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CONTROLLABILITY OF THE LINEAR SYSTEM OF THERMOELASTICITY by Enrique Zuazua Departamento de Matemática Aplicada Universidad Complutense 284 Madrid. Spain. Abstract. We prove that the linear system of thermoelasticity

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CONTROLLABILITY OF THE LINEAR SYSTEM OF THERMOELASTICITY by Enrique Zuazua Departamento de Matemática Aplicada Universidad Complutense 284 Madrid. Spain. Abstract. We prove that the linear system of thermoelasticity is controllable in the following sense: If the control time is large enough and we act in the equations of displacement by means of a control supported in a neighborhood of the boundary of the thermoelastic body, then we may control exactly the displacement and simultaneously the temperature in an approximate way. The method of proof combines: (i) A decoupling result for the system of thermoelasticity due to D. Henry, O. Lopes and A. Perissinitto, (ii) the variational approach to controllability developed recently by C. Fabre, J. P. Puel and the author and (iii) some new observability inequalities for the system of thermoelasticity. Key words and phrases: Linear system of thermoelasticity, exact controllability, approximate controllability, decoupling, observability inequalities. AMS Subject Classification: 93B5, 73C5, 35B37 1. INTRODUCTION AND MAIN RESULTS Let us consider an isotropic and homogeneous thermoelastic body occupying an open and bounded set of IR n (n 1) with boundary Γ = of class C 2. We denote by x = (x 1,..., x n ) a point of while t stands for the time variable. The displacement-vector is denoted by u = (u 1,..., u n )(u i = u i (x, t), i = 1,.., n) and the temperature by θ = θ(x, t). In the absence of exterior forces and heat sources the linear system of thermoelasticity is as follows: u tt µ u (λ + µ) divu + α θ = θ t θ + βdivu t = u = θ = u(x, ) = u (x), u t (x, ) = u 1 (x), θ(x, ) = θ (x) in (, ) in (, ) on Γ (, ) in where µ, λ are Lamé s constants and α, β the coupling parameters. It is well known that system (1.1) is well-posed in H = (H 1 ()) n (L 2 ()) n L 2 (). 1 (1.1) More precisely, for every initial data (u, u 1, θ ) H there exists a unique solution This solution is given by (u, u t, θ) C([, ); H). (u(t), u t (t), θ(t)) = S(t)(u, u 1, θ ), t where S(t) : H H, t , is the strongly continuous semigroup generated by system (1.1). We will denote by S i (t), i = 1, 2, 3 the three components of S(t). On the other hand, the energy E(t) = 1 2 [ u t 2 +µ u 2 +(λ + µ) divu 2 + αβ θ2 ] dx (1.2) satisfies de(t) dt = α β θ 2 dx We fix a control time T and a control region ; an open and non-empty subset of. We are allowed to act on the system through the equations of displacement by means of a control function f = f(x, t) (L 2 ( (, T ))) n that represents an exterior force. The support of the control is restricted to the control region. In the sequel, by χ we denote the characteristic function of the set. The thermoelastic system in the presence of the control f reads as follows u tt µ u (λ + µ) divu + α θ = fχ θ t θ + βdivu t = u = θ = u() = u, u t () = u 1, θ() = θ. in (, T ) in (, T ) on Γ (, T ) (1.3) It is well known that system (1.3) possesses an unique solution (u, u t, θ) C([, T ]; H) given in terms of the semigroup S by the variation of constants formula. The first