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EXACT CONTROLLABILITY FOR A SEMILINEAR WAVE EQUATION WITH BOTH INTERIOR AND BOUNDARY CONTROLS

BUI AN TON Received 13 May 2004

The exact controllability of a semilinear wave equation in a bounded open domain ofRn, with controls on a part of the boundary and in the interior, is shown. Feedback laws are established.

1. Introduction

The purpose of this paper is to prove the existence of the exact controllability of a semi- linear wave equation with both interior and boundary controls.

LetΩbe a bounded open subset ofRnwith a smooth boundary, let f(y) be an accre- tive mapping ofL2(0,T;H1(Ω)) intoL2(0,T;H01(Ω)) with respect to a duality mapping J,D(f)=L2(0,T;L2(Ω)) and having at most a linear growth in y. Consider the initial boundary value problem

yy+f(y)=ω inΩ×(0,T),

y(x,t)=0 onΓ0×(0,T), y(x,t)=v(u) onΓ1×(0,T), y(x, 0)=α0, y(x, 0)=α1 inΩ,

(1.1)

with

Γ0

Γ1=Ω, Γ0

Γ1= ∅, Γ1= ∅. (1.2)

The characteristic function of the subsetωofΩisχωand the control functionuis in a closed, bounded, convex subsetᐁofL2(0,T;L2(Ω)). GivenT > T0and

{α0,α1};{β0,β1} inL2(Ω)×H1(Ω), (1.3) the aim of the paper is to prove the existence of an optimal{u,v(u) }∈×L2(0,T;L21)) such that the solutionyof (1.1) satisfies

y(x,T)=β0, y(x,T)=β1 inΩ. (1.4)

Copyright©2005 Hindawi Publishing Corporation Abstract and Applied Analysis 2005:6 (2005) 619–637 DOI:10.1155/AAA.2005.619

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The exact boundary controllability of the wave equation, using the Hilbert uniqueness method of Lions [3,4], has been extensively investigated, both theoretically and numeri- cally. For the semilinear wave equation, the local controllability was established by Russell [5] and others, using the implicit function theorem. More recently, Zuazua in [8,9] in- troduced a variant of the Hilbert uniqueness method and treated the exact boundary controllability of the semilinear wave equation

(i) in the spaceL2(Ω)×H1(Ω) for asymptotically linear mappings f inWloc1,(R), (ii) in the spaceγ>0H0γ(Ω)×Hγ1(Ω) for mappings f with finL(R). The pair

{Γ0,T}is assumed to have the unique continuation property for the wave equa- tion with zero potential.

In order to handle the nonlinear term, some compactness is needed and thus the in- troduction in [9] of a smaller space for the exact controllability, where delicate estimates based on interpolation are used. A different approach is taken in this paper, it is based on the theory of accretive operators of Browder [1], Kato [2], and others. By assuming that f is accretive in the appropriate spaces, the passage to the limit can be obtained and the target space is still the largest one, namely,L2(Ω)×H1(Ω). The accretiveness hypothesis will replace the condition finL(R).

Exact controllability for the linear wave equation, with both controls in the interior and on the boundary, has been studied by the author in [6] and feedback laws were given.

Dirichlet boundary exact controllability of the wave equation has been treated by Trig- giani in [7].

Notations, the basic assumptions of the paper, and some preliminary results are given inSection 2. The exact controllability of (1.1)–(1.4) is established inSection 3. Optimal controls are shown inSection 4and feedback laws are established inSection 5.

2. Notations, assumptions, preliminary results

Throughout the paper, we will denote by (·,·) theL2(Ω) inner product as well as the pairing betweenH01(Ω) and its dualH1(Ω). LetJbe the duality mapping of the Hilbert spaceL2(0,T;H1(Ω)) into (L2(0,T;H1(Ω)))=L2(0,T;H01(Ω)) with gauge function Φ(r)=r. We have

J yL2(0,T;H01())=ΦyL2(0,T;H1())

= yL2(0,T;H1()), T

0 (J y,y)dt= y2L2(0,T;H1()), yL20,T;H1(Ω). (2.1) Definition 2.1. Letg be a mapping inL2(0,T;H1(Ω)), withD(g)=L2(0,T;L2(Ω)) and values inL2(0,T;H1(Ω)), said to be accretive with respect toJif

T

0(g(y)g(z),J(yz))dt0 y,zL20,T;H1(Ω). (2.2) We will consider mappings f ofL2(0,T;L2(Ω)) intoL2(0,T;L2(Ω)) satisfying the fol- lowing assumption.

