Stabilization of the wave equation with a delay term in the boundary or internal feedbacks

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1 Stabilization of the wave equation with a delay term in the boundary or internal feedbacks Serge Nicaise Université de Valenciennes et du Hainaut Cambrésis MACS, Institut des Sciences et Techniques de Valenciennes Valenciennes Cedex 9 France Cristina Pignotti Dipartimento di Matematica Pura e Applicata Università di L Aquila Via Vetoio, Loc. Coppito, 671 L Aquila Italy Abstract In this paper we consider, in a bounded and smooth domain, the wave equation with a delay term in the boundary condition. We also consider the wave equation with a delayed velocity term and mixed Dirichlet-Neumann boundary condition. In both cases, under suitable assumptions, we prove exponential stability of the solution. These results are obtained by introducing suitable energies and by using some observability inequalities. Some unstability examples are also given. Mathematics Subject Classification: 35L5, 93D15 Keywords and Phrases: wave equation, delay feedbacks, stabilization 1 Introduction We investigate the effect of time delay in boundary or internal stabilization of the wave equation in domains of IR n. Such effects arise in many pratical problems and it is well known, at least in one dimension, that they can induce some unstabilities, see [3, 4, 14]. To our knowledge, the analysis in higher dimension is not yet done. In this paper, we give some stability results under a sufficient condition and further we show that if this condition is not satisfied, then there exist some delays for which the system is destabilized. So, in a certain sense, our sufficient condition is also necessary in order to have a general stability result. 1

2 Let IR n be an open bounded set with a boundary Γ of class C. We assume that Γ is divided into two parts Γ D and, i.e. Γ = Γ D, with Γ D = and Γ D. In this domain, we consider the initial boundary value problem u tt (x, t) u(x, t) = in (, + ) (1.1) u(x, t) = on Γ D (, + ) (1.) u ν (x, t) = µ 1u t (x, t) µ u t (x, t τ) on (, + ) (1.3) u(x, ) = u (x) and u t (x, ) = u 1 (x) in (1.4) u t (x, t τ) = f (x, t τ) in (, τ), (1.5) where ν(x) denotes the unit normal vector to the point x Γ and u is the normal ν derivative. Moreover, τ > is the time delay, µ 1 and µ are positive real numbers and the initial datum (u, u 1, f ) belongs to a suitable space. We are interested in giving an exponential stability result for such a problem. Let us denote by v, w or, equivalently, by v w the euclidean inner product between two vectors v, w IR n. We assume that there exists a scalar function v C () such that (i) v is strictly convex in, that is there exists α > such that D (v)(x)ξ, ξ α ξ, x, ξ IR n, (1.6) where D (v) denotes the Hessian matrix of v; (ii) the vector field H := v verifies H(x) ν(x), x Γ D. (1.7) For the above assumptions see [11] where some observability estimates for second order hyperbolic equations are given. It is well known that if µ =, that is in absence of delay, the energy of problem (1.1) (1.5) is exponentially decaying to zero. See for instance Chen [], Lasiecka and Triggiani [1], Lagnese [9], Komornik and Zuazua [8], Komornik [6, 7]. On the contrary, if µ 1 =, that is if we have only the delay part in the boundary condition on, system (1.1) (1.5) becomes unstable. See, for instance Datko, Lagnese and Polis [4]. Although these examples involve only one space dimension, we can expect that a similar phenomenon occurs in higher space dimension. So, it is interesting to seek a stabilization result when µ 1 and µ are both nonzero. In this case, the boundary feedback is composed of two parts and only one of them has a delay. This problem has been studied in one space dimension by Xu, Yung and Li [14]. After a spectral analysis the authors have proved a stability result if µ < µ 1. In their paper it is also shown that if µ > µ 1 the system is unstable and if µ 1 = µ some unstabilities may occur. Here, coherently with [14], assuming that µ < µ 1 (1.8)

3 we obtain a stabilization result in general space dimension, by using a suitable observability estimate. This is done by applying inequalities obtained from Carleman estimates for the wave equation by Lasiecka, Triggiani and Yao in [11] and by using compactnessuniqueness arguments. If µ 1 = µ, we further show that there exists a sequence of arbitrary small (and large) delays such that unstabilities occur. In the case µ > µ 1, we also obtain delays which destabilize the system. In this paper we also study the problem for the wave equation with an internal feedback. In particular, we consider the system u tt (x, t) u(x, t) + a(x)[µ 1 u t (x, t) + µ u t (x, t τ)] = in (, + )(1.9) u(x, t) = on Γ D (, + ) (1.1) u ν (x, t) = on (, + ) (1.11) u(x, ) = u (x) and u t (x, ) = u 1 (x) in (1.1) u t (x, t τ) = g (x, t τ) in (, τ), (1.13) where a L () is a function such that and a(x) a. e. in, (1.14) a(x) > a >, a. e. in ω, (1.15) where ω is an open neighbourhood of. Exponential stability results for the above problem in the case of µ =, that is without delay, have been obtained by several authors. See for instance Zuazua [15], Liu [13]. On the contrary, at least for the one dimensional case, Datko [3] has shown that wave equation with a velocity term and mixed Dirichlet Neumann boundary condition is destabilized by a time delay in the velocity term. In this paper, in the case µ < µ 1, we show that the energy is exponentially decaying to zero. This is done, as for the problem with boundary feedback, by using a suitable observability estimate. If µ µ 1, we obtain an explicit sequence of arbitrary small delays that destabilize the system. Remark 1.1 In [11] the authors, in order to deal with variable coefficients, assume that there exists a scalar function v strictly convex with respect to the Riemannian metric induced by the spatial operator. Here, we are principally interested in the effect of the delay term in the boundary or internal feedback. So, in order to avoid technicalities, we consider constant coefficients. Actually, our stability results hold even for variable coefficients under the assumption of [11]. The paper is organized as follows. Well posedness of the problems is analysed in section using semigroup theory. In subsection.1 we study the well-posedness of problem 3