controllability problem one may consider is the exact controllability one: Find sufficient conditions on (, T ) (control region and control time) such that for every initial and final data (u, u 1, θ ), (v, v 1, η ) H there exists a control f (L 2 ( (, T ))) n, such that the solution of (1.3) satisfies u(t ) = v, u t (T ) = v 1, θ(t ) = η. 2 Due to the irreversibility of the system of thermoelasticity and the regularizing effect of the heat equation that the temperature satisfies this exact controllability property may not hold. Therefore, it is natural to relax the problem to the following exact-approximate controllability one:given (u, u 1, θ ), (v, v 1, η ) H and ε , to find a control f such that the solution of (1.3) satisfies { u(t ) = v, u t (T ) = v 1 θ(t ) η (1.4) L2 () ε. In other words, we request the exact controllability of the displacement and the approximate controllability of the temperature. In the sequel, if this property holds we will say that system (1.3) is exact-approximately controllable. We have the following result. Theorem 1. Let be a neighborhood of the boundary Γ in, i. e. = Θ where Θ is a neighborhood of Γ in IR n. Suppose that T diam( \ )/ µ. Then, system (1.3) is exact-approximately controllable in time T. One of the main ingredients of the proof of Theorem 1 is the following observability inequality for the adjoint system of thermoelasticity: ψ t ψ αdivϕ = ϕ =, ψ = ϕ tt µ ϕ (λ + µ) div ϕ + β ψ t = ϕ(x, T ) = ϕ (x), ϕ t (x, T ) = ϕ 1 (x) ψ(x, T ) = ψ (x) in (, T ) in (, T ) on (, T ) where ϕ = (ϕ 1,..., ϕ n ) is the adjoint displacement variable and ψ the temperature. in in (1.5) Proposition 1. Under the assumptions of Theorem 1, for every bounded set B of L 2 () there exists δ = δ(b) such that δ holds for every solution of (1.5) with initial data such that ϕ 2 dxdt (1.6) (ϕ, ϕ 1 + β ψ ) (L2 ()) n (H 1 ()) n 1, ψ B. By suitably adapting the methods developed by J. L. Lions [Li3] and C. Fabre, J. P. Puel and the author in [FPuZ1,2,3,4] we will show that Theorem 1 is a consequence of Proposition 1 and Hölmgren s Uniqueness Theorem. To prove Proposition 1 we will combine multiplier techniques, compactness arguments, Hölmgren s Uniqueness Theorem and the following deep result due to D. Henry, O. Lopes and A. Perissinitto [HeLP]: 3 Theorem A [HeLP]. Let P be the orthogonal projection from (L 2 ()) n into F = { ϕ : ϕ H 1 ()} and let us denote by {S (t)} t the strongly continuous semigroup in H associated to the following decoupled system u tt µ u (λ + µ) divu + αβp u t = θ t θ + βdivu t = u = θ = u() = u, u t () = u 1, θ() = θ in (, ) in (, ) on Γ (, ) in Then, S(t) S (t) : H C([, T ]; H) is continuous and compact. (1.7) We will denote by Si (t), i = 1, 2, 3 the three components of S (t). Let us consider now the one-dimensional problem. Suppose that = (, L) IR and = (, l 2 ) (, L). The system of thermoelasticity in the presence of a control f reeds now as follows: u tt u xx + αθ x = fχ, x L, t T θ t θ xx + βu xt =, x L, t T (1.1) u(, t) = u(l, t) = θ(, t) = θ(l, t) =, t T u(x, ) = u (x), u t (x, ) = u 1 (x), θ(x, ) = θ (x), x L Notice that u is now a scalar-valued function. In this one-simensional case we have the following result: Theorem 2. Suppose that T 2 max(, L l 2 ), then system (1.1) is exact-approximately controllable at time T. The proof of Theorem 2 is simpler since the decoupling operator P