Stabilization of second order evolution equations with unbounded feedback with delay

Transcription

1 Stabilization of second order evolution equations with unbounded feedback with delay Serge Nicaise, Julie Valein December, 8 Abstract We consider abstract second order evolution equations with unbounded feedback with delay Existence results are obtained under some realistic assumptions Sufficient and explicit conditions are derived that guarantee the exponential or polynomial stability Some new examples that enter into our abstract framework are presented Keywords second order evolution equations, wave equations, delay, stabilization functional Introduction Time-delay often appears in many biological, electrical engineering systems and mechanical applications [,, ], and in many cases, in particular for distributed parameter systems, even arbitrarily small delays in the feedback may destabilize the system, see eg [8, 9,,, 5, 6, 7,, 3] The stability issue of systems with delay is, therefore, of theoretical and practical importance We further remark that some techniques developed recently [6, 7] in order to obtain some existence results and decay rates have some similarities We therefore propose to consider an abstract setting as large as possible in order to contain a quite large class of problems with time delay feedbacks In a second step we prove existence and stability results in this setting under realistic assumptions Finally in order to show the usefulness of our approach, we give some examples where our abstract framework can be applied For a similar approach, we refer to the paper in preparation [] Without delay such an approach was developed in [4] Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 956, Institut des Sciences et Techniques of Valenciennes, F Valenciennes Cedex 9 France, SergeNicaise@univ-valenciennesfr Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 956, Institut des Sciences et Techniques of Valenciennes, F Valenciennes Cedex 9 France, JulieValein@univ-valenciennesfr

2 Before going on, let us present our abstract framework Let H be a real Hilbert space with norm and inner product denoted respectively by H and (, H Let A : D(A H be a self-adjoint positive operator with a compact inverse in H Let V := D(A be the domain of A Denote by D(A the dual space of D(A obtained by means of the inner product in H Further, for i =,, let U i be a real Hilbert space (which will be identified to its dual space with norm and inner product denoted respectively by Ui and (, Ui and let B i L(U i, D(A We consider the system described by ( ω(t + Aω(t + B u (t + B u (t τ =, t > ω( = ω, ω( = ω, u (t τ = f (t τ, < t < τ, where t [, represents the time, τ is a positive constant which represents the delay, ω : [, H is the state of the system and u L ([,, U, u L ([ τ,, U are the input functions Most of the linear equations modelling the vibrations of elastic structures with distributed control with delay can be written in the form (, where ω stands for the displacement field In many problems, coming in particular from elasticity, the input u i are given in the feedback form u i (t = Bi ω(t, which corresponds to colocated actuators and sensors We obtain in this way the closed loop system ω(t + Aω(t + B B ω(t + B B ω(t τ =, t > ( ω( = ω, ω( = ω, B ω(t τ = f (t τ, < t < τ The first natural question is the well posedness of this system In section we will give a sufficient condition that guarantees that this system ( is well-posed, where we closely follow the approach developed in [6] for the wave equation Secondly, we may ask if this system is dissipative We show in section 3 that the condition (3 < α <, u V, B u U α B u U guarantees the energy is decreasing; under this condition, using a result from [5] (see also [] we pertain a necessary and sufficient condition for the decay to zero of the energy Note that this last condition is independent of the delay and therefore under the condition (3, our system is strongly stable if and only if the same system without delay is strongly stable Note further that if (3 is not satisfied, there exist cases where some instabilities may appear (see [6, 7, 3] for the wave equation Hence this assumption seems realistic In a third step, again under the condition (3 and a certain boundedness assumption from [4] between the resolvent operator of A and of the operators B and B, see condition (, we prove that the exponential decay of the system ( follows from a certain observability estimate Again this observability estimate is independent of the delay term B B ω(t τ and therefore, under the

