s2)

jvw E(L2(52))3}

weak formulation of (3.4)-(3.6). 4.

GELDER'S ALGORITHM FOR SUBSONIC FLOW g21r2 = 0

functional

(1-kIv412)a+ldx

Eo(4) = - tsa

we shall say that

(4.2)

is

Note that it contains (3.7).

For notational convenience we suppose

(4.1)

C1(52)

then (3.9) is called a

Hol(2)

4s

=

is a stationary point of

Eo

on

lml rl = 0}

if

6Eo = Eo(o+6o,) - E0(4) = o(o) V6,k E Ho1(Q)

.

Consider the

268

M. 0. BR ISTEAU et al.

Since, from (4.1) (4.3)

6E0 = f2 2k(a+1)(1-kIvhI2)avov64,dx + o(6o)

any stationary point of fo

Hol(y)

on

satisfies

f2(1-kIv4,I2)avovwdx = 0 Vw ( Hol(2) Thus all stationary points of

Eo

on

such that

Hol(52)

0Irl

(holrl

=

and which satisfy (3.8) are solutions of our problem. Let us look at 2

2

2ka V v6

2k(a+l)

1 dx

(1-kjvj

d22

)

with our notation the mach number is such that

M2 = 2ka(1-kFv(pI2)-1

therefore, if

Iv(p I2

is the angle between

o

and

vo

v6,

2 dE

_ - 2k(a+l) f2 P(1-M2 cos2

e)Iv6(k I2 dx

This shows that if in some part of the fluid

M > 1

,

E is not convex

and the solution of (3.4)-(3.8) is only a saddle point of E. other hand, if

M < 1

in

then

2

(3.4)-(3.8) is a minimum of

E

.

E

This fact was utilized by Gelder

(1971) and Periaux (1975) for constructing The functional the

H1(s2)-norm

mn+l E H1(st):

E ;

On the

is convex and the solution of

a solution of (3.4)-(3.8).

is minimized by a gradient method with respect to i . e .

(, k

-

is constructed by solving for

f2 Pn VOn+l vwds2 = 0 VW E Ho1(2)

,

(On+l -'hl) I rl = 0


269

This method works very well (less than 15 iterations in most cases) and it is desirable to construct a method as near to it as possible, for supersonic flows. 5.

FORMULATION VIA OPTIMAL CONTROL Along the line of §5 we shall look for functionals which have the

solution of (3.4)-(3.8) for minimum.

Several functionals were studied

in Glowinski-Pironneau (1975) and Glowinski-Periaux-Pironneau (1976). In this presentation we shall study the following functional:

(5.1)

where

E() = fS2 P(IvEI2)Iv4-E)I2 dx, P(IvEI2) = (14177 J2)a

0 = 0(E)

(5.2)

is the solution in

f2 P(IVEJ2) vovwdx = 0

H1(52)

of

Vw E Hol(S2)

, 0Irl = 41

Suppose that (3.4)-(3.8) has a solution.

Proposition 1.

Given e > 0, small,the problem

min (E() It E 9}

(5.3) where

E _

{r,

(

H1(52)) tIrl = 01

has at least one solution and if solution of (3.4)-(3.8).

JvE(x)I o

of (5.3) has a subsequence which satisfies (3.5)-(3.7) and

(1-kJvtnJ2)avx

lim fn

vwdx = 0

Vw E HQ1(52)

Proof: the first part of the theorem is obvious. Let

that

{En}

be a minimizing sequence of

E

then

En E c

implies

Ilv nll2 < k-1(i-e)2 f2 dx , therefore a subsequence (denoted

also) converging towards a

Ilv(on-tn)II

0.

i

E s

can, be extracted.

Therefore

fQ Pn v(4n-En) Vwdx = f2 Pn Vtn vwdx -+ 0

Furthermore

{En}

M. 0.8RISTEAU et at.

270

for every subsequence such that

pn

converges in the

L-(2) weak star

topology. Remark.

