open loop feedback laws for the wave equation

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OPEN LOOP FEEDBACK LAWS FOR THE WAVE EQUATIONBui An Ton aa Department of Mathematics , University of British Columbia , Vancouver, B.C., V6T 1Z2,CanadaPublished online: 31 Aug 2006.

To cite this article: Bui An Ton (2002) OPEN LOOP FEEDBACK LAWS FOR THE WAVE EQUATION, Numerical Functional Analysisand Optimization, 23:3-4, 401-417, DOI: 10.1081/NFA-120006701

To link to this article: http://dx.doi.org/10.1081/NFA-120006701

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©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.

MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016

OPEN LOOP FEEDBACK LAWS FOR THE

WAVE EQUATION

Bui An Ton

Department of Mathematics, University of British Columbia,Vancouver, B.C. Canada, V6T 1Z2

E-mail: [email protected]

ABSTRACT

The dispersion coefficient and the source of the wave equation are deter-mined from partial measurements of the solution. The sought functionsare considered as controls and the problem is treated as a multicontrolopen loop problem. Feedback laws are established.

Key Words: Inverse problems; N-person control; Open loops;Feedback; Wave equation

1991 Mathematics Subject Classification: 35L99; 49J20; 49N45; 93B52

1. INTRODUCTION

Let � be a bounded open subset of R3 with a smooth boundary @� andconsider the initial boundary-value problem

ytt � �r:fu1ryg ¼ �u2 in �� ½0,T � , � 2 ð0, 1�

yð :, tÞ ¼ 0 on @�� ð0,TÞ ð1:1Þ

f yð :, 0Þ, ytð :, 0Þg ¼ f!1,!2g ¼ ! in �:

The control u ¼ ðu1, u2Þ is in a given bounded subset ofðL2

ð0,T;L2ð�ÞÞÞ

2 with u1 � 1 for all ðx, tÞ 2 �� ½0,T �: We associate with

401

Copyright # 2002 by Marcel Dekker, Inc. www.dekker.com

NUMER. FUNCT. ANAL. AND OPTIMIZ., 23(3&4), 401–417 (2002)

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Eq. (1.1) the cost functionals

Jjð�;y;!;uÞ ¼ f1þ �ð2� jÞg

Z T

�

fjðyð :, tÞ,y0ð :, tÞÞ þ kjðujÞ

� �dt, j ¼ 1, 2

ð1:2Þ

where fj, kj are Lipschitz continuous, convex functions of L2ð0,T;L2

ðGÞÞ�

L2ð0,T;L2

ð�ÞÞ into R2þ and G are bounded subsets of �:

The problem is to find fu1, u2g in U ¼ U1 � U2 such that the solution y ofEq. (1.1) is close to the observed function h in the subregions G with

f2ð yÞ ¼ k yð :, tÞ � hð :, tÞkL2ðGÞ:

As we have a multicontrol optimal problem, open and closed loops ofEqs. (1.1)–(1.2) are two different notions. A control ~uu 2 U is said to be anopen loop of Eqs. (1.1)–(1.2) if

Jjð�; ~yy;!; ~uu1, ~uu2Þ � Jjð�; y;!; vj; ~uukÞ, 8 vj 2 U j, k 6¼ j, j ¼ 1, 2 ð1:3Þ

where y is the unique solution of Eq. (1.1) with control vj, ~uuk, k 6¼ j:A characteristic feature of Eqs. (1.1)–(1.2) is that the control u1 is the

coefficient of the top order derivative of the equation. In the literature, feed-back laws have been obtained for linearly distributed system by Popa[3,4] andby the writer.[7] In Ref. [8], we have established the closed loop feedback lawsfor Eqs. (1.1)–(1.2) when kj ¼ 0, j ¼ 1, 2: Feedback laws for open loop pro-blems are almost non existent in the literature. One of the main difficulty isthat the set of open loops, Uopen, depends on the initial data. Since the valuefunction Vjð�;!Þ of Eqs. (1.1)–(1.3) is taken over the set Uopen, the usualargument establishing the Lipschitz continuity of Vj is no longer availableand difficulties arise as one needs the Clarke subgradients in the feedbacklaws. In this paper, we shall circumvent the difficulty and establish the feed-back laws for Eqs. (1.1)–(1.3).

The problem of identifying the source in the wave equation has beenstudied by Puel and Yamamoto,[5,6] by Yamamoto,[9] using exact controll-ability and the Hilbert uniqueness method. An inverse problem for the waveequation with controls on the moving boundary was investigated by Lenhart,Protopopescu and Yong.[1] By making a change of variables as in Ref. [1],one can study the optimal shape and the reflection coefficient of the waveequation, using the approach of this paper.

In Section 2, we shall prove some preliminary results. The existence ofan open loop and the value function are studied in Section 3, a nonlinearproblem is considered in Section 4 and feedback laws are established inSection 5.

