EXACT CONTROLLABILITY FOR A NONLINEAR STOCHASTIC WAVE EQUATION

EXACT CONTROLLABILITY FOR A NONLINEAR STOCHASTIC WAVE EQUATION

BUI AN TON

Received 2 July 2005; Accepted 7 September 2005

The exact controllability for a semilinear stochastic wave equation with a boundary con- trol is established. The target and initial spaces areL²(G)×H⁻¹(G) withGbeing a bound- ed open subset ofR³and the nonlinear terms having at most a linear growth.

Copyright © 2006 Bui An Ton. This is an open access article distributed under the Cre- ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let (Ω,Ꮽ,P) be a probability space withΩbeing completely regular and letwbe a stan- dard Wiener process on the space. LetGbe a bounded open subset ofR³with a smooth boundary and consider the initial boundary value problem

d y1=y2dt inQa.s., d y2−Δy1dt−du=ft,y1

dw+gt,y1

dt inQa.s., y1(·,t,ω)=0 onΓ0×(0,T) a.s.,

y1(·,t,ω)=v onΓ1×(0,T) a.s., y(·, 0,ω)=α=

α0,α1

inGa.s.

(1.1)

The functionsf,gare inC¹(R) with f,ginL^∞(R) and Q=G×(0,T), ∂G=Γ0

Γ1, Γ0

Γ1= ∅, Γ1= ∅. (1.2)

The given surfaceΓ1is closed. For givenα,βinL²(G)×H⁻¹(G), one wishes to find con- trols:

{u,v} ∈L²Ω,Ꮽ,P;C0,T;L²(G)×L²Ω,Ꮽ,P;L²0,T;L²Γ1

(1.3)

Hindawi Publishing Corporation Abstract and Applied Analysis

Volume 2006, Article ID 74264, Pages1–14 DOI10.1155/AAA/2006/74264

such that a solutiony=(y1,y2) of (1.1) takes the value

y(·,T,ω)=β inGa.s. (1.4)

for some largeT > T0.

The exact controllability of deterministic linear wave equations has been extensively investigated both theoretically and numerically, using the Hilbert uniqueness method of Lions [4].The problem arises in applied sciences when one wishes for example to steer a large complex structure to a specified target (cf. Lions [4], Russell [5]). The result was extended by Zuazua [8] to semilinear wave equations when the initial and target spaces areH^γ(G)×H^−1+γ(G) for someγ >0.Using the notion of accretive mapping, the author has established in [6], the exact controllability of a semilinear wave equation with at most a linear growth in the nonlinear term, the initial and target spaces being the natural ones, that is,L²(G)×H⁻¹(G).The literature on the exact controllability of the stochastic wave equation seems scarce. In [7], the author has shown the exact controllability of (1.1)–

(1.4) when the white noise is independent of the state and it is the purpose of this paper to treat the general case with f depending ony1.

2. Notations, assumption and some preliminary results

Throughout the paper we will assume thatΩis completely regular and by abuse of no- tations, we will denote by (·,·) theL²(G) inner product as well as the pairings ofH₀¹(G) with its dualH⁻¹(G).

Assumption 2.1. Let f,gbeC¹((0,T)×R) functions. We assume that ∂ f

∂ξ(t,·)

L^∞(R)

, ∂g

∂ξ(t,·)

L^∞(R)≤C, ∀t∈[0,T], f(t,y) + g(t,y) ≤C1 +|y|

, ∀{t,y} ∈(0,T)×R.

(2.1)

With initial and target spaces inL²(G)×H⁻¹(G), approximating solutions of the lin- earized version of (1.1)–(1.4) do not have enough regularity to allow the use of a com- pactness argument for the nonlinear terms f andg.The difficulty is circumvented by using an argument of the theory of monotone operators involving accretive mappings introduced by Browder [2], Kato [3].

LetJ be the duality mapping ofL²(0,T;H⁻¹(G)) into its dual,L²(0,T;H0¹(G)), with gauge functionΦ(r)=r.Then it is known that

_T

0(J y,y)dt= y _L²_(0,T;H⁻¹_(G)) J y _L²_(0,T;H0¹_(G))

= y ²_L2(0,T;H⁻¹(G)), ∀y∈L²0,T;H⁻¹(G).