of Theorem A is now: P ϕ = ϕ 1 L L ϕdx, ϕ L 2 (, L). Let us compare our results with those existing in the literature. To our knowledge, the first results on controllability of thermoelastic systems are due to K. Narukawa [N]. But in [N] as well as in the more recent works by J. Lagnese [La], J. Lagnese and J. L. Lions [LaLi] and J. L. Lions [Li2], only partial controllability results are established. More precisely, in these works various models of thermoelasticity are considered and it is proved that, when the coupling parameters are small enough, one may control exactly the displacement by means of one control acting in the equations of displacement but nothing is said about the controllability of the temperature. The results of the present paper prove that, without any restriction on the size of the coupling parameters, in addition to the exact controllability of the displacement, one may achieve the approximate controllability of the temperature by means of one sole control. More recently, S. Hansen [H] has proved the null controllability of the system of the thermoelasticity in one space dimension by means of one sole boundary-control, for various boundary conditions and without restrictions in the size of the coupling parameters. In [H] the controllability of both displacement 4 and temperature is proved. The methods of [H] are based on moment problems and nonharmonic Fourier series and they do not seem to extend to several space dimensions. The extension of Hansen s results to several space dimensions (i. e. boundary control problem) remains to be done. The rest of the paper is organized as follows. In Section 2 we will give some consequences of Hölmgren s Uniqueness Theorem that we will used in the proof of our results. In Section 3 we prove the observability inequality of Proposition 1. In Section 4 we proof the controllability result (Theorem 1). In Section 5 we address the one-dimensional problem. In Section 6 we comment some possible extensions of the results of this paper and some open problems. Finally in an Appendix, for the sake of completeness, we present a brief sketch of the proof of Theorem A of [HeLP] 2. UNIQUENESS RESULTS We have the following preliminary lemma. Lemma 2.1. Let (u, θ) be a solution of the system of thermoelasticity in the set A = s t/2 {B(, 1 + µs) (s, T s)}. Suppose that there exists a vector e IR n and a real number c IR such that (u, θ) = (e, c) in B(, 1) (, T ). Then (u, θ) = (e, c) in A. Remark. By B(x, r) we denote the ball of radious r centered at x IR n. This lemma shows that the level sets of solutions of the system of thermoelasticity expand as fast as in the scalar wave equation v tt µ v =. By translation invariance (in space and time) and by scaling we deduce that if (u, θ) = (e, c) in B(x, ρ) (t 1, t 2 ) then (u, θ) = (e, c) in s (t 2 t 1 )/2 {B(x, ρ + µs) (t 1 + s, t 2 s)}. Proof of Lemma 2.1. Without loss of generality we may assume that e = and c =. We observe that v=curl u satisfies v tt µ v =. Then, by Hölmgren s Uniqueness Theorem (see, for instance, J. L. Lions [Li1]) we deduce that v = in A. Therefore, there exists a scalar function Φ = Φ(x, t) such that Now we observe that w = divu and θ satisfy u = Φ in A. (2.1) { wtt (λ + 2µ) w + α θ = θ t θ + βw t = 5 in A in A. (2.2) We apply to system (2.2) an argument due to A. Haraux [Ha]. Let us denote by ϕ k and ρ k the