4 (1.1) (1.5), while in subsection. we concentrate on problem (1.9) (1.13). In section 3 and section 4 we prove the exponential stability of the problem with boundary and internal feedbacks respectively. Finally, section 5 is devoted to some unstability examples. Acknowledgment. We thank E. Zuazua who bringed our attention on reference [14] and suggested us to consider the stabilization of the wave equation with delay in general domains of IR n. Well-posedness of the problems In this section we will give well posedness results for problem (1.1) (1.5) and for problem (1.9) (1.13) using semigroup theory..1 Boundary feedback Let us set z(x, ρ, t) = u t (x, t τρ), x, ρ (, 1), t >. (.1) Then, problem (1.1) (1.5) is equivalent to If we denote by then u tt (x, t) u(x, t) = in (, + ) (.) τz t (x, ρ, t) + z ρ (x, ρ, t) = in (, 1) (, + ) (.3) u(x, t) = on Γ D (, + ) (.4) u ν (x, t) = µ 1u t (x, t) µ z(x, 1, t) on (, + ) (.5) z(x,, t) = u t (x, t) on (, ) (.6) u(x, ) = u (x) and u t (x, ) = u 1 (x) in (.7) z(x, ρ, ) = f (x, ρτ) in (, 1). (.8) U := (u, u t, z) T, U := (u t, u tt, z t ) T = ( u t, u, τ 1 z ρ ) T. Therefore, problem (.) (.8) can be rewritten as { U = AU U() = (u, u 1, f (, τ)) T (.9) where the operator A is defined by A u v z := 4 v u τ 1 z ρ,

5 with domain D(A) := { (u, v, z) T ( E(, L ()) H 1 Γ D () ) H 1 () L ( ; H 1 (, 1)) : u ν = µ 1v µ z(, 1) on ; v = z(, ) on }, (.1) where, as usual, and H 1 Γ D () = { u H 1 () : u = on Γ D }, E(, L ()) = {u H 1 () : u L ()}. Recall that for a function u E(, L ()), then u ν belongs to H 1/ ( ) and the next Green formula is valid (see section 1.5 of [5]) u wdx = uwdx + u ν ; w, w HΓ 1 D (), (.11) where ; ΓN means the duality pairing between H 1/ ( ) and H 1/ ( ). Note further that for (u, v, z) T D(A), u ν belongs to L ( ), since z(, 1) is in L ( ). Denote by H the Hilbert space Assuming that H := H 1 Γ D () L () L ( (, 1)). (.1) we will show that A generates a C semigroup on H. Let ξ be a positive real number such that Note that, from (.13), such a constant ξ exists. Let us define on the Hilbert space H the inner product u v z, ũ ṽ z H := µ µ 1, (.13) τµ ξ τ(µ 1 µ ). (.14) { u(x) ũ(x) + v(x)ṽ(x)}dx + ξ 1 z(x, ρ) z(x, ρ)dρdγ. (.15) Theorem.1 For any initial datum U H there exists a unique solution U C([, + ), H) of problem (.9). Moreover, if U D(A), then U C([, + ), D(A)) C 1 ([, + ), H). 5

6 Proof. Take U = (u, v, z) T D(A). Then, (AU, U) = v u τ 1 z ρ = So, by Green s formula, (AU, U) =, u v z H { v(x) u(x) + v(x) u(x)}dx ξτ 1 1 Integrating by parts in ρ, we get 1 that is z ρ (x, ρ)z(x, ρ)dρdγ. u 1 1 (x)v(x)dγ ξτ z ρ (x, ρ)z(x, ρ)dρdγ. (.16) ν 1 z ρ (x, ρ)z(x, ρ)dρdγ = z ρ (x, ρ)z(x, ρ)dρdγ + {z (x, 1) z (x, )}dγ, 1 Therefore, from (.16) and (.17), z ρ (x, ρ)z(x, ρ)dρdγ = 1 {z (x, 1) z (x, )}dγ. (.17) 1 u ξτ (AU, U) = (x)v(x)dγ {z (x, 1) z (x, )}dγ ν 1 ξτ = (µ 1 v(x) + µ z(x, 1))v(x)dΓ {z ΓN (x, 1) z (x, )}dγ = µ 1 v Γ 1 ξτ (x)dγ µ z(x, 1)v(x)dΓ z N ΓN ΓN (x, 1)dΓ + from which follows, using Cauchy-Schwarz s inequality, (AU, U) ( µ 1 + µ Now, observe that from (.14), 1 ξτ v (x)dγ, ) 1 ξτ ( ) + v 1 µ ξτ (x)dγ + z (x, 1)dΓ. (.18) µ 1 + µ + ξτ 1, µ ξτ 1 Then, (AU, U), which means that the operator A is dissipative. Now, we will show that λi A is surjective for a fixed λ >. Given (f, g, h) T H, we seek U = (u, v, z) T D(A) solution of (λi A) u v z = f g h,. 6