(2.2)

Moreover J is monotone and is continuous from the strong topology of L²(0,T;

H⁻¹(G)) to the weak topology ofL²(0,T;H0¹(G)).

Let F be a mapping of L²(0,T;H⁻¹(G)) intoL²(0,T;H⁻¹(G)) with D(F)=L²(0,T;

L²(G)). The mappingFis said to be accretive with respect toJif _T

0

F(y)−F(z),J(y−z)dt≥0, ∀y,z∈D(F). (2.3)

Lemma 2.2. Let f be inC¹((0,T)×R) and suppose thatAssumption 2.1is satisfied. Then λI+f is accretive with respect to the duality mappingJfor large,λ > λ0.

Proof. We have _T

0

f(t,y)−f(t,z),J(y−z)dt= _T

0

(y−z)f(t,ξ),J(y−z)dt

≤(y−z)f_L2(0,T;H⁻¹(G))J(y−z)_L2(0,T;H0¹(G))

≤(y−z)f_L2(0,T;H⁻¹(G))y−z_L2(0,T;H⁻¹(G))

(2.4)

for ally,zinL²(0,T;L²(G)).On the other hand, it is clear that (y−z)f_L2(0,T;H⁻¹(G))= inf

ϕ _{L2 (0,T;H}1 0 (G))≤1

_T

0

(y−z)f,ϕdt

≤ f _L^∞_{((0,T)×R) T}

0

|y−z|,|ϕ| dt

≤ f _L^∞_((0,T)×R) y−z _L²_(0,T;H⁻¹_(G)) ϕ _L²_(0,T;H0¹_(G))

(2.5)

for ally,zinL²(0,T;L²(G)).Thus, (y−z)f L²(0,T;H⁻¹(G))

≤ f L^∞((0,T)_×R) y−z L²(0,T;H⁻¹(G)), ∀y,z∈L²0,T;L²(G). (2.6) Takeλ > f _L^∞_((0,T)×R)and we have

_T

0

(λI+f)(y−z),J(y−z)dt

≥λ y−z ²_L2(0,T;H⁻¹(G))− _T

0

f(y)−f(z),J(y−z)dt

≥

λ− f L^∞((0,T)×R)

y−z ²_L2(0,T;H⁻¹(G))

≥0, ∀y,z∈L²0,T;L²(G).

(2.7)

The lemma is proved.

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

Lemma 2.3. Let f be as inAssumption 2.1and suppose that

y_n−→y inL²0,T;L²(G)_weak∩L²0,T;H⁻¹(G), (2.8)

then

ft,y_nj−→f(t,y) inL²0,T;L²(G)_weak. (2.9) With f as inAssumption 2.1, it follows fromLemma 2.2thatλI+f is accretive with respect to the duality mappingJ. Since

Jy_n−→J(y) weakly inL²0,T;H₀¹(G),

J(y_n)_L²_(0,T;H0¹_(G))=y_nL2(0,T;H⁻¹(G))−→ y _L²_(0,T;H⁻¹_(G))=J(y)_L2(0,T;H0¹(G))

(2.10) and sinceL²(0,T;H₀¹(G)) is a Hilbert space, we have

Jy_n−→J(y) inL²0,T;H₀¹(G). (2.11) Now using an argument of the theory of monotone operators applied to accretive mappings, the author has shown in [6] the weak convergence inL²(0,T;L²(G)) of the sequence{f(t,y_nj)}.

3. Main result

The main result of the paper is the following theorem.

Theorem 3.1. Let (Ω,Ꮽ,P) be a probability space withΩbeing completely regular and let wbe a standard Wiener process on the space. Let f,gbe as inAssumption 2.1and letα,βbe inL²(G)×H⁻¹(G).Then forT > T0, there exists a weak solution{u,v, y}in

L²Ω,Ꮽ,P;C0,T;L²(G)×L⁴Ω,Ꮽ,P;L²0,T;L²Γ1

×L⁴Ω,Ꮽ,P;C0,T;L²(G)×L⁴Ω,Ꮽ,P;C0,T;H⁻¹(G), (3.1) of (1.1)–(1.4). Moreover,

Ey1⁴

C(0,T;L²(G))

+Ey2⁴

C(0,T;H⁻¹(G))

+E v ⁴_L²_(0,T;L²_(Γ1))≤C1 +Ᏹ(α,β) (3.2)

with

Ᏹ(α,β)= α ⁴_L²_(G)×H−1(G)+ β ⁴_L²_(G)×H−1(G). (3.3) The proof of the theorem can be split into three steps.