3 conditions (3 and (, the exponential decay of the system ( follows from the exponential decay of the same system without delay Nevertheless we give the dependence of the decay with respect to the delay, in particular we show that if the delay increases the decay decreases This is the content of section 4 A similar analysis for the polynomial decay is performed in section 5 by weakening the observability estimate Again we show that if the delay increases the decay decreases In view of some applications, section 6 is devoted to the proof of these two observability estimates by using a frequency domain method and a reduction to some conditions between the eigenvectors of A and the feedback operator B Finally we finish this paper by considering in section 7 different examples where our abstract framework can be applied To our knowledge, all the examples, with the exception of the first one, are new Well posedness of the system We aim to show that system ( is well-posed For that purpose, we use semigroup theory and an idea from [6] (see also [7] Let us introduce the auxiliary variable z(ρ, t = B ω(t τρ for ρ (, and t > Note that z verifies the transport equation for < ρ < and t > (4 (5 τ z t + z ρ = z(, t = B ω(t z(ρ, = B ω( τρ = f ( τρ Therefore, the system ( is equivalent to ω(t + Aω(t + B B ω(t + B z(, t =, t > τ z t + z ρ =, t >, < ρ < ω( = ω, ω( = ω, z(ρ, = f ( τρ, < ρ < z(, t = B ω(t, t > If we introduce then U satisfies U = ( ω, ω, ż T = U := (ω, ω, z T, ( ω, Aω(t B B ω(t B z(, t, T z τ ρ Consequently the system ( may be rewritten as the first order evolution equation { U (6 = AU U( = (ω, ω, f ( τ, where the operator A is defined by u A = Aω B Bu B z(, ω u z 3 τ z ρ

4 with domain D(A := {(ω, u, z V V H ((,, U ; z( = B u, Aω+B B u+b z( H} Now, we introduce the Hilbert space H = V H L ((,, U equipped with the usual inner product (7 ω u, ω ( ũ = A ω, A ω z z Let us suppose now that H + (u, ũ H + (8 < α, u V, B u U α B u U (z(ρ, z(ρ U dρ Under this condition, we will show that the operator A generates a C -semigroup in H For that purpose, we choose a positive real number ξ such that (9 ξ α This constant exists because < α We now introduce the following inner product on H ω u, ω ( ũ = A ω, A ω + (u, ũ H + τξ H z z H (z(ρ, z(ρ U dρ This new inner product is clearly equivalent to the usual inner product on H (7 Theorem Under the assumption (8, for an initial datum U H, there exists a unique solution U C([, +, H to system (6 Moreover, if U D(A, then U C([, +, D(A C ([, +, H Proof By Lumer-Phillips theorem, it suffices to show that A is m-dissipative (see Definition 33 and Theorems 43 and 46 of [8] We first prove that A is dissipative Take U = (ω, u, z D(A Then AU, U H = = u Aω B Bu B z(, ( A u, A ω H τ z ρ ω u z (Aω + B B u + B z(, u H ξ H ( z ρ (ρ, z(ρ U dρ 4

5 Since Aω + B Bu + B z( H, we obtain ( AU, U H = A u, A ω Aω, u V H, V B Bu, u V, V B z(, u V, V ( z ξ (ρ, z(ρ dρ ρ U = Aω, u V, V Aω, u V, V B u U (z(, Bu U ( z ξ (ρ, z(ρ dρ, ρ U by duality By intregrating by parts, we obtain ( z ( (ρ, z(ρ dρ = z(ρ, z ρ U ρ (ρ dρ + ( z( U z( U, U and thus ( z (ρ, z(ρ dρ = ρ U ( z( U Bu U Therefore, by Cauchy-Schwarz s inequality, we find AU, U H = B u U (z(, B u U ξ z( U + ξ B u U By (8, we obtain B u U + ( ξ z( U + ( + ξ B u U AU, U H ( α + ξα B u U + ( ξ z( U with α + ξα and ξ because ξ satisfies condition (9 This shows that AU, U H and then the dissipativeness of A Let us now prove that λi A is surjective for some λ > Let (f, g, h T H We look for U = (ω, u, z T D(A solution of or equivalently ( (λi A ω u z = f g h λω u = f λu + Aω + B Bu + B z( = g λz + z τ ρ = h Suppose that we have found ω with the appropriate regularity Then, we have u = f + λω V 5