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Assumption 2.2. Let f be a Lipschitz continuous mapping of L2(0,T;L2(Ω)) into L2(0,T;L2(Ω)). Suppose that

(i)f(y)L2(0,T;L2(Ω))C{1 +yL2(0,T;L2(Ω))}for allyL2(0,T;L2(Ω));

(ii)λI+f is accretive in the sense ofDefinition 2.1for someλ > λ0>0.

Lemma2.3. Let f be as inAssumption 2.2and suppose that

yn,fyn −→ {y,ψ} (2.3)

in{L2(0,T;H1(Ω))(L2(0,T;L2(Ω)))weak} ×(L2(0,T;L2(Ω)))weak. Thenψ=f(y).

Proof. (1) From the definition of accretiveness, we get T

0

λ(ynz+fyn

f(z),Jynz)dt0, zL20,T;L2(Ω). (2.4) It is well known that the duality mappingJ is monotone and continuous from the strong topology ofL2(0,T;H1(Ω)) to the weak topology ofL2(0,T;H01(Ω)).Thus,

Jynz−→J(yz) inL20,T;H01(Ω)weak. (2.5) On the other hand,

JynzL2(0,T;H10(Ω))

=ynzL2(0,T;H1())−→ yzL2(0,T;H1(Ω))

= J(yz)L2(0,T;H10()).

(2.6)

ButL2(0,T;H01(Ω)) is a Hilbert space, and thus

Jynz−→J(yz) inL20,T;H01(Ω),zL20,T;H1(Ω). (2.7) (2) Since

ynz2L2(0,T;H1(Ω))= T

0

ynz,Jynzdt, (2.8)

we obtain T

0

[λ+µ]ynz+ fyn

f(z),Jynzdt

µynz2L2(0,T;H1()), µ >0,zL20,T;L2(Ω).

(2.9)

Letn→ ∞, and we have T

0

[λ+µ](yz) +ψf(z),J(yz)dt

µyz2L2(0,T;H1()), zL20,T;L2(Ω).

(2.10)

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Since f is Lipschitz continuous, a simple argument using the method of successive ap- proximations shows thatR([λ+µ]I+ f)=L2(0,T;L2(Ω)) for largeλ >0, and ([λ+µ]I+ f) is 1-1. Therefore, ([λ+µ]I+f)1exists and mapsL2(0,T;L2(Ω)) intoL2(0,T;L2(Ω)).

Thus for a givenαL2(0,T;L2(Ω)), there exists a uniquezεsuch that

zε=([λ+µ]I+f)1{[λ+µ]y+ψεα}. (2.11) Then (2.9), withz=zε, becomes

T

0

εα,Jyzε

dt0, αL20,T;L2(Ω). (2.12) We have

([λ+µ]I+f)1(y+ψεα)=zε−→([λ+µ]I+f)1(y+ψ) (2.13) inL2(0,T;H1(Ω))(L2(0,T;L2(Ω)))weakas

µzεzν2L2(0,T;H1(Ω))(ε+ν)αL2(0,T;L2())J(zεzν)L2(0,T;H01(Ω))

(ε+ν)αL2(0,T;L2(Ω))zεzνL2(0,T;H1(Ω)). (2.14) We get

limε0

T

0

α,J(yzε)dt=lim

ε−→0

T

0

α,Jy

[λ+µ]I+f1[λ+µ]y+ψεαdt

= T

0

α,Jy

[λ+µ]I+f1[λ+µ]y+ψdt

0, αL20,T;L2(Ω).