Step 1. Lety be in

L⁴Ω,Ꮽ,P;C0,T;L²(G)×L⁴Ω,Ꮽ,P;C0,T;H⁻¹(G) (3.4) and consider the exact controllability of the linear wave equation

dz1=z2dt inQ, dz2−Δz1dt=0 inQ,

z1=0 onΓ0×(0,T), z1=v onΓ1×(0,T), z(·, 0)=α, z(·,T)=β−y( ·,T) inG.

(3.5)

The following result is well known.

Lemma 3.2. Suppose all the hypotheses of Theorem 3.1are satisfied, let y be in C(0,T;

L²(G))×C(0,T;H⁻¹(G)). Then forT > T0, there exists{v, z}in L²0,T;L²Γ1

×C0,T;L²(G)×C0,T;H⁻¹(G), (3.6) weak solution of (3.5). Moreover,

v _L²_(0,T;L²_(Γ1))+z1

C(0,T;L²(G))+z2

C(0,T;H⁻¹(G))

≤C1 + α L²(G)×H⁻¹(G)+ β L²(G)×H⁻¹(G)+y(·,T)_L2(G)×H⁻¹(G)

. (3.7)

The constantCis independent ofy, α,β.

We will now consider the case wheny is in

L⁴Ω,Ꮽ,P;C0,T;L²(G)×L⁴Ω,Ꮽ,P;C0,T;H⁻¹(G). (3.8) Let

Sy=

{v, z}:{v, z}solution of (3.5). (3.9) It follows fromLemma 3.2thatSyis nonempty.

Lemma 3.3. Let Sy be as in (3.9), then it is a closed, bounded convex subset of L²(0,T;

L²(Γ1))×L²(0,T;L²(G))×L²(0,T;H⁻¹(G)).

Proof. Since the problem (3.5) is linear, it is clear thatSy is a convex subset ofL²(0,T;

L²(Γ1))×L²(0,T;L²(G))_×L²(0,T;H⁻¹(G)). It follows fromLemma 3.2 that the set is bounded. Suppose that

vn, zn

−→ {v, z} (3.10)

in

L²0,T;L²Γ1

×L²0,T;L²(G)×L²0,T;H⁻¹(G) (3.11) with{v_n, zn} ∈Sy.It is trivial to check that indeed{v, z} ∈Sy.

Letπ_Sybe the projection of L²0,T;L²Γ1

×L²0,T;L²(G)×L²0,T;H⁻¹(G) (3.12) onto the closed convex setSy, defined by

{v,x} −π_Sy{v,x} =inf {v,x} − {v,x} :∀{v,x} ∈Sy

(3.13)

with

{v,x} ={v,x}

L²(0,T;L²(Γ1))×L²(0,T;L²(G))×L²(0,T;H⁻¹(G)). (3.14)

ThenπSyis uniquely defined and let

{v,z} =π_Sy(0) (3.15)

be the unique element of minimal L²(0,T;L²(Γ1))×L²(0,T;L²(G))×L²(0,T;H⁻¹(G)) norm ofS.

Let

Λ(y) =

{v,z}:{v,z}solution of (3.5) as in (3.15), (3.16) then it maps

C0,T;L²(G)×C0,T;H⁻¹(G)

−→L²0,T;L²(Γ1)×C0,T;L²(G)×C0,T;H⁻¹(G). (3.17) Lemma 3.4. LetΛbe as in (3.16), then its graph is closed.

Proof. It is an immediate consequence ofLemma 3.2.

LetΦ(ω) be the mapping

Φ(ω)=

y(·,ω) (3.18)

ofΩintoC(0,T;L²(G))×C(0,T;H⁻¹(G)).