6 We can then determine z, indeed z satisfies the differential equation λz + τ z ρ = h and the boundary condition z( = B u = B f + λb ω Therefore z is explicitely given by ρ z(ρ = λbωe λτρ Bfe λτρ + τe λτρ e λτσ h(σdσ This means that once ω is found with the appropriate properties, we can find z and u In particular, we have ( z( = λbωe λτ Bfe λτ + τe λτ e λτσ h(σdσ = λbωe λτ + z, where z = B fe λτ + τe λτ eλτσ h(σdσ is a fixed element of U depending only on f and h It remains to find ω By (, ω must satisfy and thus by (, λ ω + Aω + λb B ω + B z( = g + B B f + λf, λ ω + Aω + λb B ω + λe λτ B B ω = g + B B f + λf B z =: q, where q V We take then the duality brackets, V, V with φ V : Moreover: λ ω + Aω + λb B ω + λe λτ B B ω, φ V, V = q, φ V, V λ ω + Aω + λb B ω + λe λτ B B ω, φ V, V = λ ω, φ V, V + ( Aω, φ V, V + λ( B Bω, φ V, V + e λτ B Bω, φ V, V = λ (ω, φ H + A ω, A φ + λ((bω, Bφ U + e λτ (Bω, Bφ U H because ω V H Consequently, we arrive at the problem ( ( λ (ω, φ H + A ω, A φ +λ((bω, Bφ U +e λτ (Bω, Bφ U = q, φ V, V H The left hand side of ( is continuous and coercive on V Indeed, we have ( λ (ω, φ H + A ω, A φ + λ((bω, Bφ U + e λτ (Bω, Bφ U H λ ω H φ H + A H A H ω φ + λ( Bω U Bφ U + e λτ Bω U Bφ U Cλ ω V φ H + A ω V φ V +λ( B L(V, U ω V φ V + e λτ B L(V, U ω V φ V C ω V φ V, 6

7 and for φ = ω V λ ω H (A + ω, A ω A ω C ω V H H + λ( B ω U + e λτ B ω U Therefore, this problem ( has a unique solution ω V by Lax-Milgram s lemma Moreover Aω + B B u + B z( = g + λf λ ω H In summary, we have found (ω, u, z T D(A satisfying ( Remark We deduce from Theorem that D(A is dense in H (see [8] Remark 3 For initial data (ω, ω, f ( τ T in D(A, we easily show that the solution (ω(t, u(t, z(t T = T (t(ω, ω, f ( τ T, where T (t is the semigroup generated by A, is indeed solution of ( in the sense that u(t = ω(t, and and ω satisfies ( z(ρ, t = B (t τρ, 3 The energy We now restrict the hypothesis (8 to obtain the decay of the energy Namely, we suppose that (3 holds, namely < α <, u V, B u U α B u U For an initial datum (ω, ω, f ( τ T H, Theorem guarantees the existence of a weak solution (ω(t, u(t, z(t T = T (t(ω, ω, f ( τ T Hence the associated energy (which corresponds to the inner product on H is defined by E(t :=:= ( A ω(t + u(t H + τξ z(ρ, t U H dρ, where ξ is a positive constant satisfying (3 < ξ < α, that exists because < α < Note that by Remark 3 for initial data in D(A, this energy takes the form (4 E(t := ( A ω H + ω H + τξ B ω(t τρ U dρ 7

8 3 Decay of the energy Proposition 3 If (3 holds, then for all (ω, ω, f ( τ T D(A, the energy of the corresponding regular solution of ( (ie (ω, ω, B ω(t τρ T C([, +, D(A C ([, +, H is non-increasing and there exist two positive constants C and C depending only on α and ξ such that (5 ( ( C B ω(t U + B ω(t τ U E (t C B ω(t U + B ω(t τ U Proof Deriving (4, we obtain E (t = ( A ω, A ω H + ( ω, ω H + τξ = Aω, ω V,V ( ω, Aω + B B ω + B B ω(t τ H +ξτ (B ω(t τρ, B ω(t τρ U dρ (B ω(t τρ, B ω(t τρ U dρ = Aω, ω V,V ω, Aω + B B ω + B B ω(t τ V, V +ξτ (B ω(t τρ, B ω(t τρ U dρ, because Aω + B B ω + B B ω(t τ H Then E (t = Aω, ω V,V ω, Aω V, V ω, B B ω V, V ω, B B ω(t τ V, V +ξτ (B ω(t τρ, B ω(t τρ U dρ = B ω U (B ω, B ω(t τ U +ξτ (B ω(t τρ, B ω(t τρ U dρ Moreover, recalling that z(ρ, t = B ω(t τρ, we see that (B ω(t τρ, B ω(t τρ U dρ = = τ because z z ρ (ρ, t = τ t (ρ, t Then, we have ( z(ρ, t, z t ( (ρ, t dρ U z(ρ, t, z (ρ, t ρ (B ω(t τρ, B ω(t τρ U dρ = τ ρ z(ρ, t U dρ U dρ, = τ ( z(, t U z(, t U = τ ( B ω(t τ U B ω(t U 8