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2. NOTATIONS, PRELIMINARY RESULTS

Throughout the paper, we shall denote by H ¼ L2ð�Þ, the Hilbert space

with the inner product ð : , :Þ and the norm k : k : The usual Sobolev spacesH1

0 ð�Þ,Hkð�Þ are written as H1

0 , Hk respectively.Let X1 be the Hilbert space

kxkX1¼ kxkL2ð0,T;H3Þ þ kx0

kL2ð0,T;H1Þ þ kx00kL2ð0,T;HÞ ð2:1Þ

with

U1 ¼ fu1 : 1 � u1ð�, tÞ, ku1kX1� c1g

Denote by X2 the Hilbert space

kxkX2¼ kxkL2ð0,T;H1Þ þ kx0

kL2ð0,T;HÞ þ kx00kL2ð0,T;HÞ ð2:2Þ

with

U2 ¼ fu2ð�, tÞ : ku2kX2� c2g:

It is clear that U j are bounded convex subsets of L2ð0,T;HÞ: From the

Sobolev imbedding theorem and from Aubin theorem, we deduce thatU ¼ U1 � U2 is a compact subset of L2

ð0,T;H2Þ � L2

ð0,T;HÞ:

Assumption 2.1.We assume that there exists continuous 1-1 functions gj of U j

into R, j ¼ 1:2:

Throughout the paper, Mj is the natural injection mapping of Xj intoL2

ð0,T;HÞ and its adjoint M�j is the projection of L2

ð0,T;HÞ onto Xj:For the initial boundary-value problem (1.1), the following result is

known.

Theorem 2.1. Let fu,!g be in U � ðH10 � HÞ, then there exists a unique

solution y of Eq. (1.1) with

ky0k2L2ð0,T;HÞ þ kyk2L2ð0,T;H10Þ � C 1þk!1k

2H1

0þk!2k

2H þku1k

2X1

þku2k2X2

n o:

If ! is in fH10 \ H2

g � H1, then the solution y is in L1ð0,T;H1

0 Þ \

L2ð0,T;HÞ and

ky00k2L2ð0,T;HÞ þky0k2L1ð0,T;H1Þ �C 1þk!1k2H1

0\H2 þk!2k

2H1

0þku02k

2L2ð0,T;HÞ

n

þku1k2L1ð0,T;H2Þ

þku01k2L1ð0,T;H1Þ

o:

The generic constant C is independent of u,!:

OPEN LOOP FEEDBACK LAWS 403

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The proof is almost the same as the one given in Lions’ book.[2]

Let kj be Lipschitz continuous, convex functions from L2ð0,T;HÞ into

Rþ and let k�j , the conjugate of kj, be given by

k�j ðvjÞ ¼ sup ðwj , vjÞL2ð0,T;HÞ � kjðwjÞ : 8wj 2 U j

n o, j ¼ 1, 2: ð2:3Þ

Taking a maximizing sequence, one can show easily that

k�j ðvjÞ ¼ ðw�

j , vjÞL2ð0,T;HÞ � kjðw�j Þ, j ¼ 1, 2

for some w�j 2 U j: It is clear that k�

j is convex and l.s.c. on L2ð0,T;HÞ and

thus, the subgradient @k�j is defined.

Lemma 2.1. Let k�j be as in Eq. (2.3), then @k�

j is continuous fromðL2

ð0,T;HÞÞweak into ðL2ð0,T;HÞÞ:

Proof. We have

k�j ðvjÞ ¼ ðw�

j , vjÞL2ð0,T;HÞ � kjðw�j Þ:

Thus,

k�j ðujÞ�k�

j ðvjÞ � ð�,ujÞL2ð0,T;HÞ � ðw�j ,vjÞL2ð0,T;HÞ 8� 2 U j,8uj 2L2

ð0,T;HÞ

� ðw�j ,uj � vjÞL2ð0,T;HÞ8uj 2L2

ð0,T;HÞ:

Hence w�j 2 @k�

j ðvjÞ. Suppose that vnj ! vj in ðL2

ð0,T;HÞÞweak and let

k�j ðv

nj Þ ¼ ðwn

j , vnj ÞL2ð0,T;HÞ � kjðw

nj Þ:

Since wnj 2 U j, it follows from the definition of U j that there exists a

subsequence such that wnkj ! wj in L2

ð0,T;HÞ. Hence

k�j ðv

nkj Þ ! ðwj, vjÞL2ð0,T;HÞ � kjðwjÞ:

On the other hand

k�j ðvjÞ � lim inf k�

j ðvnj Þ

� ðwj, vjÞL2ð0,T;HÞ � kjðwjÞ:

It follows from the definition of k�j that

k�j ðvjÞ ¼ ðwj, vjÞL2ð0,T;HÞ � kjðwjÞ,

i.e. wj 2 @k�j ðvjÞ and the lemma is proved. œ

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3. OPEN LOOP AND THE VALUE FUNCTION

Consider the initial boundary-value problem

y0 � �r:fu1ryg ¼ �v2 in �� ð�,TÞ, � 2 ð0, 1�

y ¼ 0 on @�� ð�,TÞ ð3:1Þ

yð :, �Þ, y0ð :, �Þ� �

¼ f!1,!2g ¼ ! in �

From Theorem 2.1, there exists a unique solution y of Eq. (3.1) and

V1ð�;!; v2Þ ¼ inffJ1ð�; y;!; u1; v2Þ : 8 u1 2 U1g ð3:2Þ

with J1 as in Eq. (1.2), is well-defined. Similarly we define

V2ð�;!; v1Þ ¼ inffJ2ð�; x;!; u2; v1Þ : 8 u2 2 U2g: ð3:3Þ

The main results of the section are the following theorems.