7 that is verifying λu v = f λv u = g λz + τ 1 z ρ = h Suppose that we have found u with the appropriated regularity. Then, (.19) and we can determine z. Indeed, by (.1), and, from (.19), v := λu f (.) z(x, ) = v(x), for x, (.1) λz(x, ρ) + τ 1 z ρ (x, ρ) = h(x, ρ), for x, ρ (, 1). (.) Then, by (.1) and (.), we obtain So, from (.), ρ z(x, ρ) = v(x)e λρτ + τe λρτ h(x, σ)e λστ dσ. ρ z(x, ρ) = λu(x)e λρτ f(x)e λρτ + τe λρτ h(x, σ)e λστ dσ, on (, 1), (.3) and, in particular, with z L ( ) defined by z(x, 1) = λu(x)e λτ + z (x), x, (.4) 1 z (x) = f(x)e λτ + τe λτ h(x, σ)e λστ dσ, x. (.5) By (.) and (.19), the function u verifies that is Problem (.6) can be reformulated as (λ u u)wdx = Integrating by parts, λ(λu f) u = g, λ u u = g + λf. (.6) (g + λf)wdx, w H 1 Γ D (). (.7) (λ u u)wdx = (λ u uw + u w)dx ν wdγ = (λ uw + u w)dx + (µ 1 vw + µ z(x, 1))wdΓ = (λ uw + u w)dx + {µ 1 (λu f)w + µ (λue λτ + z )w}dγ, 7

8 where we have used (.) and (.4). Therefore, (.7) can be rewritten as (λ uw + u w)dx + (µ 1 + µ e λτ )λuwdγ = (g + λf)wdx + µ 1 fwdγ µ z wdγ, w HΓ Γ 1 D (). N (.8) As the left-hand side of (.8) is coercive on H 1 Γ D (), the Lax-Milgram lemma guarantees the existence and uniqueness of a solution u H 1 Γ D () of (.8). If we consider w D() in (.8), we have that u solves in D () λ u u = g + λf, (.9) and thus u E(, L ()). Using Green s formula (.11) in (.8) and using (.9), we obtain from which follows (µ 1 + µ e λτ )λuwdγ + u ν ; w = µ 1 fwdγ µ z wdγ, u ν + (µ 1 + µ e λτ )λu = µ 1 f µ z on. (.3) Therefore, from (.3), u ν = µ 1v µ z(, 1) on, where we have used (.) and (.4). So, we have found (u, v, z) T D(A) which verifies (.19). Now, the well posedness result follows from the Hille Yosida theorem.. Internal feedback Setting problem (1.9) (1.13) is equivalent to z(x, ρ, t) = u t (x, t τρ), x, ρ (, 1), t >, (.31) u tt u + a(x)[µ 1 u t (x, t) + µ u t (x, t τ)] = in (, + ) (.3) τz t (x, ρ, t) + z ρ (x, ρ, t) = in (, 1) (, + ) (.33) u(x, t) = on Γ D (, + ) (.34) u ν (x, t) = on (, + ) (.35) z(x,, t) = u t (x, t) on (, ) (.36) u(x, ) = u (x) and u t (x, ) = u 1 (x) in (.37) z(x, ρ, ) = g (x, ρτ) in (, 1). (.38) 8

9 then If we denote by U := (u, u t, z) T, U := (u t, u tt, z t ) T = ( u t, u a(µ 1 u t + µ z(, 1, )), τ 1 z ρ ) T. Therefore, problem (.3) (.38) can be rewritten as { U = A U U() = (u, u 1, g (, τ)) T (.39) where the operator A is defined by u A v := z with domain v u aµ 1 v aµ z(, 1) τ 1 z ρ, D(A ) := { (u, v, z) T ( H () H 1 Γ D () ) H 1 () L (; H 1 (, 1)) : u ν = on ; v = z(, ) in }. Denote by H the Hilbert space (.4) H := H 1 Γ D () L () L ( (, 1)), (.41) equipped with the inner product u ũ 1 v, ṽ := { u(x) ũ(x) + v(x)ṽ(x)}dx + ξ z(x, ρ) z(x, ρ)dρdx, z z H (.4) where ξ is a fixed positive number satisfying (.14). Arguing analogously to the previous case, we can show that the operator A generates a C semigroup on H. Consequently we have the following well-posedness result. Theorem. For any initial datum U H there exists a unique solution U C([, + ), H ) of problem (.39). Moreover, if U D(A ), then U C([, + ), D(A )) C 1 ([, + ), H ). 3 Boundary stability result In this section, in order to prove an exponential stability result for problem (1.1) (1.5), we assume (1.8). Let us define the energy as E(t) := 1 {u t (x, t) + u(x, t) }dx + ξ 9 1 u t (x, t τρ)dρdγ, (3.1)