9 Consequently, E (t = B ω U (B ω, B ω(t τ U ξ B ω(t τ U + ξ B ω(t U Cauchy-Schwarz s inequality yields and E (t B ω U + ( + ξ B ω(t U + ( ξ B ω(t τ U E (t B ω U + ( + ξ B ω(t U ( + ξ B ω(t τ U Therefore, by (3, these estimates leads to E (t C ( B ω(t U + B ω(t τ U with C = min {( ξα α, ( ξ } which is positive according to the assumption (3, and to E (t C ( B ω U + B ω(t τ U with which is also positive { C = max, ξ + } Remark 3 Integrating the expression (5 between and T, we obtain ( B ω(t U + B ω(t τ U dt C (E( E(T C E( This estimate implies that B ω( L ((, T, U and B ω( τ L ((, T, U Remark 33 If (3 is not satisfied, there exist cases where instabilities may appear, see [6, 7, 3] for the wave equation Hence this condition appears to be quite realistic 3 Decay of the energy to We give a necessary and sufficient condition that guarantees the decay to of the energy 9

10 Proposition 34 Assume that (3 holds Then, for all initial data in H, we have (6 lim t E(t = if and only if for any (non zero eigenvector ϕ D(A of A, we have (7 B ϕ Remark 35 Notice that this necessary and sufficient condition is the same than in the case without delay (see [] and therefore, the system ( with delay is strongly stable (ie the energy tends to zero if and only if the system without delay (ie for B = is strongly stable Proof Let us show that (7 implies (6 For that purpose we closely follow [] First, we show that A has no eigenvalue on the imaginary axis If it is not the case, let iω be an eigenvalue of A where ω R Let ϕ be an eigenvector associated with iω Then ϕ is of the form ϕ = z iωz iωe iωτρ Bz, with (8 ω z + Az + iωb B z + iωe iωτ B B z = It is an immediate consequence of the identity (iωi Aϕ = First we notice that ω since for ω =, the above identity reduces to Az = with z D(A Since by hypothesis A is invertible, we get z = and therefore is not an eigenvalue of A We now take the duality bracket, V, V between (8 and z V : = ω z + Az + iωb B z + iωe iωτ B B z, z V, V = ( ω I + Az, z V, V + iω B z U + iωe iωτ B z U We look at the imaginary part of this expression to obtain ω ( Bz U + cos(ωτ Bz U =, which implies, because ω, We deduce that B z U + cos(ωτ B z U = = B z U + cos(ωτ B z U B z U B z U ( α B z U,

11 by (3 with α < Consequently B z U = which implies (9 B z = Moreover, by (8, (9 and (3, we have Az = ω z Therefore, z is an eigenvector of A of associated eigenvalue ω such that B z =, which contradicts (7 Thus, A has no eigenvalue on the imaginary axis Now, we can apply the main theorem of Arendt and Batty [5]: As σ(a ir is empty (because the surjectivity of (iωi A is equivalent to the injectivity of ω + A iωb B iωe iωτ B B, we obtain (6 Let us show that (6 implies (7 For that purpose we use a contradiction argument Suppose that there exists an eigenvector ϕ of A of associated eigenvalue λ such that B ϕ = Let us set Then ω is solution of ( and satisfies ω(, t = ϕ cos(λt E(t = E( because and B ω(t U = λ sin (λt B ϕ U = B ω(t τ U = λ sin (λ(t τ Bϕ U αλ sin (λ(t τ Bϕ U =, by (3 This means that we have obtained a solution of system ( with a constant energy, which contradicts (6 4 The exponential stability 4 A priori estimate In order to obtain the characterization of decay properties of the damped system via observality inequalities for the conservative system we will use the following