Theorem 3.1. Let ! be in H10 � H and suppose that Assumption 2.1 is satisfied.

Then the set, Uopen, of open loops of Eqs. (1.1)–(1.3) is non-empty. LetVjð�;!; vkÞ, k 6¼ j be as in Eqs. (3.2)–(3.3). Then

Vjð�;!; ~uukÞ ¼ Jjð�; ~yy; ~uuj; ~uukÞ, k 6¼ j, j ¼ 1, 2

with ~uu 2 Uopenð!Þ and ~yy is the solution of Eq. (1.1) with control ~uu, initial value!: Furthermore for u 2 U

jVjð�;!; ukÞ � Vjð�; �; ukÞj � Cfk!1 � �1kH10þ k!2 � �2kg

8!, � 2 H10 � H, k 6¼ j:

The constant C is independent of !, �, u.

With Theorem 3.1, we are led to the definition of the value function forEqs. (1.1)–(1.3). Let

Vjð�;!Þ ¼ inffVjð�;!; ~uukÞ : 8 ~uu 2 Uopenð!Þ, k 6¼ jg, j ¼ 1, 2: ð3:4Þ

We now have the following theorem.

Theorem 3.2. Let ! be in H10 � H and let Vjð�;!Þ be as in Eq. (3.4). Then the

Clarke subgradients

. @1Vjð�; : ,!2Þ exists and is in L2ð0,T;H�1

Þ.. @2Vjð� ;!1; : Þ exists and is in L2ð0,T ;HÞ:


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Moreover

q1j��

L2ð0,T;H�1Þþ q2j��

L2ð0,T;HÞ� C,

8 q1j , q2j

� �2 @1Vjð�; :;!2Þ � @2Vjð�;!1; : Þ:

The constant C is independent of �, !: Furthermore f@1Vjð�;!Þ,@2Vjð�;!Þg is continuous from fL2

ð0,T;H10 � HÞg

2 to ðL2ð0,T;H�1

ÞÞweak�

ðL2ð0,T;HÞÞweak:

Let f!, �, vg 2 ðH10 � HÞ

2� U and let Vjð�;!; vkÞ be as in Eqs. (3.2)–(3.3).

Lemma 3.1. Suppose all the hypotheses of Theorem 3.1 are satisfied. Then

jVjð�;!; vkÞ � Vjð�; �; vkÞj � Cfk!1 � �1kH10þ k!2 � �2kg

8!, � 2 H10 � H, 8 v 2 U:

The constant C is independent of �.

Proof. (1) Let V1ð�;!; v2Þ be as in Eq. (3.2). Then there exists a minimizingsequence f y1n, u

n1g such that

J1ð�; yn1;!; v2Þ �

1

n� V1ð�;!; v2Þ:

We denoted by yn1 the unique solution of

ð yn1Þ

00� �r: un

1ryn1

� �¼ �v2 in �� ð0,TÞ

yn1 ¼ 0 on @�� ð0,TÞ

yn1ð :, 0Þ , yn

1ð Þ0ð :, 0Þ

� �¼ f!1 ,!2 g:

From the estimates of Theorem 2.1 and from our hypotheses on U1, weget by taking subsequences

f yn1, ð y

n1Þ

0, un1g ! f yy1, ð yyÞ

01, uu1g

in

L2ð0,T;HÞ \ ðL1

ð0,T;H10 ÞÞweak�

� ðL1ð0,T;HÞÞweak� � ðL2

ð0,T;H2Þ \ ðL2

ð0,T;H3ÞÞweak:

It is trivial to check that f yy1, uu1g is the solution of Eq. (1.1) with controlsuu1, v2 and we have

J1ð�; yn1;!; u

n1; v2Þ ! J1ð�; yy1;!; uu1, v2Þ:

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Hence

V1ð�;!; v2Þ ¼ J1ð�; yy1;!; uu1, v2Þ

for some uu1 2 U1: Similarly

V2ð�;!; v1Þ ¼ J2ð�; yy2;!; v1; uu2Þ

for some uu2 2 U2 and yy2 is the solution of Eq. (1.1) with controls v1, uu2:(2) Let !, � be in H1

0 � H, then we have

V1ð�; �; v2Þ � V1ð�;!; v2Þ ¼ V1ð�; �; v2Þ � J1ð�; yy1;!; uu1, v2Þ

� J1ð�; yy1; �; uu1; v2Þ � J1ð�; y;!; uu1; v2Þ

� C

Z T

�

j f1ð yÞ � f1ð yy1Þj dt

� Ckyy1 � ykL2ð0,T;HÞ þ kyy01 � y0kL2ð0,T;HÞ:

y is the solution of Eq. (1.1) with controls uu1, v2 and initial value � ¼ ð�1, �2Þ:From Theorem 2.1 we obtain

k yy1 � ykL2ð0,T;HÞ þ kyy01 � y0kL2ð0,T;HÞ � Cfk!1 � �1kH10þ k!2 � �2kg:

Thus,

V1ð�; �; v2Þ � V1ð�;!; v2Þ � Cfk!1 � �1kH10þ k!2 � �2kg:

Reversing the role played by !, � and we get

V1ð�;!; v2Þ � V1ð�; �; v2Þ � Cfk!1 � �1kH10þ k!2 � �2kg

Hence

jV1ð�;!; v2Þ � V1ð�; �; v2Þj � Cfk!1 � �1kH10þ k!2 � �2kg

The constant C is independent of �, v2, !, �.An identical argument gives the stated result for V2: œ

Lemma 3.2. Suppose all the hypotheses of Theorem 3.1 are satisfied. Then theset, Uopen, of open loop controls of Eqs. (1.1)–(1.3), is non-empty. Moreover

Vjð�;!; ~uukÞ ¼ Jjð�; ~yy;!; ~uuj; ~uukÞ, k 6¼ j, j ¼ 1, 2

for ~uu 2 Uopenð!Þ and ~yy is the solution of Eq. (1.1) with control ~uu, initial-value !:

Proof. From Lemma 3.1, we know that for a given v2 2 U2, there existsuu1 2 U1 such that

V1ð�;!; v2Þ ¼ J1ð�; yy1;!; uu1; v2Þ: ð3:5Þ


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Let

K1 ¼ uu1 : V1ð�;!; v2Þ ¼ J1ð�; yy1;!; uu1; v2Þ� �

and let g1 be as in Assumption 2.1, a 1-1 continuous function on U1. Then

inffg1ðu1Þ : u1 2 K1g

exists. With a minimizing sequence we can show that

g1ðu�1Þ ¼ inffg1ðu1Þ : u1 2 K1g:

Since by hypothesis g1 is 1-1, the minimum is unique.In a similar fashion, for a given v12 U1, there exists a unique u�2 with

g2ðu�2Þ ¼ inffg2ðu2Þ : 8 u22 K2g

and K2 is the set of uu2 for which

V2ð�;!; v1Þ ¼ J2ð�; yy2; v1; uu2Þ:

(2) We define the mapping S of U into U by the equation

SðvÞ ¼ uu ¼ fuu1, uu2g:

The mapping S is well-defined and we now show that it has a fixedpoint. Since U is a compact convex subset of L2

ð0,T;HÞ, to apply theSchauder fixed point theorem it suffices to check the continuity of themapping.

Suppose that vn! v in L2

ð0,T;HÞ with vn2 U and let Sðvn

Þ ¼ uun:From Theorem 2.1, we have

yynj

�� L2ð0,T;H1Þ

þ ðyynj Þ

0��

L2ð0,T;HÞ� C:

With uunj 2 U j, we get by taking subsequences

yynj , ð yy

jnÞ

0, uunj

� �! yyj, yy

0j, ~uuj

� �in

L2ð0,T;HÞ \ ðL1

ð0,T;H10 Þweak�

� ðL2ð0,T;HÞÞweak � L2

ð0,T;H2Þ \ ðL1

ð0,T;H2ÞÞweak� :

It is trivial to check that

yy001 � �r: ~uu1ryy1� �

¼ �v2 in �� ð0,TÞ

yy1 ¼ 0 on @�� ð0,TÞ

yy1ð :, 0Þ , yy01ð :, 0Þ

� �¼ !:

Similarly for yy2:

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(3) Since\n

vn2 : kv

n2 � v2kL2ð0,T;HÞ � "

n o� fv2g,

we have

J1 �;xn1;!; uu

n1; v2ð Þ � J1 �; yyn

1;!; uun1; v

n2ð Þ

� J1 �; xxn1;!; u1; v

n2ð Þ, 8 u1 2 U1

Thus,

J1ð�;x1;!; ~uu1; v2Þ � J1ð�; xx1;!; u1; v2Þ, 8 u1 2 U1

i.e.

V1ð�;!; v2Þ ¼ J1ð�;x1;!; ~uu1; v2Þ:

Since the Eq. (1.1) has a unique solution for each given control f ~uu1, v2gand a given inital data !, we get x1 ¼ yy1 and

V1ð�;!; v2Þ ¼ J1ð�; yy1; ~uu1; v2Þ:

Therefore ~uu1 2 K1:A similar argument holds for ~uu2 and thus, SðvÞ ¼ ~uu:(4) The Schauder fixed point theorem yields the existence of ~uu 2 U

such that Sð ~uuÞ ¼ ~uu: Therefore

Vjð�;!; ~uukÞ ¼ Jjð�; ~yy;!; ~uuj; ~uukÞ

� Jjð�; yj;!; uj; ~uukÞ, 8 uj 2 U j, k 6¼ j

where yj is the solution of Eq. (1.1) with controls fuj, ~uukg and initial value !:It follows from the definition of open loops that ~uu is an open loop of