10 where ξ is a positive constant verifying τµ < ξ < τ(µ 1 µ ). (3.) Proposition 3.1 For any regular solution of problem (1.1) (1.5) the energy is decreasing and there exists a positive constant C such that Proof. Differentiating (3.1) we obtain E (t) = E (t) C {u t (x, t) + u t (x, t τ)}dγ. (3.3) {u t u tt + u u t }dx + ξ and then, applying Green s formula, Now, observe that and Therefore, 1 E u (t) = u t ν Γ dγ + ξ N 1 1 u t (x, t τρ) = τ 1 u ρ (x, t τρ), u tt (x, t τρ) = τ u ρρ (x, t τρ). u t (x, t τρ)u tt (x, t τρ)dρdγ = τ 3 Integrating by parts in ρ, we obtain that is 1 u t (x, t τρ)u tt (x, t τρ)dρdγ, u t (x, t τρ)u tt (x, t τρ)dρdγ. (3.4) 1 u ρ (x, t τρ)u ρρ (x, t τρ)dρdγ. (3.5) 1 u ρ (x, t τρ)u ρρ (x, t τρ)dρdγ = u ρ (x, t τρ)u ρρ (x, t τρ)dρdγ + {u ρ(x, t τ) u ρ(x, t)}dγ, 1 u ρ (x, t τρ)u ρρ (x, t τρ)dρdγ = 1 Then, from (3.5) and (3.6), 1 u t (x, t τρ)u tt (x, t τρ)dρdγ = τ 3 = τ 3 ΓN {u ρ(x, t) u ρ(x, t τ)}dγ = τ 1 {u ρ(x, t τ) u ρ(x, t)}dγ. (3.6) 1 u ρ (x, t τ)u ρρ (x, t τρ)dρdγ {u t (x, t) u t (x, t τ)}dγ. (3.7) 1

11 Using (3.4), (3.7) and the boundary condition (1.3) on, we have E (t) = µ 1 u Γ t (x, t)dγ µ u t (x, t)u t (x, t τ)dγ N 1 ξτ + u ΓN t 1 ξτ (x, t)dγ u t (x, t τ)dγ. (3.8) From (3.8), applying Cauchy-Schwarz s inequality, we obtain E (t) which implies with ( µ 1 + µ ) 1 ξτ ( ) + u 1 µ ξτ t (x, t)dγ + u t (x, t τ)dγ, E (t) C {u t (x, t) + u t (x, t τ)}dγ, C = min { (µ1 µ 1 ξτ ) ( µ, 1 ξτ ) } +. Since ξ is choosen satisfying assumption (3.), the constant C is positive. We can write E(t) = E(t) + E N (t), where E(t) is the standard energy for the wave equation and E(t) := 1 E N (t) := ξ {u t (x, t) + u(x, t) }dx, (3.9) With a change of variable we can rewrite E N (t) = ξ τ 1 u t (x, t τρ)dρdγ. (3.1) t t τ u t (x, s)dsdγ. (3.11) We can now give a boundary observability inequality which we will use to prove the exponential decay of the energy E(t). Proposition 3. There exists a time T > such that for all times T > T there exists a positive constant C (depending on T ) for which T E() C for any regular solution u of problem (1.1) (1.5). {u t (x, t) + u t (x, t τ)}dγdt, (3.1) 11

12 Proof. From Proposition 6.3 of [11], for T greater than a sufficiently large time T, and any ε >, we have T E() c {( u) } + u ν t dγdt + c u H 1/+ε ( (, T )), (3.13) for a suitable constant c (depending on T ). Estimate (3.13) is obtained by Carleman estimates under the assumption that there exists a function v of class C satisfying (1.6) and (1.7). The function v is needed to construct a suitable weight function for Carleman estimates (see the proof of Proposition 4. below). Then, by (3.13) and the boundary condition (1.3), we have T E() c {u t (x, t) + u t (x, t τ)}dγdt + c u H 1/+ε ( (, T )), (3.14) for a suitable positive constant c. From (3.11) we have that By a change of variable in (3.15) we obtain, for T τ, E N () c u t (x, s)dsdγ. (3.15) τ T E N () c u t (x, t τ)dγdt. (3.16) Denote by T := max{τ, T }. Then, from (3.14) and (3.16), for any T > T we have E() = E() + E N () T c {u t (x, t) + u t (x, t τ)}dγdt + c u H 1/+ε ( (, T )), (3.17) for a suitable positive constant c depending on T. In order to obtain (3.1) we need to absorb the lower order term u H 1/+ε ( (, T )). To do this, we argue by contradiction. Suppose that (3.1) is not true. Then, there exists a sequence {u n } n of solutions of problem (1.1) (1.5) such that, denoting by E n () the energy E related to u n at the time, T E n () > n {u nt(x, t) + u nt(x, t τ)}dγdt. (3.18) From (3.17), we have { T } E n () c {u nt(x, t) + u nt(x, t τ)}dγdt + u n H 1/+ε. (3.19) ( (, T )) Then, from (3.18) and (3.19), T n {u nt(x, t) + u nt(x, t τ)}dγdt { T } < c {u nt(x, t) + u nt(x, t τ)}dγdt + u n H 1/+ε, ( (, T )) 1

13 that is (n c) T {u nt(x, t) + u nt(x, t τ)}dγdt < c u n H 1/+ε ( (, T )). (3.) Renormalizing, we obtain a sequence {w n } n of solutions of problem (1.1) (1.5) verifying and T w n H 1/+ε = 1, (3.1) ( (, T )) {w nt(x, t) + w nt(x, t τ)}dγdt < c n c. (3.) From (3.1), (3.) and (3.19), it follows that the sequence {w n } n is bounded in H 1 ( (, T )). Since H 1 ( (, T )) is compactly embedded in H 1/+ε ( (, T )), there exists a subsequence which, for simplicity of notation, we still denote by {w n } n, such that Then, from (3.1), Moreover, by (3.), Therefore, we have that w n w strongly in H 1/+ε ( (, T )). T w H 1/+ε = 1. (3.3) ( (, T )) {w t (x, t) + w t (x, t τ)}dγdt =. w t = on (, T ), and w ν = on (, T ). Putting v := w t, v solves in a distributional sense v v = in (, T ), with v v = on Γ (, T ), ν = on (, T ). Therefore, from Holmgren s uniqueness theorem (see [1] Chap. I, Th. 8., page 9 ) v. This implies that w(x, t) = w(x). Thus, w verifies w = in w = on Γ D w ν = on, and so w. This is in contradiction with (3.3). Then, the observability inequality (3.1) is proved. From (3.1) easily follows the stability result. 13