12 assumption from [4]: If β > is fixed and C β = {λ C Rλ = β }, the function ( λ C β H(λ = λb (λ I + A B L(U is bounded, where B L(U, V with U a Hilbert space We consider the evolution system ( { ÿ(t + Ay(t = Bv(t y( = ẏ( =, and the following conservative system { φ(t + Aφ(t = ( φ( = ω, φ( = ω Let us recall the two following results proved in [4]: Lemma 4 Suppose that v L (, T ; U and that the solutions φ of ( are such that B φ( H (, T ; U and there exists a constant C > such that (B φ ( L (,T ;U C (ω, ω V H (ω, ω V H Then the system ( admits a unique solution having the regularity y C(, T ; V C (, T ; H Proposition 4 Suppose that v L (, T ; U and that the system ( admits a unique solution having the regularity y C(, T ; V C (, T ; H Then hypothesis ( holds if and only if B y( H (, T ; U and there exists a constant C > independent of T such that (B y ( L (,T ;U CeβT v L (,T ;U Let ω C(, T ; V C (, T ; H be the solution of ( with (ω, ω, f ( τ T D(A Then it can be split up in the form ω = φ + ψ, where φ is solution of the system without damping (, and ψ satisfies (3 { ψ(t + Aψ(t = B B ω(t B B ω(t τ ψ( =, ψ( =

13 We now set( B = (B B L(U, V where U = U U It is easy to verify B that B = B L(V, U Therefore ψ is solution of (4 where v(t = with and by Remark 3 ( B ω(t B ω(t τ v = { ψ(t + Aψ(t = Bv(t ψ( =, ψ( =, In other words, ψ is solution of system ( B = (B B ( B ω( B ω( τ L ((, T, U Then ψ = ω φ C(, T ; V C (, T ; H Suppose that the hypothesis ( is satisfied for B = (B B and U = U U By applying Proposition 4, we obtain (B ψ U dt CeβT v(t U dt, which is equivalent to ( (Bψ U + (Bψ U dt Ce βt ( B ω(t U + B ω(t τ U dt In particular, we have (Bψ U dt Ce βt ( B ω(t U + B ω(t τ U dt Therefore, since ω = φ + ψ, we have ( T T (Bφ (t U dt (Bω (t U dt + (Bψ (t U dt Ce βt Thus, we have proved the following result: ( (B ω(t U + (B ω(t τ U dt Lemma 43 Suppose that the assumption ( is satisfied for B = (B B, U = U U Then the solutions ω of ( and φ of ( satisfy (Bφ (t U dt Ce βt ( B ω(t U + B ω(t τ U dt, with C > independent of T 3

14 4 The stability result Theorem 44 Assume that the hypotheses (3 and ( are verified for B = (B B, U = U U If there exist a time T > and a constant C > such that the observability estimate (5 A ω H + ω H C (Bφ (t U dt holds, where φ is solution of (, then the system ( is exponentially stable in the energy space: there exist C > independent of τ and ν > such that, for all initial data in H, (6 E(t CE(e νt t > Proof Let ω be a solution of ( with initial data (ω, ω, f ( τ D(A Without loss of generality, we can always assume that (5 holds with T > τ and C independent of τ Integrating the inequality (5 of Proposition 3 between and T, we obtain E( E(T C ( B ω(t U + B ω(t τ U dt C ( B ω(t U + B ω(t τ U dt + C B ω(t τ U dt ( T T Ce βt (Bφ (t U dt + B ω(t τ U dt by Lemma 43 By assumption (5, we obtain E( E(T Ce βt ( A ω H T + ω H + B ω(t τ U dt, with C independent of τ As T > τ, by change of variables, we have B ω(t τ U dt = τ τ τ B ω(t U dt B ω(t U dt = τ B ω( τρ U dρ The two previous inequalities and (3 directly imply that E( E(T C e βt ( A ω H + ω H + ξτ with C = C/(/α This means that for T > τ, we have E( E(T C e βt E( B ω( τρ U dρ, 4