(2.15) Therefore,

y=([λ+µ]I+ f)1([λ+µ]y+ψ), i.e., [λ+µ]y+f(y)=[λ+µ]y+ψ;f(y)=ψ.

(2.16)

The lemma is proved.

Remark 2.4. Suppose that f is a continuous mapping ofL2(0,T;L2(Ω)) into itself and that fis inL(R) with

sup

R |f| ≤c. (2.17)

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Then (λI+ f) is accretive inL2(0,T;H1(Ω)), with respect to the duality mappingJ, forλ > c. Indeed, we have

T

0(λ(yz) +f(y)f(z),J(yz))dt

T

0(λ(yz),J(yz))dt

cyzL2(0,T;H1(Ω))J(yz)L2(0,T;H10(Ω))

c) T

0(yz,J(yz))dt

=c)yz2L2(0,T;H1())0

(2.18)

for ally,zinL2(0,T;L2(Ω)).

3. Existence theorem

The main result of the section is the following theorem.

Theorem3.1. Let f be as inAssumption 2.2, let α=

α01 , β=

β01 be inL2(Ω)×H1(Ω); uinᐁ. (3.1) Then forTT0, there exists a solutionyof (1.1)–(1.4). Moreover,

yC(0,T;L2(Ω))+yC(0,T;H1(Ω))Ᏹ(u,α,β) (3.2) with

Ᏹ(u;α,β)=

uL2(0,T;L2(Ω))+α0

L2()+α1

H1()+β0

L2()+β1

H1()

. (3.3) The constantCis independent ofu,α,β.

Consider the exact controllability of the linear wave equation y1 y1=ω inΩ×(0,T),

y1(x,t)=0 onΓ0×(0,T), y1(x,t)=v1(u) onΓ1×(0,T), y1(x, 0)=α0, y1(x, 0)=α1 inΩ,

y1(x,T)=β0, y1(x,T)=β1 inΩ.

(3.4)

The following result has been proved by the author in [6].

Lemma3.2. Let uand let{α,β}be inL2(Ω)×H1(Ω),then forTT0, there exist v1(u)L2(0,T;L21))and a unique solutiony1of (3.4). Moreover,

y1

C(0,T;L2())+y1C(0,T;H1())+v1

L2(0,T;L2(Γ1))CᏱ(u;α,β). (3.5) The constantCis independent ofu,α,β.

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Consider the initial boundary value problem

y2∆y2=0 inΩ×(0,T),

y2(x,t)=0 onΓ0×(0,T), y2(x,t)=v2 onΓ1×(0,T), y2(x, 0)=0, y2(x, 0)=0 inΩ,

(3.6)

wherev2=n· ∇ϕwithϕbeing the unique solution of the initial boundary value problem ϕϕ=0 inΩ×(0,T),

ϕ=0 on∂Ω×(0,T), ϕ(x,T)=g0, ϕ(x,T)=g1 inΩ.

(3.7)

We have the following known result.

Lemma3.3. Let{g0,g1}be inH01(Ω)×L2(Ω), then there exists a unique solutiony2of (3.6).

Moreover, y2

C(0,T;L2(Ω))+y2C(0,T;H1(Ω))+v2

L2(0,T;L21))Cg0

H10(Ω)+g1

L2(Ω)

. (3.8)

The constantCis independent ofg0,g1.

LetΛbe the mapping ofH01(Ω)×L2(Ω) into its dualH1(Ω)×L2(Ω), defined by Λ(g)=

y2(x,T),y2(x,T) . (3.9)

It is well known in the Hilbert uniqueness method that Λ is an isomorphism of H01(Ω)×L2(Ω) ontoH1(Ω)×L2(Ω).

We now consider the nonlinear initial boundary value problem y3y3= −f(y1+y2+y3) inΩ×(0,T),

y3(x,t)=0 on∂Ω×(0,T), y3(x, 0)=0, y3(x, 0)=0 inΩ.

(3.10)

Lemma3.4. Let f be as inAssumption 2.2and let{y1,y2}be as in Lemmas3.2and3.3.