Eqs. (1.1)–(1.3). œ

Proof of Theorem 3.1. Let u 2 U, then from Lemma 3.1 we get

jV1ð�;!; u2Þ � jV1ð�; �; u2Þj � Cfk!1 � �1kH10þ k!2 � �2k,

8!, � 2 H10 � H: ð3:6Þ

The constant C is independent of u, !, �: In particular, we have

V1ð�;!; u2Þ � V1ð�; ~��; u2Þ�� Ck!1 � �1kH1

08 ~�� ¼ ð�1,!2Þ:

Thus, the Clarke subgradient of V1 with respect to !1 exists and we shallwrite it as @1V1ð�; :;!2; ~uu2Þ. Moreover

q11ð :,!2; ~uu2Þ��

L2ð0,T;H�1Þ� C , 8 q11 2 @1V1ð�; :,!2; ~uu2Þ:


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With � ¼ ð!1, �2Þ in Eq. (3.6), we have the Clarke subgradient of V1 withrespect to !2 and we shall write it as @2V1ð�;!1; :; u2Þ: Furthermore

q21ð :,!1; u2Þ��

L2ð0,T;HÞ� C 8 q21 2 @2V1ð�;!; u2Þ:

The constant C is independent of u,!, �:A similar argument holds for @1V2ð�; :,!2; u1Þ, @2V2ð�;!1; :, ; u1Þ:The theorem is proved.

Proof of Theorem 3.2. (1) From Theorem 3.1, we know that the set, Uopen,of open loop controls ~uu of Eqs. (1.1)–(1.3) is non-empty. Moreover

Vjð�;!; ~uukÞ ¼ Jjð�; ~yy;! ; ~uuj ; ~uukÞ

� Jjð�; yj ;! ; uj; ~uukÞ, 8 uj 2 U j, k 6¼ j: ð3:7Þ

Let

Vjð�;!Þ ¼ inffVjð�;!; ~uukÞ : 8 ~uu 2 Uopenð!Þ, k 6¼ jg, j ¼ 1, 2 ð3:8Þ

be the value function of Eqs. (1.1)–(1.3).(2) Let f ~uun

kg be a minimizing sequence of the optimization problem (3.8)with

Vjð�;!Þ � Vjð� ;!; ~uunkÞ �

1

n

¼ Jjð�; ~yyn;!; ~uun

j , ~uunkÞ �

1

nð3:9Þ

where ~yyn is the solution of Eq. (1.1) with control ~uun and initial value !:With U as in Section 2 and with the estimates of Theorem 2.1, we get

by taking subsequences

f ~yyn, ð ~yynÞ0, ~uun

j g ! f yy, yy0, uujg

in

L2ð0,T;HÞ \ L2 0,T;H1

0

weak

� ðL2ð0,T;HÞÞweak � L2

ð0,T;HÞ \ ðL1ð0,T;H1

ÞÞweak� :

It follows from Eq. (3.9) that

Jj �; yy;!; uuj; uukÞ � Vjð�;!

ð3:10Þ

From the definition of open loop, we have

Jj �; ~yyn;!; ~uun

j ; ~uunk

� Jj �;x

nj ;!; uj; ~uu

nk

, 8 uj 2 U j:

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An argument as above shows that

Jj �; yy; uuj; uuk

� Jjð�; xj;!; uj; uukÞ, 8 uj 2 U j, j 6¼ k:

xj is the solution of Eq. (1.1) with control fuj, uukÞ and initial value !:Therefore uu is an open loop control of Eqs. (1.1)–(1.3). Thus, Eq. (3.10)implies that

Vjð�;!Þ ¼ Vjð�;!; uukÞ, uu 2 Uopenð!Þ k 6¼ j, j ¼ 1, 2: ð3:11Þ

(2) Let !, � be in H10 � H and consider

Vjð�;!Þ � Vjð�; �Þ ¼ Vjð�;!; uukÞ � Vjð�; �Þ, k 6¼ j ð3:12Þ

with uu 2 Uopenð!Þ, as in Eq. (3.11).Since uu is in Uopenð!Þ, we have two cases: (i) uu =2 Uopenð�Þ or (ii)

uu 2 Uopenð�Þ:. Suppose that uu =2 Uopenð�Þ: Since

Vjð�; �Þ ¼ inffVjð�; �; vkÞ : 8 v 2 Uopenð�Þg,

we have two cases. In the first case

Vjð�; �; uukÞ � Vjð�; �Þ: ð3:13Þ

It follows from Eqs. (3.12)–(3.13) and from Lemma 3.1 that

Vjð�;!Þ � Vjð�; �Þ � Vjð�;! ; uukÞ � Vjð�; � ; uukÞ

� Cfk!1 � �1kH10þ k!2 � �2kHg ð3:14Þ

Reversing the role played by !, � and we get

Vjð�; �Þ � Vjð�;!Þ � Cfk!1 � �1kH10þ k!2 � �2kg

8!, � 2 H10 � H: ð3:15Þ

Hence

jVjð�;!Þ � Vjð�; �Þj � Cfk!1 � �1kH10þ k!2 � �2kg

8!, � 2 H10 � H ð3:16Þ

The constant C is independent of !, �, �:. Suppose that uu =2 Uopenð�Þ and that

Vjð�; �; uukÞ � Vjð�; �Þ:


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Then

Vjð�; �Þ � Vjð�;!Þ ¼ Vjð�; �Þ � Vjð�;!; uukÞ

� Vjð�; �; uukÞ � Vjð�;!; uukÞ

� Cfk!1 � �1kH10þ k!2 � �2kg

Reversing the role played by !, � and combining with the above esti-mate, we obtain

jVjð�; �Þ � Vjð�;!Þj � Cfk!1 � �1kH10þ k!2 � �2kg:

. Suppose that uu 2 Uopenð!Þ \ Uopenð�Þ, then

Vjð�; �; uukÞ � Vjð�;!; uukÞ � Cfk!1 � �1kH10þ k!2 � �2kg:

Thus,

Vjð�; �Þ � Vjð�;!Þ � Cfk�1 � !1kH10þ k!2 � �2kg

Reversing the role played by !, � and we get Eq. (3.16).It follows from Eq. (3.16) that the Clarke subgradients @1Vjð�; :, ;!2Þ,

@2Vjð�;!1; :Þ of Vj exist. Moreover

q1j��

L2ð0,T;H�1Þþ q2j��

L2ð0,T;HÞ� C, 8 q1j , q

2j

� �2 @1Vjð�;!Þ � @2Vjð�;!Þ:

(3) It remains to show that f@1Vj, @2Vjg are continuous from H10 � H to

ðL2ð0,T;H�1

ÞÞweak � ðL2ð0,T;HÞÞweak: Suppose that !n

! ! in H10 � H:

Then

q1j ð!nÞ

�� L2ð0,T;H�1Þ

þ q2j ð!nÞ

�� L2ð0,T;HÞ

� C

for all fq1j ð!nÞ, q2j ð!

nÞg 2 @1Vjð�;!

nÞ � @2Vjð�;!

nÞ:

From the weak compactness of the unit ball in a Hilbert space, we getby taking subsequences

q1j ð!nÞ, q2j ð!

nÞ

� �! ~qq1j , ~qq

2j

� �in ðL2

ð0,T;H�1ÞÞweak � ðL2

ð0,T;HÞÞweak:

On the other hand, the definition of subgradients givesZ T

0

fVjð�;�1;!n2Þ�Vjð�;!

n1;!

n2Þgd� �

Z T

0

< q1j ð!nÞ,�1�!n

1 > dt ð3:17Þ

for all �1 2 H10 : The pairing between H1

0 and its dual H�1 is denoted by< :, : > :

It follows from Eqs. (3.16)–(3.17) thatZ T

0

fVjð�; �1;!2Þ � Vjð�;!1;!2Þg dt �

Z T

0

< q1j , �1 � !1 > dt

for all �1,!1 in H10 :

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Hence q1j 2 @1Vjð�;!1;!2Þ: A similar argument for q2j :The theorem is proved.

4. A NONLINEAR PROBLEM

Let Vj be as in Theorem 3.2 and consider the initial boundary-valueproblem

y00 � r:f ~uu1ryg ¼ ~uu2 in �� ð0,TÞ

y ¼ 0 on @�� ð0,TÞ

yð :, 0Þ , y0ð :, 0Þ� �

¼ ! in� ð4:1Þ

~uuj 2 @k�j M�

j @2Vjðt; yðtÞ; y0ðtÞÞ

� � , j ¼ 1, 2

k�j ðM

�j f@2Vjðt; ~yyðtÞ, ~yy

0ðtÞÞgÞ ¼ ðM�

j f@2Vjðt; ~yyðtÞ, ~yy0ðtÞÞg , ~uujÞL2ð0,T;HÞ � kjð ~uujÞ

Mj is as in Section 2 and k�j is the conjugate of kj:

Theorem 4.1. Suppose all the hypotheses of Theorem 3.1 are satisfied. Supposefurther that ! 2 fH1

0 \ H2g � H1

0 , Let Vj be as in Theorem 3.2 and let Mj, kj

be as in Section 2, then there exists a solution ~yy of Eq. (4.1) with

f ~yy, ~yy0 ~yy00g 2 L2ð0,T;H1

0 \ H2Þ � L2

ð0,T;H10 Þ � L2

ð0,T;HÞ:

~uuj is the unique element of the closed convex set @k�j ðM

�j �

f@2Vjðt; ~yyðtÞ, ~yy0ðtÞÞgÞ with minimal L2

ð0,T;HÞ-norm.Let

BC ¼

�x : kxkL2ð0,T;H2\H1

0Þ þ kx0

kL2ð0,T;H10Þ þ kx00

kL2ð0,T;HÞ � Cf1þk!1k2H2 :

þk!2kH1g,xð :, 0Þ ¼ !1,x0ð :, 0Þ ¼ !2

�:

With x 2 BC, the expressions @2Vjðt;xðtÞ,x0ðtÞÞ are defined by Theorem

3.2. Consider the initial boundary-value problem

y00 � �r:fu1ryg ¼ �u2 in �� ð0,TÞ

y ¼ 0 on @�� ð0,TÞ

f yð :, 0Þ , y0ð :, 0Þg ¼ ! ¼ ð!1 ,!2Þ ð4:2Þ

ujðtÞ 2 @k�j ðM

�j f@2Vjðt;xðtÞ, x

0ðtÞÞ, j ¼ 1, 2

k�j ðM

�j f@2Vjð :; x; x

0ÞgÞ ¼ ðMjuj, @2Vjð :; x, x

0Þ ÞL2ð0,T;HÞ � kjðujÞ


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Since @2Vjðt;xðtÞ;x0ðtÞÞ is a closed convex subset of L2

ð0,T;HÞ, beingthe image of a subgradient, there exists a unique element pjðxÞ of @2Vjðt; x; x

0Þ

with minimal L2ð0,T;HÞ-norm. With @k�

j ðM�j pjðxÞÞ being the image of a

subgradient, there exists a unique element u�j of minimal L2ð0,T;HÞ-norm.

Consider the problem

y00 � �r:fu�1ryg ¼ u�2 in �� ð0,TÞ

y ¼ 0 on @�� ð0,TÞ ð4:3Þ

f yð :, 0Þ , y0ð :, 0Þg ¼ ! in �

There exists a unique solution y of Eq. (4.2) and we define a nonlinearmapping S of BC into BC by

Sx ¼ y

It follows from Theorem 2.1 and the estimate of the theorem that for Csufficiently large, y is in BC:

Proof of Theorem 4.1. To prove the theorem, it suffices to show that S has afixed point in BC: Since BC is a compact convex subset of L2

ð0,T;HÞ, inorder to apply Schauder’s fixed point theorem we need only to show thecontinuity of S:

(1) Suppose that xn 2 BC and that xn ! x in L2ð0,T;HÞ: Set

yn ¼ Sxn, then from the estimates of Theorem 2.1 we obtain, by takingsubsequences

xn, yn, x0n, y

0n,x

00n, y

00n, p

nj ðxnÞ, u

�j, n

� �! x, y, x0, y0, x00, y00, pj, u

�j

� �

in

fL2ð0,T;H1

Þ \ ðL1ð0,T;H2

ÞÞweak�g2

� fL2ð0,T;HÞ \ ðL1

ð0,T;H1ÞÞweak�g

2

� fðL2ð0,T;HÞÞweakg

2� ðL2

ð0,T;HÞÞweak

� L2ð0,T;H2

Þ \ ðL2ð0,T;H1

ÞÞweak:

It is clear that fx, yg 2 BC � BC:(2) Let pn

j ðxnÞ be the minimal L2ð0,T;HÞ-norm element of

@2Vjðt;xnðtÞ; x0nðtÞÞ: From the definition of subgradients, we have

Z T

0

fVjðt;xnðtÞ;x0�ðtÞÞ�Vjðt;xnðtÞ;x

0nðtÞÞgdt�

Z T

0

pnj ðxnÞ,x

0�ðtÞ�x0

nðtÞ

dt

for all x� 2 L2ð0,T;HÞ:

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The Lipschitz continuity in H1� H of Vj given by Theorem 3.2

implies that

Z T

0

Vjðt; xðtÞ; x0�ðtÞÞ � Vjðt; xðtÞ; x

0ðtÞÞ

� �dt �

Z T

0

pj, x0�ðtÞ � x0

ðtÞ

dt

for all x0� 2 L2

ð0,T;HÞ: Hence pj 2 @2Vjðt; xðtÞ; x0ðtÞÞ:

We now show that pj is the element of @2Vjðt;x, x0Þ with minimal

L2ð0,T;HÞ-norm. Let

B"ðxÞ ¼nx" : x"

2 BC, kx � x"kL2ð0,T;HÞ

þ kðx � x"Þ0kL2ð0,T;H1Þ þ kðx � x"

Þ00kL2ð0,T;HÞ � "

o:

Then\"

@2Vjðt; x"ðtÞ; ðx"

Þ0ðtÞÞ : x"

2 B"ðxÞ� �

� @2Vjðt; xðtÞ; x0ðtÞÞ

since xn 2 B" for n � n0:Hence

pnj ðxnÞ

�� L2ð0,T;HÞ

� kq"j kL2ð0,T;HÞ 8 q"j 2 @2Vjðt; x"; ðx"

Þ0Þ, x"

2 B"ðxÞ,8 ":

In particular, we have

pnj ðxnÞ

�� L2ð0,T;HÞ

� kqjkL2ð0,T;HÞ 8 qj 2 @2Vjðt;x;x0Þ:

Therefore

kpjkL2ð0,T;HÞ � kqjkL2ð0,T;HÞ 8 qj 2 @2Vjðt; x; x0Þ:

(3) By definition, we have

k�j ðM

�j p

nj Þ ¼ ðM�

j pnj , u

nj ÞL2ð0,T;HÞ � kjðu

nj Þ:

Thus,

k�j ðM

�j pjÞ ¼ ðM�

j pj, ujÞL2ð0,T;HÞ � kjðujÞ

and uj 2 @k�j ðM

�j pjÞ:

It is now easy to check that uj is the element of minimal L2ð0,T;HÞ-

norm of @k�j : We shall not reproduce it. Therefore SðxÞ ¼ y:

Applying the Schauder fixed point theorem and we deduce the existenceof ~yy with Sð ~yyÞ ¼ ~yy:

From the definition of S, we deduce that ~uuj is the unique element of theclosed convex set @k�

j ðM�j f@2Vjðt; ~yyðtÞ, ~yy

0ðtÞÞgÞ:

The theorem is proved.