15 This estimate is equivalent to E( E(T C e βt E(T, because the energy is decreasing, which leads to E(T γe(, where γ = +C e βt < Applying this argument on [(m T, mt ], for m =,, (which is valid because the system is invariant by translation in time, we will get Therefore, we have E(mT γe((m T γ m E( E(mT e νmt E(, m =,, with ν = T ln γ = T ln( + C e βt > which depends on T and thus on τ (because T > τ For an arbitrary positive t, there exists m N such that (m T < t mt and by the non-increasing property of the energy, we conclude E(t E((m T e ν(m T E( γ e νt E( Hence the energy decays exponentially with the decay rate ν = T ln γ = T ln(+ C e βt < τ ln( + C e βτ, because T > τ Notice that the constant C of (6 can be chosen such that C + C (which is independent of τ because γ = + C e βt + C By density of D(A into H, we deduce that (6 holds for any initial data in H Remark 45 In the previous theorem, we have seen that the decay rate is ν = T ln γ = T ln( + C e βt < τ ln( + C e βτ because T > τ and where C is independent of τ Therefore, when τ becomes larger, the decay is slower Remark 46 Notice that the sufficient condition (5 for the exponential decay of the energy is the same than the case without delay (see [4] Therefore, if the hypothesis ( holds and if the dissipative system without delay (ie with B = is exponentially stable, then the system ( is exponentially stable 5 The polynomial stability In some cases, the decay of the energy is not exponential, but can be polynomial Our aim here is to give a sufficient condition that yields the explicit decay rate The proof of this stability result requires the next technical Lemma proved in [3, Lemma 5] 5

16 Lemma 5 Let (ε k k be a sequence of positive real numbers satisfying (7 ε k+ ε k Cε +µ k+ k, where C > and µ > are constants Then there exists a positive constant M (depending on µ and C such that ( with M > 4 (+µc +µ ε k M ( + k +µ k, Moreover we recall the following interpolation result Lemma 5 For (ω, ω D(A V, we have where C > ω m+ D(A C ω m D(A ω D(A m, ω m+ H C ω m D(A ω D(A m, Proof If we denote by {λ kn } n the eigenvalues of A counted without multiplicity, l n the multiplicity of the eigenvalue λ kn and {ϕ kn +j} the j ln orthonormal eigenvectors associated with the eigenvalue λ kn, this lemma is a direct consequence of the equivalence u D(A s ln n j= u k n+j λ 4s k n for all s R, when u = ln n j= u k n+jϕ kn+j and of Hölder s inequality with p = + m and q = m + As for (ω, ω, f ( τ D(A, ω is not necessarily in D(A, we can not apply Lemma 5 to ω, and therefore we need to make the following hypothesis: there exists C > such that for all (ω, ω, z D(A, we have (8 ω m+ V C (ω, ω, z m D(A ω D(A m Theorem 53 Let ω be a solution of ( with (ω, ω, f ( τ D(A Assume that the hypotheses (3, ( and (8 are verified for B = (B B, U = U U If there exist a positive real number m, a time T > and a constant C > such that ( (9 (Bφ (t U dt C ω D(A m + ω D(A m holds where φ is solution of (, then the energy decays polynomially, ie, there exists C > depending on m and τ such that, for all initial data in D(A, C (3 E(t ( + t m (ω, ω, f ( τ D(A t > 6

17 Proof As the hypothesis ( is satisfied for B = (B B, U = U U, by using Lemma 43, we obtain ( ( B ω(t U + B ω(t τ U dt Ce βt ω D(A m + ω D(A m On the other hand, integrating the inequality (5 of Proposition 3 between and T for T large enough: T max(t, τ, we have E( E(T C T ( B ω(t U + B ω(t τ U dt C ( B ω(t U + B ω(t τ U dt + C B ω(t τ U dt Ce ( ω βt D(A m + ω D(A m + τ B ω( τρ U dρ, by change of variable (because T > τ Therefore (3 E(T E( K e ( ω βt D(A m + ω D(A m + τξ B ω( τρ U dρ, for some K > independent of T and τ Therefore, by (8, the previous interpolation result of Lemma 5 and a convexity inequality, we have: ( (ω, ω m+ V H C ω m+ V + ω m+ H ( C (ω, ω, f ( τ m D(A ω m + ω D(A m D(A ω D(A m C (ω, ω, f ( τ ( m ω D(A m + ω D(A D(A m Denoting by X m = D(A m D(A m, we have shown that (3 (ω, ω X m C Now introduce the modified energy (ω, ω m+ V H (ω, ω, f ( τ m D(A Ẽ(t = U(t D(A = ( U(t H + AU(t H As in Proposition 3, this energy Ẽ is decaying Combining the estimates (3 and (3, we obtain E(T E( K e βt ( (ω, ω m+ V H Ẽ( m + ξτ B ω( τρ U dρ, 7