14 Theorem 3.3 Assume that (1.8) holds. There exist positive constants C 1, C such that, for any regular solution of problem (1.1) (1.5), Proof. From (3.3), we have E(T ) E() C E(t) C 1 E()e C t, t. (3.4) T By (3.5) and the observabilty estimate (3.1), we obtain T E(T ) E() C then {u t (x, t) + u t (x, t τ)}dγdt. (3.5) {u t (x, t) + u t (x, t τ)}dγdt C C 1 (E() E(T )), E(T ) CE(), with C < 1. This easily implies the stability estimate (3.4), since our system (1.1) (1.5) is invariant by translation and the energy E is decreasing. Remark 3.4 Analogous arguments apply if we have more than one delay term in the boundary feedback, that is if condition (1.3) is substituted by u ν (x, t) = µ u t (x, t) k µ i u t (x, t τ i ), on (, + ), i=1 with µ, µ i, τ i, i = 1,..., k, positive parameters. In this case, the right energy for our problem is E(t) := 1 {u t (x, t) + u(x, t) }dx + with suitable positive constants ξ i, i = 1,..., k. Indeed, if choosing ξ i such that k i=1 k µ > µ i, i=1 µ i < ξ i τ 1 i, i = 1,..., k, and k i=1 ξ i ξ i τ 1 i we can prove that the energy is exponentially decaying to zero. 1 u t (x, t ρτ i )dρdγ, k < µ µ i, i=1 Remark 3.5 If Γ D, the strong solution of (1.1) (1.5) has a singular behaviour along Γ D since u(t) E(, L ()) with u ν L ( ) and u = on Γ D (see for instance [5] for D domains). Therefore the results from [11] cannot be invoked. For standard feedback law (i.e. the case µ = ), the multiplier method has been used as an alternative in [1] to obtain stability results under strong geometrical conditions. Unfortunately their approach cannot be directly applied here because for µ >, we only have u ν L ( ) which forbids the use of the multiplier identity. 14

15 4 Internal stability result In this section, under the assumption (1.8), we want to prove exponential stability for problem (1.9) (1.13). Let us define the energy as F(t) := 1 {u t (x, t) + u(x, t) }dx + ξ where ξ is a positive constant verifying (3.). We first show that the energy F is decreasing. 1 a(x) u t (x, t τρ)dρdx, (4.1) Proposition 4.1 For any regular solution of problem (1.9) (1.13) the energy is decreasing and there exists a positive constant C such that F (t) C Proof. Differentiating (4.1) we obtain F (t) = {u t u tt + u u t }dx + ξ and then, applying Green s formula, F (t) = u t (u tt u)dx + ξ a(x){u t (x, t) + u t (x, t τ)}dx. (4.) 1 a(x) u t (x, t τρ)u tt (x, t τρ)dρdx, where we have used the boundary conditions (1.1) and (1.11). As in the proof of Proposition 3.1, we can compute 1 a(x) u t (x, t τρ)u tt (x, t τρ)dρdx, (4.3) 1 a(x) u t (x, t τρ)u tt (x, t τρ)dρdx 1 = τ 3 a(x) u ρ (x, t τ)u ρρ (x, t τρ)dρdx = τ 3 a(x){u ρ(x, t) u ρ(x, t τ)}dx = τ 1 a(x){u t (x, t) u t (x, t τ)}dγ. Now, using (4.4) and equation (1.9) in identity (4.3), we have F (t) = µ 1 1 ξτ + a(x)u t (x, t)dx µ a(x)u t (x, t)dx 1 ξτ a(x)u t (x, t)u t (x, t τ)dx a(x)u t (x, t τ)dx. (4.4) (4.5) From (4.5), applying Cauchy-Schwarz s inequality and recalling (3.), we obtain estimate (4.). 15

16 We can write F(t) = E(t) + E (t), where E(t) is the standard energy for the wave equation defined in (3.9) and E (t) := ξ With a change of variable we can rewrite E (t) = ξ τ 1 a(x) u t (x, t τρ)dρdx. (4.6) t a(x) u t (x, s)dsdx. (4.7) t τ Let w be the solution of the homogeneous problem for the wave equation with mixed Dirichlet Neumann boundary condition, w tt (x, t) w(x, t) = in (, + ) (4.8) w(x, t) = on Γ D (, + ) (4.9) w ν (x, t) = on (, + ) (4.1) w(x, ) = w (x) and w t (x, ) = w 1 (x) in. (4.11) Denote by E w (t) the standard energy for the wave equation corresponding to w, that is E w (t) = 1 {wt (x, t) + w(x, t) }dx. (4.1) Note that E w (t) is constant. We can give an observality inequality for problem (4.8) (4.11). Proposition 4. There exists a time T such that for all times T > T there exists a positive constant C 1 (depending on T ) for which T E w () C 1 for any regular solution w of problem (4.8) (4.11). ω w t (x, t)dxdt, (4.13) Proof. Inequality (4.13) easily follows from some estimates of [11] and standard arguments with multipliers. We give the proof for reader s convenience. Let ω, ω 1 be open neighbourhoods of such that Let ϕ be a smooth function such that ω ω ω 1. (4.14) ϕ(x) 1, ϕ on \ ω, ϕ 1, on ω 1. (4.15) 16