Note that if

solution of (5.3).

is a weak limit of

t-

{gn)

,

may not be a

This, however, does not seem to create problems in

practice. Proposition 2 If

t1r1 = 01

6Ir1 = 0 , then

(5.6) E(?+6ts)-E(

) =

(M12

2fQ p(jvtf2)(l+

2p'p

1IV'012

= +2k,(1-klvml2)-IIvm12)

Proof From (5.1) and (5.2) (5.7)

E(+6r)-E(ti) = 2fS[2a'vg-m1V(d- 0 12-pv(,6-t;)v6t;+pV(O-g) v6,b]dx

+ o(6) + 0(61;) where ka(1-kIVFl2)a-1

p' = -

From (5.3)

(5.8)

fQ pv6ovwdx = - f2 2p'vt;-v6EvO-vwdx+o(6t;)

and since

p(1v(t;+6)12)

Nw E Ho1 (9)

is bounded from below by a positive number,

if a > 2 . there exists K such that IIv60I) 5 in (5.8), (5.7) becomes Therefore, by letting w = d-t 6E = - 2f [ pV(O- ) V6t + P,

I V4, 12-1 v2;12) vt;' vbt;] dx

IQ

and from (5.2) the term

pvov6t;

disappears.


271

Corollary 1 If

is a stationary point of

i,

E

,

it satisfies:

2

(5.9)

in

(1-Iv Zl2 Iv01-2) vZ] = 0

v-[p(1 +

2

2

(5.10)

(1 +

= 0

(1-Iv I2 Iv- -2) a

P

;

lr1 = l

Ir2

Remark: It should be noted that in most cases (5.3) has no other stationary point than the solutions of (3.4)-(3.7).

Indeed let

(xC,yC,zt) be a curvilinear system of coordinate such that

,o,o)

vi:=(a C

Then, from (5.9), (5.10)

(5.17)

ax

[p(l +

e

(1- IvZI2 Ivm1-2)

or

r2(1- IvZ12 IVml-2)

] = 0,

an

Ir2 = 0

Ir2 = -2, ZIrl = 01

This system looks like the one dimensional transonic equation for a, compressible fluid with density

P (1 + 2

(1- IcEI2 IvOl-2))

Therefore, if the t-stream lines meet two boundaries and the shocks and

1+ then

- Z

.

Z (l- IvE12 Iv4s l -2) > 0

at < + m at

M. 0. BRISTEAU et al.

272

DISCRETIZATION AND NUMERICAL SOLUTIONS

6.

Let

be a set of triangles ortetrahedra of 2 where

Th

T.

is

which approximate

rl

Suppose that

the length of the greatest side.

U T c Q ,

h

n T2 =

or a vertex

VT1,T2 E Th

TETh

0

Let

and

2h = U T

parts of ash

rlh' r2h

h

and

r2

Let

Hh

. be an approximation of

is completely determined by the values

Note that any element of Hh that it takes at the nodes of

i

N = n+p+m

nodes

Pi

Th

with

E ]n,n+p[ , and if we define

wi = 1

(6.2)

Then any function

T VT E Th}

linear on

Hh = {wh ( C°(2h)I wh

(6.1)

has

Hl(2):

.

Therefore if we assume that if

Pi E rlh

wi c Hh

i > n+p, Pi

by

at node 1 and zero at all other nodes

w E Hh

(6.3)

0 = Eaiwi

Algori thm

1

is written as

N

Let

Ioh =

E

i;1w.

, then (5.2) becomes

i=1 f9(1-k{VEhI2)aVOhVwidx = 0 (6.4)

Oh =

n+p i E m wi + i=1

N E

n+p+l

i

01wi

i=1,...,n+p

E r2h

Th if


273

and (5.6) becomes

(6.5)

6Eh =

2 (6.6)

o(8 )

E i=1

6Eh = f2[P-P'(Iv0h1 2-Ivdhi2)]v

dx

Consider the following algorithm Step 0

Choose

Step 1

Compute

0hj

Step 2

Compute

{6Ehj,

Step 3

Compute

6C

set

-r,, t,o

j=0

by solving (6.4) with

Ch = Chj

by (6.6)

n+p = h

(6.7)

E 6C 'w. i=1

by solving

p6hvwidx = 6Ehj, i=1,...,n+p h

Step 4 (6.8)

Compute an approximation

min XE[0,1 ]

jr

2h

P()d v(Ch (x)

of the solution of

S,j

-

h

x))I2dx

where N

gh(x) = 1z

Step 5

Set

(4j-x6gh)wi

4hj+1 = 4h(aj), j=j+l

and go to step 1.

Proposition 3 Let

{ghj}j-,.0

be a sequence generated by algorithm 1 such that

vFj(x)I s k-1/2 Vx, vi

.