Set

Θ(ω)=Λ◦Φ(ω)=

{v(·,ω),z(·,ω)}: solution of (3.5) as in (3.15). (3.19) Lemma 3.5. The mappingΛhas a universally measurable sectionσ and the application σ◦Θof

Ω−→L²0,T;L²(Γ1)×C0,T;L²(G)×C0,T;H⁻¹(G) (3.20) is a measurable section ofΘ.Furthermore

Ez1²

C(0,T;L²(G))

+Ez2²

C(0,T;H⁻¹(G))

+Ev²_L2(0,T;L²(Γ1))

≤C1 + α ²_L²_(G)×H−1(G)+ β ²_L²_(G)×H−1(G)+Ey(·,T)²_L2(G)×H⁻¹(G)

. (3.21) Proof. The mappingΛhas nonempty closed images inL²(0,T;L²(Γ1))×C(0,T;L²(G))× C(0,T;H⁻¹(G)) with a closed graph. It follows from a theorem of Von Neumann that there exists a universally measurable sectionσofΛ(cf. [1, Theorem 3.1]).

SinceP is a Radon measure on a completely regular space and sinceΦis a random variable and therefore measurable in the sense of Lusin, we deduce that for each positive integerkthere exists a compact setK_kofΩsuch that

P Ω

K_k

≤1

k (3.22)

and the restrictionΦk=Φ|KkofΦtoKkis continuous.

We may assume that{Kk}is an increasing sequence.

The measure P induces on Kk a Radon measurePk and Φk(Pk) is a Radon mea- sure onL²(0,T;L²(Γ1))×C(0,T;L²(G))×C(0,T;H⁻¹(G)). Since σ isΦk(P_k) measur- able, it follows that σ◦Φk is P measurable on Kk in L²(0,T;L²(Γ1))×C(0,T;L²(G))

×C(0,T;H⁻¹(G)). Let

v_k(·,ω), z_k(·,ω)₌

⎧⎨

⎩

(σ◦Φ)(·,ω), ifω∈K_k,

0, ifω /∈Kk. (3.23)

The functions {vk(·,ω), zk(·ω)}from ΩintoL²(0,T;L²(Γ1))×C(0,T;L²(G))×C(0,T;

H⁻¹(G)) are measurable. SinceP(_kKk)=1, we have v_k, zk

−→ {v, z}, a.s. with{v, z} ∈Λ(y). (3.24) Thus,{v, z}is measurable in the sense of Lusin and therefore is a random variable. The stated estimate is an immediate consequence of that ofLemma 3.2.

Step 2. We now consider the linear stochastic system

d y1=y2dt inQa.s., d y2−Δy1dt=

f(t,z1+y1

−ft,z1

dw +{g(t,z1+y1)−gt,z1

dt inQa.s., y1=0 on∂G×(0,T) a.s.,

y(·, 0)=0 inGa.s.

(3.25)

Lemma 3.6. Suppose all the hypotheses ofTheorem 3.1are satisfied and let z be as in (3.15).

Then there exists a unique y, solution of (3.25). Moreover,

Ey(·,t)⁴_L2(G)

+Ey1 4

L²(0,t;H0¹(G))

≤C1 + expEy1⁴

L²(0,t;L²(G))

, ∀t∈[0,T]. (3.26)

Furthermore,

Ey₁²_L2(0,T;L²(G))

≤C1 + expEy1²

L²(0,T;L²(G)

, E

sup

n

1 νn

sup

|s|≤δn

_t

0

y2(r+s)−y2(r)²_H−1(G)dr

≤C1 +Ᏹ(α,β) + expEy1⁴

L²(0,t;L²(G))

(3.27)

with{νn,δn} →0⁺and_nδn/νn<∞. The constantCis independent oftand ofz1. Proof. The existence of a unique solution of (3.25) is well known. we will now establish the estimate of the lemma.

(1) A standard argument gives y(·,t)²_L2(G)+y1²

L²(0,t;H¹0(G))≤ _t

0

fs,z1+y1

−f·,z1

,y2

dw +

_t

0

gs,z1+y1

−g·,z1²

L²(G)ds.