17 Then, the function ϕw verifies (ϕw) tt (ϕw) = F (w), where F (w) = w ϕ + ϕ w, with the same boundary conditions as w. Therefore, we can apply to ϕw the result of Proposition 4..1 in [11]. Let us recall some notations from [11]. Without loss of generality we can suppose that the function v satisfying assumptions (1.6) and (1.7) is non negative on. Denote, ( ) 1/ maxx v(x) T =, (4.16) α with α as in (1.6). Define the function φ : IR IR by φ(x, t) := v(x) c ( t T ), (4.17) where T > T is fixed and the constant c is choosen as follows. From (4.16), there exists a constant δ > such that αt > 4 max v(x) + 4δ. x For fixed δ there is c such that c T > 4 max x v(x) + 4δ, c (, α). (4.18) Note that Set, φ(x, ) < δ and φ(x, T ) < δ uniformly in. (4.19) T ( ) w dγdt ν BT w Γ (,T ) = 1 e γφ H ν Γ D + 1 T e γφ H ν(wt T w )dγdt, (4.) where T w denotes the tangential gradient of w. Then, from Proposition 4..1 of [11], using (4.19) and recalling that E w (t) is constant, we have { T T BT w Γ (,T ) c e γφ (ϕw) dxdt + T ϕ wt dxdt } + ϕ w dxdt + e γδ E w () { T T (4.1) c e γφ w dxdt + ω T wt dxdt ω } + w dxdt + e γδ E w (), for a suitable positive constant c, where the parameter γ can be choosen sufficiently large in order to have the desired inequality. 17

18 Now, consider another smooth cut off function ψ such that ψ(x) 1, ψ on \ ω, ψ 1, on ω. (4.) Integrating by parts, T [ ] T (w tt w)ψwe γφ dxdt = ψww t e γφ dx T T + w (ψwe γφ )dxdt w t (ψwe γφ ) t dxdt [ ] T T = ψww t e γφ dx ψw t (w t e γφ + we γφ γφ t )dxdt T T + ψe γφ w dxdt + w w (ψe γφ )dxdt. Then, from (4.3), recalling that w satisfies (4.8), we have T ψe γφ w dxdt = [ ψww t e γφ dx T ] T T T (ψwt e γφ + ww t ψe γφ γφ t )dxdt ψw w ( ψ)e γφ dxdt ψw w e γφ dxdt. (4.3) (4.4) Since E w (t) is constant, using Cauchy-Schwarz s inequality and Poincaré s theorem, we can estimate [ ] T ψww t e γφ dx ce δγ E w (), and so, from (4.4), we obtain T T ψe γφ w dxdt ce δγ E w () + 1 { T +c wt dxdt + for a suitable positive constant c. By (4.5) we deduce T { T T e γφ w dxdt c wt dxdt + ω ω ω T ψe γφ w dxdt } w dxdt w dxdt + e δγ E w (), }, (4.5) which, used in (4.1), gives { T T } BT w Γ (,T ) c wt dxdt + w dxdt + e δγ E w (). ω (4.6) Then, from (4.6) and Theorem 3.4 of [11] (Carleman estimate (3.14)), taking γ sufficiently large, we obtain T E w () c wt (x, t)dxdt + c w L ω ( (,T )). Now, estimate (4.13) follows from compactness uniqueness arguments. 18

19 Proposition 4.3 There exists a time T such that for all times T > T there exists a positive constant C (depending on T ) for which T F() C for any regular solution u of problem (1.9) (1.13). a(x){u t (x, t) + u t (x, t τ)}dxdt, (4.7) Proof. Following Zuazua [15], we can decompose the solution u of problem (1.9) (1.13) as u = w + w, where w solves (4.8) (4.1) with initial condition and w verifies w(x, ) = u (x), w t (x, ) = u 1 (x) in, w tt w = a(x)[µ 1 u t (x, t) + µ u t (x, t τ)] in (, + ) (4.8) w(x, t) = on Γ D (, + ) (4.9) w ν (x, t) = on (, + ) (4.3) w(x, ) = and w t (x, ) = in. (4.31) Then, from (4.6) and (4.1), F() = E() + E () = E w () + ξ 1 a(x) u t (x, ρτ)dρdx. (4.3) If we take T > T := max{t, τ}, from (4.3) with a change of variable we obtain and then, from (4.13), F() c c T F() E w () + c a(x) u t (x, t τ)dtdx, T a(x) {wt (x, t) + u t (x, t τ)}dtdx a(x) T { w t (x, t) + u t (x, t) + u t (x, t τ)}dtdx, (4.33) for a suitable positive constant c. Therefore, from standard energy estimates for w, we obtain T F() C a(x){u t (x, t) + u t (x, t τ)}dxdt. Now, using estimate (4.7), as in the case of boundary feedback we obtain the exponential stability result. 19