Then there exists a solutiony3of (3.10). Moroever,

y3

C(0,T;H01(Ω))+y3C(0,T;L2(Ω))CᏱ(u;α,β) +g0

H01(Ω)+g1

L2(Ω)

. (3.11)

The constantCis independent ofu,α,β0,g.

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Proof. (1) Consider the system

yy= −fy1+y2+z inΩ×(0,T), y=0 on×(0,T),

y(x, 0)=0, y(x, 0)=0 inΩ.

(3.12)

Letzbe an element of the set ᏮC=

z:zL2(0,t;H1())+zL2(0,t;L2())

CᏱ(u;α,β) +g0

H01(Ω)+g1

L2(Ω)

exp(Ct);t[0,T]. (3.13)

Clearly, there exists a unique solutionyof the above initial boundary value problem with

y(·,t)H01()+y(·,t)L2()

Cy1

L2(0,t;L2())+y2

L2(0,t;L2())+zL2(0,t;L2(Ω)). (3.14) Taking into account the estimates of Lemmas3.2and3.3, we obtain

y(·,t)H01(Ω)+y(·,t)L2(Ω)

C

Ᏹ(u;α,β) +g1L2(Ω)+ t

0z(·,s)L2(Ω)ds

. (3.15)

Sincezis inᏮC, it follows that

y(·,t)H01()+y(·,t)L2()

CᏱ(u;α,β) +g0

H01()+g1

L2()

exp(Ct) (3.16)

for allt[0,T], and thusyC.

(2) LetᏭbe the nonlinear mapping ofᏮC, considered as a closed convex subset of L2(0,T;L2(Ω)) intoL2(0,T;L2(Ω)) defined by

Ꮽ(z)=y. (3.17)

We will show thatᏭsatisfies the hypotheses of the Schauder fixed point theorem.

Let{zn}be inᏮC and let yn=Ꮽ(zn). From Aubin’s theorem we get subsequences, denoted again by{yn,zn}such that{yn,zn} → {y,z}in

L20,T;L2(Ω)

L0,T;H1(Ω)weak

2

. (3.18)

Since f is a continuous mapping ofL2(0,T;L2(Ω)) intoL2(0,T;L2(Ω)), we getᏭ(z)= y. It follows from the Schauder fixed point theorem that there existsy3inᏮC, solution of (3.10). With f being Lipschitz continuous, the solution is unique and the lemma is

proved.

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Letᏸbe the nonlinear mapping ofH01(Ω)×L2(Ω) intoH1(Ω)×L2(Ω) defined by ᏸg0,g1 =ᏸ(g)=

y3(x,T),y3(x,T) . (3.19) SinceΛis an isomorphism ofH01(Ω)×L2(Ω) ontoH1(Ω)×L2(Ω), its inverseΛ1is well defined. We consider the nonlinear mapping

=Λ1. (3.20)

It is clear that᏷is a nonlinear mapping ofH01(Ω)×L2(Ω) intoH01(Ω)×L2(Ω). We will now show that᏷has a fixed point

᏷(g)=g, i.e., ᏸ(g)=Λ(g), (3.21)

and thus

y3(x,T),y3(x,T) =

y2(x,T),y2(x,T) . (3.22) LetᏮCbe the set

C= g:g=

g0,g1 ;g0

H01(Ω)+g1

L2(Ω)Ᏹ(u;α,β). (3.23) It follows from the Sobolev embedding theorem thatᏮCis a compact convex subset of L2(Ω)×H1(Ω).

SinceΛis an isomorphism ofH01(Ω)×L2(Ω) ontoH1(Ω)×L2(Ω), we have

chH01(Ω)×L2(Ω)Λ(h)H1(Ω)×L2(Ω)ChH01(Ω)×L2(Ω) (3.24) for allhH01(Ω)×L2(Ω).