eigenfunctions and eigenvalues of the Laplace-Beltrami operator on the sphere S n of IR n. Multiplying in (2.2) by ϕ k and integrating on S n we deduce that w k = w k (r, t) = S n wϕ k dσ and θ k = θ k (r, t) = S n θϕ k dσ satisfy in [ w k,tt (λ + 2µ) [ θ k,t w k,rr + n 1 r θ k,rr + n 1 θ k,r ρ k r r 2 θ k B = s t/2 w k,r ρ ] k r 2 w k ] + βw k,t =. [ + α {(, 1 + µs) (s, T s)}. θ k,rr + n 1 θ k,r ρ k r r 2 θ k ] = (2.3) Since (w k, θ k ) = (, ) for r 1 and t T it is sufficient to consider the system (2.3) in the region B = s t/2 {(1/2, 1 + µs) (s, T s)} where its coefficients are analytic. The characteristic lines of system (2.3) are as follows t = constant, t = ± r + r. λ + 2µ They correspond to the underlying heat and wave equations of system (2.3), respectively. By Hölmgren s Uniqueness Theorem (see F. John [J]) we deduce that (w k, θ k ) = (, ) in B (and, thus, in B) for all k IN. Thus (divu, θ) = (w, θ) = (, ) in A. (2.4) Therefore, we already have θ = in A. But, on the other hand, by (2.1) and (2.4): w = divu = Φ = in A. (2.5) Since u = Φ and u = in B(, 1) (, T ) we have that Φ = f(t) in B(, 1) (, T ). In virtue of (2.5) we deduce that Φ = f(t) in A, but then u = Φ = in A. As an immediate consequence of Lemma 2.1 we have the following result. Corollary 2.2. Suppose that u = in B(, 1) (, T ). such that (u, θ) = (, c) in A. Then, there exists some c IR Proof: Note that since u = in B(, 1) (, T ) then, by the equations of displacement and temperature: θ = θ t θ = in B(, 1) (, T ). 6 This implies the existence of c IR such that θ = c in B(, 1) (, T ). The corollary is now a direct consequence of Lemma 2.1. Now we state the main uniqueness result for the system of thermoelasticity: Proposition 2.3. Suppose that T diam( \ )/ µ. Let (u, θ) be a solution of the Dirichlet problem for the system of thermoelasticity such that u = in (, T ). Then u θ in (, T ). Proof. In view of Corollary 2.2 (see J. L. Lions [Li1], Chapter 1) we deduce the existence of c IR such that (u, θ) = (, c) in (diam( \ )/ µ, T diam( \ )/ µ). Since θ = on Γ (, T ) we deduce that c =. By forward uniqueness we obtain that (u, θ) = (, ) in (diam( \ )/ µ, T ). By backward uniqueness (see J. L. Lions and B. Malgrange [LiM]) we deduce that This concludes the proof of the proposition. (u, θ) = (, ) in (, diam( \ )/ µ). In the one-dimensional frame of Theorem 2 we have the following uniqueness result: Proposition 2.4. Suppose that T 2 max(, L l 2 ). Let (u, θ) be a solution of (1.1) with f such that u = in (, l 2 ) (, T ). Then u θ in (, L) (, T ). Let us now consider the adjoint system: ϕ tt µ ϕ (λ + µ) divϕ + β σ t = σ t σ αdivϕ = ϕ = σ = in (, T ) in (, T ) on Γ (, T ). (2.6) The following uniqueness result will be one of the main ingredients of the proof of the controllability result: Proposition 2.5. Suppose that T diam( \ )/ µ. Let (ϕ, σ) be a solution of system (2.6) such that ϕ = in (, T ). Then ϕ σ in (, T ). Proof: Reversing the time variable we get ϕ tt µ ϕ (λ + µ) div ϕ β σ t = σ t σ αdiv ϕ = ϕ = σ = where ϕ(x, t) = ϕ(x, T t) and σ(x, t) = σ(x, T t). 