20 Theorem 4.4 Let the assumption (1.8) be satisfied. Then there exist positive constants C 1, C such that, for any regular solution of problem (1.9) (1.13), F(t) C 1 F()e C t, t. (4.34) Remark 4.5 Analogous arguments apply if we have more than one delay term in the internal feedback, that is if equation (1.9) is replaced by u tt (x, t) u(x, t) + a(x) [ k µ u t (x, t) + µ i u t (x, t τ i ) ] = in (, + ), i=1 with µ, µ i, τ i, i = 1,..., k, positive parameters. In this case, if k µ > µ i, i=1 the right energy to consider, in order to prove exponential decay, is E(t) := 1 {u t (x, t) + u(x, t) }dx + with constants ξ i, i = 1,..., k, choosen as in Remark 3.4. k i=1 ξ i 1 a(x) u t (x, t ρτ i )dρdx, Remark 4.6 In the case Γ D, since for internal feedbacks we have u ν = on, we can use the multiplier identity from [1], and then obtain stability results under the same geometrical conditions than those from [1]. 5 Some unstability examples In this section we will give some unstability examples in the case µ µ Boundary feedback In this subsection we consider the problem with boundary feedback (1.1) (1.5). Let us consider the spectral problem for the system u tt (x, t) u(x, t) = in (, + ) u(x, t) = on Γ D (, + ) u ν (x, t) = µ 1u t (x, t) µ u t (x, t τ) on (, + ). We seek a solution of (5.1) in the form (5.1) u(x, t) = e λt ϕ(x), λ lc.

21 Then, ϕ has to be a solution of the eigenvalue problem ϕ + λ ϕ = in ϕ = on Γ D ϕ ν = (µ 1 + µ e λτ )λϕ on, (5.) which can be reformulated, in a variational form, as ϕ vdx + λ ϕvdx + (µ 1 + µ e λτ )λ ϕvdγ =, v HΓ 1 D (). (5.3) We want to find a solution for λ := ib, with b IR. For this choice of λ the problem (5.3) can be rewritten as Assume that ϕ vdx b ϕvdx + (µ 1 + µ e ibτ )ib ϕvdγ =, v HΓ 1 D (). (5.4) cos(bτ) = µ 1 µ. (5.5) Note that, since we are considering the case µ µ 1, there exist b, τ such that (5.5) holds. Then, we choose µ sin(bτ) = µ µ 1. (5.6) Under these assumptions, (5.4) becomes ϕ vdx b ϕvdx + b µ µ 1 ϕvdγ =, v HΓ 1 D (). (5.7) In particular, for v = ϕ, (5.7) gives ϕ dx b Without loss of generality we can assume ϕ := and then, the identity (5.8) can be rewritten as where ϕ dx + b µ µ 1 ϕ dγ =. (5.8) ϕ dx = 1 (5.9) b b µ µ 1q (ϕ) q 1 (ϕ) =, (5.1) q (ϕ) := ϕ dγ, q 1 (ϕ) := ϕ dx. (5.11) Now we distinguish two cases. Case (a) µ 1 = µ. In this case, under our assumptions, (5.1) becomes b = q 1 (ϕ). (5.1) 1

22 Define b := min w H 1 Γ D () w = 1 q 1 (w) (5.13) If ϕ verifies q 1 (ϕ) = min w H 1 Γ D () w = 1 q 1 (w), then it easy to see that ϕ is a solution of (5.4) with b as in (5.13). Then, ϕ verifies (5.) and so u(x, t) := e ibt ϕ(x) (5.14) is a solution of problem (5.1). Therefore, we have found a solution of our boundary problem whose energy is constant. Indeed, an easy computation show that, for the function u defined in (5.14), ( u(x, t) + u t (x, t) )dx = b >, t. Note that, from our assumptions (λ = ib, cos(bτ) = 1, sin(bτ) = ), problem (5.) becomes the classical eigenvalue problem for the Laplace operator with mixed Dirichlet- Neumann boundary condition. So, we can take a sequence {b n } n of positive real numbers defined by b n = Λ n, n IN, where Λ n, n IN, are the eigenvalues for the Laplace operator. Then, putting b n τ = (l + 1)π, l IN, we obtain a sequence of delays τ n,l = (l + 1)π b n, l, n IN, which become arbitrary small (or large) for suitable choices of the indices n, l IN. Therefore, in the case µ 1 = µ, we have found a set of time delays for which problem (1.1) (1.5) is not asymptotically stable. Case (b) µ > µ 1. In this case, from (5.1) we have b = 1 ( ) µ µ 1q (ϕ) ± (µ µ 1)q(ϕ) + 4q 1 (ϕ). Define b := 1 min w H 1 Γ D () w = 1 ( ) µ µ 1q (w) + (µ µ 1)q(w) + 4q 1 (w). (5.15)