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5. FEEDBACK LAWS

We shall now establish the feedback laws in the case when k1 ¼ k2 ¼ 0 :The general case is open.

Theorem 5.1. Suppose all the hypotheses of Theorem 4.1 are satisfied andsuppose further that k1 ¼ k2 ¼ 0: Let ~yy be as in Theorem 4.1. Then ~uu ofTheorem 4.1 is an open loop control of Eqs. (1.1)–(1.3) and

Vjð!Þ ¼

Z T

0

fjð ~yy, ~yy0Þ dt ¼ J1ð ~yy;!; ~uu1 , ~uu2Þ, j ¼ 1, 2:

Vj, ~uuj are as in Theorem 4.1 and ~yy is the solution of Eq. (1.1) with controls ~uu:

The formula is the same as in the case of linearly distributed systems butwith different meanings for the value functions.

Consider the initial boundary-value problem

y00 � �r:fv1ryg ¼ �v2 in �� ðt,TÞ

y ¼ 0 on @�� ðt,TÞ ð5:1Þ

f yð :, tÞ , y0ð :, tÞg ¼ f ~yyð :, tÞ, ~yy0ð :, tÞg in �

with ~yy as in Theorem 4.1. Let Vjðt; ~yyðtÞ, ~yy0Þ be the value functions associated

with Eq. (5.1), defined by Theorems 3.1–3.2. From Theorem 3.1, we knowthat Eqs. (5.1)–(1.2)–(1.3) has an open loop control uu 2 Uopenð ~yyÞ and that

Vjðt; ~yyðtÞ , ~yy0ðtÞÞ ¼ Vjðt; ~yyðtÞ , ~yy

0ðtÞ; uukÞ

¼ Jjðt; yy ; ~yyðtÞ, ~yy0ðtÞ; uu1 , uu2Þ, k 6¼ j, j ¼ 1, 2: ð5:2Þ

where yy is the solution of Eq. (5.1) with control uu and initial value f ~yy, ~yy0g:

Proof of Theorem 5.1. From Eq. (5.2), we have

Vjðt; ~yyðtÞ, ~yy0ðtÞÞ ¼ Jjðt; yy; ~yyðtÞ, ~yy

0ðtÞ; uu1, uu2Þ:

Hence

d

dtVjðt; ~yyðtÞ , ~yy

0ðtÞÞ ¼ �fjð yyðtÞ, yy

0ðtÞÞ ¼ �fjð ~yyðtÞ, ~yy

0ðtÞÞ: ð5:3Þ

Integrating with respect to t from 0 to T and we obtain

VjðT; ~yyðTÞ; ~yy0ðTÞÞ � Vjð0; ~yyð0Þ; ~yy0ð0ÞÞ ¼ �

Z T

0

fjð ~yyðsÞ, ~yy0ðsÞÞ ds:

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Therefore

Vjð!Þ ¼

Z T

0

fjð ~yy, ~yy0Þ dt ¼ Jjð ~yy;!; ~uu1; ~uu2Þ, j ¼ 1, 2: ð5:4Þ

The theorem is proved.

REFERENCES

1. Lenhart, S.; Protopopescu, V.; Yong, J. Identification of Boundary Shape andReflectivity in a Wave Equation by Optimal Control Techniques. Differentialand Integral Equations 2000, 13, 941–972.

2. Lions, J.L. Quelques Methodes de Resolution des Problemes aux Limites NonLineaires; Dunod: Paris, 1969.

3. Popa, C. Feedback Laws for Nonlinear Distributed Control Problems viaTrotter-Type Product Formulae. SIAM J. Control Optimi. 1995, 33, 971–999.

4. Popa, C. The Relationship Between the Maximum Principle and DynamicProgramming for the Control of Parabolic Variational Inequalities. SIAM J.Control Optimi. 1997, 35, 1711–1738.

5. Puel, J.; Yamamoto, M. Applications of Exact Controllability to Some InverseHyperbolic Problems; C.R. Acad. Sci.: Paris, Series I, 320, 1171–1176.

6. Puel, J.; Yamamoto, M. On a Global Estimate in a Linear Inverse HyperbolicProblem. Inverse Problems 1996, 12, 995–1002.

7. Bui An Ton. Closed Loop Control for Evolution Inclusions. NumericalFunctional Analysis and Optimization 1999, 20, 983–1001.

8. Bui An Ton. An Inverse Problem for the Wave Equation: Feedback Laws.Submitted for publication.

9. Yamamoto, M. Stability, Reconstruction Formula, and Regularization for anInverse Source Hyperbolic Problem by a Control Method. Inverse Problems1995, 11, 481–496.


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open loop feedback laws for the wave equation

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