7 in (, T ) in (, T ) on Γ (, T ) Note that ( ϕ, σ t ) are solutions of the system of thermoelasticity (1.3) where α ( resp. β) has been replaced by β (resp. α). Thus, as a consequence of Proposition 2.3 we get ϕ = and σ t =. On the other hand, it is easy to see that if ϕ = and σ = σ(x) then, necessarily, σ =. This concludes the proof of this proposition. Let us finally consider the adjoint system in one space dimension: ϕ tt ϕ xx + βσ xt = in (, L) (, T ) σ t σ xx αϕ x = in (, L) (, T ) ϕ(, t) = ϕ(l, t) = σ(, t) = σ(l, t) = for t (, T ) The corresponding uniqueness result is as follows: (2.7) Proposition 2.6. Suppose that T 2 max(, L l 2 ). Then, if (ϕ, σ) solves (2.7) and ϕ = in (, l 2 ) (, T ) we have (ϕ, σ) = (, ) in (, L) (, T ). Remark. In the statements above we have not made any regularity assumption on the solutions. These results apply to any weak solutions in the sense of distributions (of course, the boundary conditions, if any, have to be incorporated in the weak formulation). In the sequel we will apply these uniqueness results to solutions of the adjoint system of thermoelasticity with data (at time t = T ) in (L 2 ()) n (H 1 ()) n L 2 (). 3. THE OBSERVABILITY INEQUALITY This section is devoted to the proof of Proposition 1. We will use the notation H = (L 2 ()) n (H 1 ()) n L 2 () First we observe that if (ϕ, ψ) solves (1.5), then (φ, ψ), where φ(x, t) = ϕ(x, s) ds + χ(x) t with { µ χ (λ + µ) divχ = ϕ 1 β ψ χ = solves φ tt µ φ (λ + µ) divφ + β ψ = ψ t ψ αdivφ t = φ =, ψ = φ(x, T ) = χ(x), φ t (x, T ) = ϕ (x), ψ(x, T ) = ψ (x) in on, in (, T ) in (, T ) on (, T ) in (3.1) Taking into account that χ (H 1 ()) and n ϕ1 +β ψ (H 1 ()) n are equivalent norms, we see that Proposition 1 is equivalent to the following one: Proposition 3.1. Under the assumptions of Theorem 1, for every bounded set B of L 2 () there exists δ = δ(b) such that δ φ t 2 dxdt (3.2) 8 holds for every solution of (3.1) with initial data such that (χ, ϕ ) (H 1 ()) n (L 2 ()) n 1, ψ B. (3.3) Thus, it is sufficient to prove Proposition 3.1. To prove Proposition 3.1, first we introduce the decoupled system associated to (3.1): φ tt µ φ (λ + µ) div φ αβp φ t = ψ t ψ αdiv φ t = φ =, ψ = φ(x, T ) = χ(x), φ t (x, T ) = ϕ (x), ψ(x, T ) = ψ (x) in (, T ) in (, T ) on (, T ) in (3.4) and consider the subsystem that φ satisfies: φ tt µ φ (λ + µ) div φ αβp φ t = φ = φ(x, T ) = χ(x), φ t (x, T ) = ϕ (x) in (, T ) on (, T ) in. (3.5) We have the following observability inequality for system (3.5). Proposition 3.2. Suppose that T diam( \ )/ µ. Then, there exists a constant C and a semi-norm X : (H 1 ()) n (L 2 ()) n IR + such that χ 2 (H 1 ())n + ϕ 2 (L 2 ()) n C[ φ t 2 dxdt + X 2 (χ, ϕ )] (3.6) holds for every solution of (3.5), X : (H 1 ()) n (L 2 ()) n IR + being continuous and compact. The proof of this proposition will be given at the end of this section. Let us now conclude the proof of Proposition 3.1 by assuming that Proposition 3.2 holds. We decompose the solution of (3.1) as (φ, ψ) = ( φ, ψ) + (ξ, η) where ( φ, ψ) solves (3.4) and (ξ, η) satisfies ξ tt µ ξ (λ + µ) divξ = αβp φ t β ψ η t η αdivξ t = ξ =, η = ξ(x, T ) = ξ t (x, T ) =, η(x, T ) = As a consequence of Proposition 3.2 we have χ 2 (H 1 ())n + ϕ 2 (L 2 ()) n C[ 9 in (, T ) in (, T ) on (, T ) in (3.7) ( φ t 2 + ξ t 2 )dxdt + X 2 (χ, ϕ )]. (3.8) We argue by contradiction. Suppose that Proposition 3.1 does not hold. Then, there exists a bounded set B of L 2 () and a sequence of initial data (χ j, ϕ j, ψ j ) with ψ j B satisfying (3.3) such that φ j,t 2 dxdt as j. (3.9) In view of (3.8) and taking into account (3.9) and that (χ j, ϕ j ) (H 1 ())n (L 2 ()) n 1 holds, we deduce that We introduce