Lemma3.5. Letbe as in (3.20), then it mapsCintoCwith

C=supc1CᏱ(u;α,β),CᏱ(u;α,β) . (3.25) Proof. (1) Letgbe inᏮC, then

ᏸ(g)=

y3(x,T),y3(x,T) , (3.26) and we obtain from the estimates ofLemma 3.4

ᏸ(g)H1(Ω)×L2(Ω)CᏱ(u;α,β). (3.27) Thus,

Λ1᏷(g)H01(Ω)×L2(Ω)c1᏷(g)L2()×H1()

c1CᏱ(u;α,β)C. (3.28)

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Lemma3.6. Letbe given by (3.20), then it has a fixed point inC.

Proof. In view of Lemma 3.5, it suffices to show that ᏷ is a continuous mapping of L2(Ω)×H1(Ω) intoL2(Ω)×H1(Ω) as the setᏮCis a compact convex subset ofL2(Ω)× H1(Ω).

LetgnC, then there exists a subsequence such that gn−→g inL2(Ω)

H01(Ω)weak×H1(Ω)

L2(Ω)weak. (3.29) Set

ᏸgn=

y3,n(·,T),y3,n(·,T) , (3.30) wherey3,nis the solution of (3.10),y2,nis the solution of (3.6) withg=gn.

It follows from the estimates of Lemmas3.3and3.4that y2,n,y2,n ,v2,n −→

y2,y2,v2 (3.31)

in

C0,T;H1(Ω)

L0,T;L2(Ω)weak×

L0,T;H1(Ω)weak

×

L20,T;L21)weak (3.32) and{y3,n,y3,n } → {y3,y3}in

C0,T;L2(Ω)

L0,T;H01(Ω)weak×

L0,T;L2(Ω)weak

L2(0,T;H1(Ω)). (3.33)

It is trivial to check that y2 is the solution of (3.6). We now useAssumption 2.2to show thaty3is the solution of (3.10). Indeed,

y2,n+y3,n−→y2+y3 inL20,T;H1(Ω)

L20,T;L2(Ω)weak (3.34) and f(y1+y2,n+y3,n)ψ weakly in L2(0,T;L2(Ω)). Since f is accretive in L2(0,T;

H1(Ω)), it follows fromLemma 2.3thatψ= f(y1+y2+y3), and hence ᏸgn−→ᏸg inL2(Ω)

H01(Ω)weak×H1(Ω)

L2(Ω)weak. (3.35) Let

᏷gn=Λ1ᏸ(gn)=hn, (3.36)

then

Λ(hn)=gn=

y3,n(·,T),y3,n(·,T) =

y2,n (·,T),y2,n(·,T) (3.37) andy2,nis the unique solution of (3.6) withg=hn.

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WithhnCand with the estimates ofLemma 3.3, we get hn,y2,n,y2,n −→

h,y2,y2 (3.38)

in

H01(Ω)weakL2(Ω)×

L2(Ω)weakH1(Ω)×

L0,T;L2(Ω)weak

×

L0,T;H1(Ω)weak. (3.39) Furthermore,

y2,n(·,T),y2,n(·,T) −→

y2(·,T),y2(·,T) inH1(Ω)×H2(Ω). (3.40) Moreover,Λ(h)= {y2(·,T),y2(·,T)}. It follows that

Λ(h)=

y2(·,T),y2(·,T) =

y3(·,T),y3(·,T) =ᏸ(g). (3.41) Hence,

hn=Λ1gn=gn−→h=Λ1ᏸ(g)=g (3.42) in (L2(Ω))×(H1(Ω)). The nonlinear mapping ᏷ satisfies the hypotheses of the Schauder fixed point theorem, and thus there existsgCsuch that

g=Λ1g=g. (3.43)

Proof ofTheorem 3.1. In view ofLemma 3.6, there existsgCsuch that

y3(·,T),y3(·,T) =

y2(·,T),y2(·,T) (3.44) withy2,y3being the unique solution of (3.6), (3.10), respectively, and withg=g.

Letuᐁ, then it is clear that

y=y1+y2+y3, v(u)=v1+v2 (3.45) are a solution of (1.1)–(1.4). The estimate of the theorem is an immediate consequence

of those of Lemmas3.2–3.6.