23 We now prove that if the minimum in the right hand side of (5.15) is attained at ϕ, that is µ µ 1q (ϕ) + (µ µ 1)q(ϕ) + 4q 1 (ϕ) := min w H 1 Γ D () w = 1 ( µ µ 1q (w) + (µ µ 1)q (w) + 4q 1 (w) then ϕ is a solution of (5.7) with b as in (5.15). To show this take, for ε IR, Then, If we denote g(ε) := w = ϕ + εv, ε v then, by definition (5.16), So, we have that g(ε) g() = that, after an easy computation, gives with v H 1 Γ D () such that w = ϕ + ε v = 1 + ε v. ), (5.16) ϕvdx =. (5.17) ( ) µ µ 1q (ϕ + εv) + (µ µ 1)q(ϕ + εv) + 4q 1 (ϕ + εv), ( ) µ µ 1q (ϕ) + (µ µ 1)q(ϕ) + 4q 1 (ϕ). dg(ε) dε =, ε= (5.18) ϕ vdx + b µ µ 1 ϕvdγ =. (5.19) Since any function ṽ H 1 Γ D () can be decomposed as ṽ = γϕ + v, γ IR, v H 1 Γ D () with ϕvdx =, from (5.19) and (5.8) we obtain that ϕ satisfies (5.7) with b defined in (5.15). So, for such positive b, ( bτ = arccos µ ) 1 + lπ, l IN, µ defines a sequence of time delays for which the problem (1.1) (1.5) is not asymptotically stable. 3

24 5. Internal feedback In this subsection we will give unstability examples for the problem with internal feedback (1.9) (1.13). Let us consider the spectral problem for the system u tt (x, t) u(x, t) + a(x)[µ 1 u t (x, t) + µ u t (x, t τ)] = u(x, t) = on Γ D (, + ) u (x, t) = ν on (, + ). We restrict our analysis to the case a(x) 1 in. We seek a solution of (5.) in the form in (, + ) (5.) u(x, t) = e λt ϕ(x), λ lc. Then, ϕ has to solve the eigenvalue problem ϕ = [λ + (µ 1 + µ e λτ )λ]ϕ in ϕ = on Γ D (5.1) ϕ ν = on. Let us consider the standard problem for the Laplace operator with mixed Dirichlet Neumann boundary condition ϕ = µ ϕ in ϕ = on Γ D (5.) ϕ ν = on. We want to show that for any Λ eigenvalue of problem (5.), there exists λ lc solution of the equation λ + (µ 1 + µ e λτ )λ = Λ. (5.3) We seek a solution λ = α + iβ, α, β IR, with Under this assumption the equation (5.3) becomes βτ = (l + 1)π, l IN. (5.4) { α + β = Λ Now, we distinguish two cases. Case (a) µ 1 = µ. In this case, from (5.5) we have µ e ατ = α + µ 1 (5.5) α =, β = Λ. 4

25 Therefore, for any Λ n eigenvalue of problem (5.), if β n IR verifies β n = Λ n, then for λ = iβ n problem (5.1) admits a non zero solution. Take β n positive. From our assumption (5.4) τ n,l = (l + 1)π β n, n, l IN, is a set of time delays that become arbitrary small (or large) for suitable choices of the indices n, l IN. For such delays the problem (1.9) (1.13) admits solutions in the form u(x, t) = e iβt ϕ(x), whose energy is constant and strictly positive. So, system (1.9) (1.13) is not asymptotically stable. Case (b) µ > µ 1. For a fixed α >, from the second equation of (5.5), we obtain and so, in order to have τ(α) >, we consider τ(α) = 1 ( ) α ln µ, (5.6) µ 1 + α < α < 1 (µ µ 1 ). From (5.4), the first equation of (5.5) becomes α + (l + 1) π τ (α) = Λ, (5.7) where τ(α) is given by (5.6). Denoting we have while g(α) := α + (l + 1) π, α (, (µ τ µ 1 )/), (α) τ(α) + and g(α) + as α + ; τ(α) + and g(α) + as α 1 (µ µ 1 ). Since g is a continuous function of α, for any fixed Λ eigenvalue of problem (5.) there exists α ( < α < (µ µ 1 )/) such that (5.7) is verified. Therefore, for such α there exists a delay τ(α) (defined by (5.6)) such that a function of the form e α+iβ ϕ(x) 5

26 solves problem (1.9) (1.13). Since α the energy of such a solution is not decaying to zero. So, this solution is not asymptotically stable. Note that, for any Λ n eigenvalue of problem (5.) and for any l IN there exist α n,l and a delay τ n,l = τ(α n,l ) such that (5.5) is verified with β n,l = (l + 1)π τ n,l. From the first equation of (5.5), (l + 1) π τ n,l Λ n. Then, for a fixed l IN, if n +, then τ n,l +. On the contrary, for a fixed n IN, for l +, then τ n,l +. Therefore, we have unstability phenomena for a sequence of arbitrary small or large time delays. References [1] R. Bey, A. Heminna and M. Moussaoui, Singularities of the solution of a mixed problem for a general second order elliptic equation and boundary stabilization of the wave equation. J. Math. Pures Appl., 78: , [] G. Chen. Control and stabilization for the wave equation in a bounded domain I-II. SIAM J. Control Optim., 17:66 81, 1979; 19:114 1, [3] R. Datko. Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM J. Control Optim., 6: , [4] R. Datko, J. Lagnese and M. P. Polis. An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim., 4:15 156, [5] P. Grisvard. Elliptic problems in nonsmooth domains, volume 1 of Monographs and Studies in Mathematics. Pitman, Boston London Melbourne, [6] V. Komornik. Rapid boundary stabilization of the wave equation. SIAM J. Control Optim., 9:197 8, [7] V. Komornik. Exact controllability and stabilization, the multiplier method, volume 36 of RMA. Masson, Paris, [8] V. Komornik and E. Zuazua. A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl., 69:33 54, 199. [9] J. Lagnese. Note on boundary stabilization of wave equations. SIAM J. Control and Optim., 6:15 156,

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