Every accumulation point of {Chj}j:-.,o

stationary point of the functional

(6.9)

Eh(Ch) = f2hIV(kh-gh)I2dx

is a

M. O. BRiSTEAU et al.

274

where

is the solution of (6.4), in

Oh -Oh(h)

Sh = {Ch ( Hhi lvth(x)l

k-1/2 tlx E 2h)

`:

Proof Algorithm I (6.9) in

`h

is the method of steepest descent applied to minimize

, with the norm

f2 o hP h dx

(6.10)

h

Therefore

{Eh(Chj)}j

decreases until

6Ehj

reaches zero.

Remark 6.1: (6.4) should be solved by a method of relaxation but (6.7) can be factorized once and for all by the method of Choleski. Remark 6.2: Problem (6.8) is usually solved by a Golden section search or a Newton method.

Remark 6.3: Step 5 can be modified so as to obtain a conjugate gradient method.

Remark 6.4: The restriction: Juh (x)j

k-1/2

in theorem 5.1 is

j

not too close to

k-1/2, otherwise one must treat

this restriction as a constraint in the algorithm.

Also, even though

theorem (5.1) ensures the computation of stationary points only, it is a common experience that global minima can be obtained by this procedure if there is a finite number of local minima.

Remark 6.5: The entropy condition numerically.

Let

M(x)

Ath < + -

can be taken into account

be a real valued function then

Ath < M(x)

becomes, from (6.7)

(6.11)

-E 7j 8Ehj 5 M(xi)

i

= 1,...,n+p

Therefore, to satisfy (6.11) at iteration 6Ehj = 0 equality.

in (6.7) for all

i

j+l

,

it suffices to take

such that (6.11) at iteration

This procedure amounts to control

w = at

j

instead of

is an t


7.

275

NUMERICAL RESULTS The method was tested on a nozzle discretized as shown on figure 1,

The Polak-Ribiere method of

(300 triangular elements, 180 nodes).

conjugate gradient was used with an initial control: At = 0

A mono-dimensional optimization subroutine based

(incompressible flow).

on a dichotomic search was given to us by Lemarechal.

Several boundary

conditions were tested 1°) Subsonic mach number

at the entrance, zero potential on

Mm = 0.63

exit, the method had already converged in 10 iterations (to be compared with the Gelder-Periaux method) giving a criterion (Eho =

EhlO = 2 10-13

10-4)

2°) Entrance and exit potential specified. For a decrease of potential of

41 - 02 = 0.7

the method had

converged in 20 iterations without including the entropy condition, giving a criterion of

Eh20 = 5.10-7 , the results are shown on

figure 2. 30) Supersonic mach number

M. = 1.25

The method computes a solution that has a shock at the first section of discretization. Another boundary condition must be added. One iteration of the method takes 3 seconds on an I811370/158 on this example.

A three dimensional nozzle is being tested: the result will be shown at the conference.

20 to 40 iterations are usually sufficient

for the algorithm to converge. the tabulated data. tested.

The results are in good agreement with

Simple and multi-bodies airfoils are also being

For them it is necessary to include the entropy condition;

80 iterations are usually more than sufficient for the convergence. 8.

CONCLUSIONS

Thus this method seems very promising.

It compares very well with

the finite differences method available and it has the advantage of allowing complicated two and three dimensional geometries.

This work

illustrates the fact that optimal control theory is a powerful tool with unexpected applications sometimes.

276



277

A

II i

i

a

ao

O

N

-


278

ACKNOWLEDGMENT

We wish to thank M. Periaux, Perrier and Poirier for allowing us to use their data files and computer,and for their valuable comments. REFERENCES 1. 2. 3.

4.

5.

6. 7.

8.

Garabedian, P. R., Korn, D. G. - Analysis of transonic airfoils. Com. Pure Appl. Math., Vol. 24, pp. 841-851 (1971). Gelder, D. - Solution of the compressible flow equation. Int. J. on Num. Meth. in Eng., Vol. 3, pp. 35-43 (1971). Glowinski, R., Periaux, J., Pironneau, 0. - Transonic flow computation by the finite element method via optimal control. CongrPs ICCAD Porto Fino, June 1976. transsonique Glowinski, R. and Pironneau, 0. - Calcul par des mEthodes finis et de contr8le optimal. Proc. Conf. IRIA, December 1975. Jameson, A. - Iterative solution of transonic flows. Conf. Pure and Applied Math. (1974). Norries, D. H. and G. de Vries - The Finite Element Method. Academic Press, New York (1973) Periaux, J. - Three dimensional analysis of compressible potential flows with the finite element method. Int. J. for Num. Methods in Eng., Vol. 9 (1975). Polak, E. - Computational methods in optimization. Academic

Press (1971).

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