4. Optimal control

We associate with (1.1)–(1.4) the cost function

᏶(y;u;α;β)= T

0

|y(x,t)|dx dt, (4.1) whereyis a solution of (1.1)–(1.4) given byTheorem 3.1. The main result of the section is the following theorem.

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Theorem4.1. Let f be as inAssumption 2.2, let

α01 ,β0,β1 be inL2(Ω)×H1(Ω), (4.2) then forT > T0, there existsuᐁ, and

{y,y,v(u) } ∈C0,T;L2(Ω)×C0,T;H1(Ω)×L20,T;L21) (4.3) such that

V(α,β)=᏶(y;u;α,β) =inf᏶(y;u;α,β) :u. (4.4) Proof. (1) Let{un,vn,yn}be a minimizing sequence of the optimization problem (4.4) with

᏶(yn;un;α,β)V(α,β) + 1/n. (4.5) From the estimates ofTheorem 3.1, we have

ynC(0,T;L2(Ω))+ynC(0,T;H1(Ω))+vnL2(0,T;L21))Cun;α,β

C{1 +Ᏹ(u;α,β)}, (4.6) asᐁis a bounded subset ofL2(0,T;L2(Ω)). Thus there exists a subsequence such that

yn,yn,un,vn −→ {y,y,u, v} (4.7) in

C0,T;H1(Ω)

L0,T;L2(Ω)weak×

L0,T;H1(Ω)weak, C0,T;H2(Ω)×

L20,T;L2(Ω)weak×

L20,T;L2Γ1)weak. (4.8) We now show that{y,u,v}is a solution of (1.1)–(1.4) and it is clear that it suffices to prove that

fyn

−→f(y) inL20,T;L2(Ω)weak. (4.9) Since f(yn)ψweakly inL2(0,T;L2(Ω)) and sincef is accretive inL2(0,T;H1(Ω)), it follows fromLemma 2.3thatψ=f(y). The theorem is now an immediate consequence

of (3.7).

Lemma4.2. LetVbe as the value function associated with (1.1)–(1.4) and the cost function (4.1). Then,

|V(α;β)V(γ;β)| ≤Cα0γ0

L2(Ω)+α1γ1

H1(Ω)

(4.10)

for allα,β,γinL2(Ω)×H1(Ω). The constantCis independent ofα,β,γ.

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Proof. Letα,βbe inL2(Ω)×H1(Ω), then it follows fromTheorem 4.1that

V(α,β)=᏶(y;u;v(u); α,β). (4.11) Then,

V(γ,β)V(α,β)᏶(z;u, v(u), γ,β)᏶(y;u, v(u), γ,β)

T

0

{|z| − |y|}dx dt

T

0

|zy|dx dt

CzyL2(0,T;L2(Ω)).

(4.12)

On the other hand, we have

(zy)∆(zy)=f(y)f(z) inΩ×(0,T), zy=0 onΓ0×(0,T), zy=vv onΓ1×(0,T),

zy|t=0=γ0α0, (zy) |t=0=γ1α1 inΩ, zy|t=0=0=(zy)|t=0 inΩ.

(4.13)

ApplyingTheorem 3.1with

y=zy, v=vv, (4.14)

we have

V(γ,β)V(α,β)Cα0γ0

L2()+α1γ1

H1()

. (4.15)

Thus,

V(γ,β)V(α,β)Cα0γ0

L2()+α1γ1

H1()

. (4.16)

Reversing the role played byα,γ, we get the stated result.

Consider the following initial boundary value poblem for the heat equation ϕ∆ϕ=h inΩ×(0,T),

ϕ(x,t)=0 on×(0,T), ϕ(x, 0)=0 inΩ. (4.17) LetSbe the linear mapping ofL2(0,T;H1(Ω)) intoL2(0,T;H01(Ω)) given by

Sh=ϕ, (4.18)

whereϕis the unique solution of (4.17). ThenSis a compact linear mapping ofL2(0,T;

H1(Ω)) intoL2(0,T;L2(Ω)) and

ϕC(0,T;L2())= ShC(0,T;L2())ChL2(0,T;H1()). (4.19)

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