STABILITY OF COUPLED SYSTEMS

The exponential and asymptotic stability are studied for certain coupled systems involving unbounded linear operators and linear infinitesimal semigroup generators. Examples demonstrating the theory are also given from the field of partial differential equations.


Introduction
In [1] and [2], the authors have studied a system which consists in a coupling of the wave equation with the heat equation.They showed that, under some conditions, one has exponential stability or asymptotic stability.Another system has been studied in [3].Stability results are also proved there.Our goal, in this paper, is to give results for the stability of more general systems than those cited.
The system we want to study is: ( where A, B and C are unbounded operators on complex Hilbert spaces which will be precisely defined in what follows.This work is divided into three parts.In the second section, sufficient conditions are given which insure the exponential stability of the semigroup associated to system (1).The third section is devoted to the study of the asymptotic stability.Applications to some examples end this paper.

Exponential stability
In this section, we set the following assumption:

Assumption (E.S) 1) (A, D(A)
) is a generator of an exponentially stable semigroup of contractions (S A (t)) on the Hilbert space X: there exist ω A > 0 and M A > 0 such that S A (t) X ≤ M A e −tω A ∀t ≥ 0; 2) (C, D(C)) is self-adjoint on the Hilbert space Y and generator of an exponentially stable semigroup (S C (t)); 3) (B, D(B)) is an operator from Y to X such that D(A) ⊂ D(B * ) and it is (−C) Proof.L is monotone since for all (u, v) ∈ D(L), one has: So, using assumptions (E.S 1 )-(E.S 2 ): The assumption (E.S 4 ) implies that L is maximal.The Lumer-Phillips theorem gives the result.

Let us denote by
, t ≥ 0, a solution of (2).One has: Theorem 2. The semigroup S L (t) generated by (L, D(L)) is exponentially stable: there exist ω > 0 and M > 0 such that Before proving the theorem, let us recall (see Pritchard and Zabczyk [11]) that a semigroup (S(t)) t generated on a Hilbert space (H, , H ) by an operator (A, D(A)) is exponentially stable if and only if there exists a bounded, positive and selfadjoint operator P on H such that: This last equation (in P ) is called the Lyapunov equation (E.L) and the solution P the Lyapunov operator.As the semigroups associated to (A, D(A)) and (C, D(C)) are exponentially stable, we denote by P A and P C their associated Lyapunov operators.One has: In fact, as the operator (C, D(C)) is selfadjoint and its semigroup exponentially stable, the associated Lyapunov operator

So we have:
Lemma 3.There exists ε 0 > 0 such that for all ε ≤ ε 0 and all Y 0 ∈ D(L), the functional Y is such that there exists D(ε) > 0 for which Proof.We will prove that, for sufficiently small ε > 0, there exists a positive constant D(ε) such that Evaluating the derivative of ρ ε (t) , we get from system (2) and from (3) Now we estimate the products using the assumptions on B and C: X , where α is a positive constant.We recall that the assumption (E.S 3 ) implies that all the quantities of the right member are bounded.
Next, we estimate the last term: Grouping these estimates together (with the use of the dissipativity of A), we get If we choose α < α 0 , where and ε < ε 0 , where and the two last inequalities give where This proves the exponential decay of ρ ε (t).Lemma 4. For ε > 0, the application Proof.One has the inequalities The first inequality is obvious.The second is (4).
Proof of the theorem.Lemmas 3 and 4 prove the theorem since we deduce from them that, ∀t ≥ 0, we have , which proves the exponential decay and gives an estimation of the decay rate.

The asymptotic stability
In this section, the assumption (E.S) is replaced by Assumption (A.S) 1) (A, D(A)) has a compact resolvent and is a generator of contraction semigroup (S A (t)) on the Hilbert space X; 2) (C, D(C)) has a compact resolvent, is self-adjoint on the Hilbert space Y and generator of an exponentially stable C 0 −semigroup (S C (t)); 3) i) D(C) ⊂ D(B) and there exist two constants α C and δ C with ) and there exist two constants α A and δ A with Remark 6.The assumption (A.S 3 ) is natural because the operator L can be written as follows Clearly D is a generator of contraction semigroup and since E is dissipative, L is also a generator of contraction semigroup (see for example [10], Corollary 3.3, p. 82). For let us denote by γ(Y 0 ) its orbit and by ω(Y 0 ) its ω − limit set : and, using the assumption (A.S 3 ), we get where K is a positive constant.This proves the proposition for Y 0 ∈ D(L 2 ) and, by a density argument, we obtain the result for all Y 0 ∈ D(L).

Corollary 8. For all
Proof.One has Since we assumed that the constants α A and α C are less than one, from the last proposition we deduce that Au X and Cv Y are bounded in X and Y respectively.As A and C have compact resolvent, we have the result.