(3.28)

With f,gas inAssumption 2.1, they are Lipschitz continuous inL²(0,T;L²(G)) and thus y(·,t)²_L2(G)+y1²

L²(0,t;H0¹(G))

≤C _t

0y1²

L²(G)ds+ _t

0

f·,z1+y1

−f·,z1

,y2

dw . (3.29) Taking the square of the two sides of (3.29) and then the mathematical expectation, we obtain by applying the martingale inequality,

Ey(·,t)⁴_L2(G)

≤C

Ey1⁴

L²(0,t;L²(G))

+ _t

0Ey2⁴

L²(G)

ds

. (3.30)

We have used the hypothesis that f isL²(0,T;L²(G)) Lipschitz continuous. An appli- cation of the Gronwall lemma yields

Ey(·,t)⁴_L2(G)

≤C1 + expEy1⁴

L²(0,t;L²(G))

(3.31) for allt∈[0,T].The constantCis independent ofz1. Now going back to (3.30) and we obtain

Ey1⁴

L²(0,t;H0¹(G))

≤C1 + expEy1⁴

L²(0,t;L²(G))

(3.32)

for allt∈[0,T].The constantCis independent ofz1.It is clear that Ey₁²_L2(0,T;L²(G))

≤C1 + expEy1²

L²(0,T;L²(G))

. (3.33)

(2) For any positive,θwe have y2(t+θ)−y2(t)_H−1(G)

≤Cθ+ _t+θ

t

y1

H0¹(G)ds+ _t+θ

t

fs,z1+y1

−fs,z1

dw

L²(G)

+C _t+θ

t y1

L²(G)ds.

(3.34)

Taking the square of the two sides, integrating with respect totfrom 0 toTand then the mathematical expectation, we obtain by applying the martingale inequality,

E

sup

|θ|≤δ

_T

0

y2(t+θ)₋y2(t)²_H−1(G)dt

≤Cδ²+Cδ²E T

0

y1²

H¹0(G)dt

+CδE T

0 y1²

L²(G)dt

≤C1δ.

(3.35)

We have used the hypothesis that f andg areL²(0,T;L²(G)) Lipschitz continuous.

Taking into account the estimates of the first part, we get the stated result.

Step 3. Let

Ꮾ=

y1,y2

:Ey(·,t)⁴_L2(G)

,Ey₁²_L2(0,t;L²(G))

≤Cexp(t),

E

sup

|θ|≤δ

_t

0

y2(r+θ)−y2(r)²_H−1(G)dr

≤Cδ,∀t∈[0,T]

.

(3.36)

Let h be an element of

L²Ω,Ꮽ,P;L²0,T;H₀¹(G)×L²Ω,Ꮽ,P;L²(G). (3.37) SinceΩis completely regular and h is a random variable and hence is measurable in the Lusin, for eachkthere exists a compact subsetKkofΩsuch that

P Ω

Kk

≤1

k (3.38)

and the restriction of h toKkis continuous onKkwith values inH₀¹(G)×L²(G), we may assume without loss of generality that theKk’s are increasing.

Lemma 3.7. LetᏮbe as above, then it is a compact convex subset of

L²Kk,Ꮽ,P;C0,T;L²(G)×L²Kk,Ꮽ,P;C0,T;H⁻¹(G). (3.39) Proof. (1) Let y_nbe inᏮ, then

Eyn⁴

C(0,T;L²(G))

+Ey_1,n⁴_L2(0,t;H0¹(G))

+Ey_1,n ⁴_L2(0,T;L²(G))

≤C, E

sup

|θ|≤δ

_T

0

y2,n(t+θ)−y2,n(t)²_H−1(G)dt

≤Cδ. (3.40)

Since y_nare random variables, they are measurable in the sense of Lusin. By hypoth- esis,Ωis completely regular, thus for each positive integerk, there exist compact subsets K_1,kⁿ ,K_2,kⁿ ofΩwith

P Ω

K_1,kⁿ

,P Ω

K_2,kⁿ

≤1

k (3.41)

and the restrictions of yn toK_1,kⁿ ×K_2,kⁿ are continuous with values inC(0,T;L²(G))∩ L²(0,T;H0¹(G))×C(0,T;L²(G)). Moreover the setsK_1,kⁿ ,K_2,kⁿ are increasing. Without loss of generality we will assume that

Kk⊂Kⁿ_j,k, j=1, 2. (3.42)

(2) Since ynare inᏮ, a simple proof by contradiction gives yn(·,ω)_C(0,T;L2(G))+y1,n(·,ω)_L2(0,T;H0¹(G))

+y_1,n(·,ω)_L2(0,T;L²(G))≤Ck(ω), a.s. inKk, sup

|θ|≤δ

_T

0

y_2,n(t+θ,ω)−y_2,n(t,ω)²_H−1(G)dt≤δC_k(ω), a.s. inK_k.