the normalized data lim inf[ ξ j,t 2 dxdt + X 2 (χ j, ϕ j)] . (3.1) j (ˆχ j, ˆϕ j, ˆψ j ) = (χ j, ϕ j, ψ j )/[ ξ j,t (L 2 ( (,T ))) n + X2 (χ j, ϕ j)] 1/2 and the corresponding solutions ( ˆφ j, ˆψ j ) and (ˆξ j, ˆη j ) of (3.1) and (3.7). We have then ˆξ j,t 2 dxdt + X 2 (ˆχ j, ˆϕ j) = 1, j 1; ˆφ j,t 2 dxdt as j. (3.11) In view of (3.8) we deduce that (ˆχ j, ˆϕ j) (H 1 ()) n (L 2 ()) n C. On the other hand, by (3.1) and taking into account that ψ j B we know that ( ˆψ j ) remains in a bounded set ˆB of L 2 (). By extracting subsequences we deduce that and { (ˆχj, ˆϕ j) (ˆχ, ˆϕ ) ˆψ j ˆψ { ˆφj,t ˆφ t ˆξ j,t ˆξ t weakly in (H 1 ()) n (L 2 ()) n weakly in L 2 () weakly in (L 2 ( (, T ))) n (3.12) weakly in (L 2 ( (, T ))) n as j, where ( ˆϕ, ˆψ), ( ˆφ, ˆψ) and (ˆξ, ˆη) are the solutions of (1.5), (3.1) and (3.7) corresponding to the limit initial data. On the other hand, in virtue of Theorem A we know that (ˆξ j,t ) is relatively compact in C([, T ]; (L 2 ()) n ) and therefore ˆξ j,t ˆξ t strongly in (L 2 ( (, T ))) n. (3.13) 1 As a consequence of (3.11) and (3.12) we deduce that ˆϕ = ˆφ t = in (, T ). (3.14) In view of (3.14) and applying Proposition 2.5 we obtain that ( ˆϕ, ˆψ) in (, T ) and therefore ( ˆϕ, ˆϕ 1, ˆψ ). (3.15) This implies that ˆξ. (3.16) However, combining (3.11), (3.13) and the fact that X : (H 1 ()) n (L 2 ()) n IR + is compact we deduce that and this contradicts (3.15)-(3.16). Proof of Proposition 3.2 ˆξ t 2 (L 2 ( (,T ))) n + X2 (ˆχ, ˆϕ ) = 1. (3.17) To simplify the notation we shall denote by φ the solution of system (3.5) and by (φ, φ 1 ) its initial data. In the sequel, by X(φ, φ 1 ) we denote a generic term in our estimates that is continuous and compact from (H 1 () n (L 2 ()) n into IR + and that may change from line to line. We proced in several steps. Step 1. Estimates near the boundary. Let ρ = ρ(x) be a non-negative smooth function such that ρ in, ρ = 1 on Γ, ρ = in \ (3.18) ρ/ρ 1/2 L (). (3.19) This function is easy to construct. It is sufficient to take ρ = ρ 2 where ρ is any function satisfying (3.18). Let us also consider a non-negative function h : [, T ] IR + such that h = 1 in [ε, T ε], h() = h(t ) = with ε such that T diam( \ )/ µ + 2ε. Multiplying in (3.5) by ρ(x)h(t)φ(x, t) and integrating by parts in (, T ) we obtain: ρ(x)h(t)(µ φ 2 +(λ + µ) divφ 2 )dxdt ρ φ t 2 dxdt + X 2 (φ, φ 1 ). This implies that for any open subset of such that cl( ) and for any ε ε we have ε (µ φ 2 +(λ + µ) divφ 2 )dxdt φ t 2 dxdt + Y 2 (φ, φ 1 ). ε 11 Since T diam( \ )/ µ + 2ε we can find and ε verifying the conditions above and such that T diam( \ )/ µ + 2ε. Denoting by for the sake of simplicity and taking into account that the system is translation invariant with respect to time, we see that it suffices to obtain the following estimate: (φ, φ 1 ) 2 (H 1 ())n (L 2 ()) n C ( φ 2 + φ t 2 )dxdt + X 2 (φ, φ 1 ) (3.2) Let us consider now a function φ = ρφ with ρ as above.then, φ verifies: φ tt µ φ (λ + µ) div φ αβp φ t = Z 1 + αβz 2 in (, T ) φ = on Γ (, T ) φ(t ) = ρφ, φ t (T ) = ρφ 1 in (3.21) with and Z 1 = µ[2 ρ φ + ρφ] + (λ + µ)[ ρdivφ + ( ρ φ)] Z 2 = ( ) 1 [ ρ φ t + 2 ρ (( ) 1 (divφ t )) + ρ( ) 1 (divφ t )] ρ(( ) 1 (divφ t )). Since system (3.21) is well posed in (H 1 () n (L 2 ()) n and Z 1 + αβz 2 2 L 1 (,T ;(L 2 ()) n ) C ( φ 2 + φ t 2 )dxdt +

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