Theorem 9. Under the assumption (A.S), if the unique solution of the system
and Since S L (t n )LY 0 is relatively compact, there exists a subsequence (t Thus, v ≡ 0 and u verifies The assumption of the theorem implies that u ≡ 0. This proves that Consequently, Y = 0 and ω(Y 0 ) = {0}.Since S L (t) is uniformly bounded and D(L 2 ) is dense in X × Y , the result follows.

Applications
As a first example, consider the problem where Ω is an open bounded subset of R n with a smooth boundary, a ∈ L ∞ (Ω) and (b, D(b)) is a linear operator on L 2 (Ω) which will be precisely specified later.The system (7) can be written formally as follows: (Ω) (the energy space equipped with the product norm denoted by .) and Then, assuming the following geometrical control property (see [4]) Ω 0 ⊂ Ω is open and there exists T > 0 with the property that every geodesic (ray) of length T meets Ω 0 .
As a direct consequence of the results of the previous sections, one has Theorem 10.The following statements are true: where Ω 0 is open and satisfies the geometrical control property and b is (−∆) Proof.From the results of [4], A generates an exponentially stable semigroup in case (i).The other assumptions of theorem 2 are easily derived.Property (ii) is a direct consequence of theorem 9.

Remark 11. (a)
This result is true, in particular, if b is a bounded operator in L 2 (Ω).In this case, one cannot expect exponential stability if a ≡ 0 and b is a constant and, in a sense, (ii) in theorem 10 is optimal.More precisely, when a and b are constants, we have (see [1] and [2]) Proposition 12.The eigenvalues (λ 3k+j ) of L satisfy the following properties (i) for sufficiently large k, one of the eigenvalues (denoted by λ 3k ) is real and the two others are complex; Re(λ 3k+j ) < − a 2 for all k and j = 1,2 (iii) L satisfies the spectrum determined growth assumption and for every b ∈ R , there exists where the µ k are the eigenvalues of (−∆) and , where d is a vector of R n , the result still holds.The interest here is that, in the one-dimensional case (the one-dimensional thermoelasticity system), the exponential stability remains even if a ≡ 0 (see [8]).
The second example deals with the boundary stabilization of the thermoelasticity system.Let Ω an open bounded set in R n with smooth boundary Γ, where u = [(u 1 , . . ., u n )] (resp.w) is the displacement (resp.the temperature) of the system, ν(x) is the exterior normal to Γ at x, α > 0 is the coupling parameter and x 0 is a fixed point in R n (n ≤ 3 for physical cases) (see [7]), We assume that and set On this space, we define the scalar product The energy space will be the Hilbert space and we denote by • the induced norm on H. |.| will denote the L 2 -norm. Let One has the following: Assumption (E.S) is satisfied if a geometrical hypothesis on Ω (see below in the proof ) is added which insures the validity of Grisvard's integral formula.
Proof.Following Grisvard [5], in the case n = 2 one has the Green formula: • τ S for all u in D(∆) (with mixed boundary conditions) where u − S∈Σ c S u S is in H 2 (Ω), u S being the first singularities of the Laplace operator.In this formula, Σ is a finite set (the "vertexes") and (Σ c , Σ s ) is a partition of Σ defined by (i) S ∈ Σ c if S is downstream of Γ 1 following the positive orientation of Γ; (ii) S ∈ Σ s if S is downstream of Γ 0 following the positive orientation of Γ.In this case, the geometric condition on Ω is This last condition implies the inequality The exponential stability of the semigroup whose generator is A is then proved as in [6].The operator B is (−∆) 1/2 -bounded and D(A) ⊂ D(B * ).Note that the last (crucial) inequality remains true in higher dimension with similar geometric restriction on Ω (see [5] for n = 3 and [M] for n ≥ 3) and thus the semigroup is also exponentially stable.
Theorem 13.Under the previous assumption, (S L (t)) is exponentially stable.
We give, in what follows, a second proof, more constructive, which insures an estimate of ω L , the constant of the exponential decay of S L (t). (10).We define, for all t ≥ 0, the function where and ∆ −1 is the inverse of ∆ considered as an operator of L 2 (Ω) with domain )), we have, for all i = 1, ..., n, From these three last relations, we get (12) On the other hand, and then, using the second equation in (10), we obtain Thus, from ( 12)-( 14), (15) where, in this last inequality, we used the notation Let us denote by I 0 the integral on Γ 0 in inequality (15).It is estimated as in [6] by ( 16) For the next terms of (15), let β > 0 any constant.One has Using first Green's formula, we obtain for the third integral With the inequalities ( 16)-( 19), we conclude that We choose β > 0 such that But, on the other hand, for all t ≥ 0, As we get the result with 2ω ≥ εC 2 (β) ( 1 2 + 1 2 ε max(C 0 , 2R + (n − 1)C 0 )) and ε verifying the condition: (1 − ε(2R + (n − 1)C 0 ) > 0.