(3.43)

SinceK_kis a compact subset ofΩ, we have Kk⊂

N j=1

Vj

ωj,δ⊂Ω. (3.44)

It follows from Aubin’s theorem that there exists a subsequence such that y_1,ns·,ω_j−→y1

·,ω_j inC0,T;L²(G). (3.45) Furthermore,y_1,n→y1in

L⁴Ω,Ꮽ,P;L²0,T;H₀¹(G)_weak∩

L⁴Ω,Ꮽ,P;L^∞0,T;L²(G)_weak^∗. (3.46) From the diagonalization process, we get

y_1,n·,ω_j−→y1

·,ω_j inC0,T;L²(G),∀j. (3.47) Since the restriction ofy1toK_kis continuous with values inC(0,T;L²(G)), we have

Kk

y1,n(·,ω)−y1(·,ω)²_C(0,T;L2(G))dP

≤ N j=1

Vj

y1,n(·,ω)−y1,n

·,ωj²

C(0,T;L²(G))

+y_1,n·,ω_j−y1

·,ω_j²_C(0,T;L2(G))

dP +y1

·,ωj

−y1(·,ω)²_C(0,T;L2(G))dP

≤Cε.

(3.48)

Hence,

y_1,n−→y1 inL²K_k,Ꮽ,P;C0,T;L²(G). (3.49) (3) Withy_2,n, we will apply a compactness theorem involving fractional time derivative instead of Aubin’s theorem. We get as before as a subsequence such that

y_2,n·,ω_j−→y2

·,ω_j inC0,T;H⁻¹(G), j=1,. . .,N,

y_2,n−→y2 inL⁴Ω,Ꮽ,P;L^∞0,T;L²(G)_weak^∗. (3.50) Again by the diagonalization process, we get a subsequence such that

y_2,n·,ω_j−→y2

·,ω_j inC0,T;H⁻¹(G), j=1,. . .,N. (3.51)

Sincey2,nandy2are continuous onKkwith values inC(0,T;H⁻¹(G)), we have

Kk

y_2,n(·,ω)−y2(·,ω)²_C(0,T;H−1(G))dP

≤ N j=1

Vj

y_2,n(·,ω)−y_2,n·,ω_j²_C(0,T;H−1(G))

+y2,n

·,ωj

−y2

·,ωj²

C(0,T;H⁻¹(G))

+y2

·,ωj

−y2(·,ω)²_C(0,T;H−1(G))

dP≤Cε.

(3.52)

Thus,

y_n−→y1 inL²K_k,Ꮽ,P;C0,T;L²(G)×L²K_k,Ꮽ,P;C0,T;H⁻¹(G) (3.53)

and the lemma is proved.

Letᏸbe the nonlinear mapping ofᏮinto

L²Ω,Ꮽ,P;C0,T;L²(G)×L²Ω,Ꮽ,P;C0,T;H⁻¹(G) (3.54) defined by

ᏸ(y)=y, (3.55)

where y is the unique solution of (3.25) given byLemma 3.6.

Lemma 3.8. The mappingᏸtakesᏮintoᏮ.

Proof. It is an immediate consequence of the estimates ofLemma 3.6and of the Gronwall

lemma.

We will considerᏮas a subset of

L²Kk,Ꮽ,P;C0,T;L²(G)×L²Kk,Ꮽ,P;C0,T;H⁻¹(G). (3.56) Lemma 3.9. Letᏸbe as in (3.70), thenᏸis continuous from

L²K_k,Ꮽ,P;C0,T;L²(G)×L²K_k,Ꮽ,P;C0,T;H⁻¹(G) (3.57) into itself.

Proof. (1) Letyn∈Ꮾand letᏸ(yn)=ynwith ynbeing a solution of (3.25). Let znbe the solution of (3.5) withy replaced by y_n.

Suppose that

yn, yn

−→ {y, y} (3.58)

in

L²K_k,Ꮽ,P;C0,T;L²G)×L²K_k,Ꮽ,P;C0,T;H⁻¹(G)². (3.59)