18 for some K > independent of T and τ, or equivalently (33 ( E(T E( K e βt (ω, ω m+ V H + ξτ f ( τ Ẽ( m L ((,, U Using the trivial estimate (ξτ m+ f ( τ m+ L ((,, U = ξτ f ( τ L ((,, U (ξτm f ( τ m L ((,, U τξ f ( τ L ((,, U Ẽ(m the above inequality (33 becomes E(T E( K e βt E( K βt E(m+ e Ẽ(, m (ω, ω m+ V H + (ξτm+ f ( τ m+ Ẽ( m L ((,, U with K > independent of T and τ Since the energy of our system is decaying, we obtain (34 E(T E( K e βt E(T m+ Ẽ( m We now follow the method used in [3] The estimate (34 being valid on the intervals [kt, (k + T ], for any k, we have (35 E((k + T E(kT K e βt E((k + T m+ Ẽ(kT m Setting and dividing (35 by ε k = E(kT Ẽ(, Ẽ(, we obtain (36 ε k+ ε k K e βt ε m+ k+, because Ẽ(kT Ẽ( By Lemma 5 with µ = m > (as m >, there exists a constant M > (depending on m and K e βt, and verifying ( M 4e > βt m mk such that or equivalently M ε k ( + k m M k, E(kT Ẽ( ( + k m This estimate and again the decay of the energy lead to the estimate (3, where C = M ( + T m 8

19 Remark 54 Since the proof of the above theorem reveals that C = M ( + ( T m with M 4e > βt m mk and T > τ, the constant C depends on τ and when τ becomes larger, the decay becomes slower 6 Checking the observability inequalities In this section, we show how to obtain the observability inequalities used in Theorems 44 and 53 Our method is based on the generalized gap condition Before giving spectral conditions to obtain exponential or polynomial decay, we recall some results about Ingham s inequality 6 Preliminaries about Ingham s inequality Let {λ k } k be the set of eigenvalues of A counted with their multiplicities (ie we repeat the eigenvalues according to their multiplicities We further rewrite the sequence of eigenvalues {λ k } k as follows : λ k < λ k < < λ ki < where k =, k is the lowest index of the second distinct eigenvalue, k 3 is the lowest index of the third distinct eigenvalue, etc For all i N, let l i be the multiplicity of the eigenvalue λ ki, ie λ ki < λ ki = λ ki + = = λ ki +l i < λ ki +l i = λ ki+ We have k =, k = l, k 3 = l + l, etc Let {ϕ ki +j} j li be the orthonormal eigenvectors associated with the eigenvalue λ ki We assume that the following generalized gap condition holds : (37 M N, γ >, k, λ k+m λ k Mγ Fix a positive real number γ γ and denote by A k, k =,, M the set of natural numbers k m satisfying (see for instance [6] λ km λ km γ λ kn λ kn < γ for m + n m + k, λ km+k λ km+k γ Then one easily checks that the sets A k + j, j =,, k, k =,, M form a partition of N Notice that some sets A k may be empty because, for the generalized gap condition, the choice of M takes into account multiple eigenvalues Now for k m A k, we recall that the finite differences e m+j (t, j =,, k, corresponding to the exponential functions e iλ k t m+j, j =,, k are given by e m+j (t = m+j p=m (λ kp λ kq e iλ kp t m+j q=m q p 9

20 Write for shortness, e n (t the same finite differences functions corresponding to λ kn Now we are ready to recall the next inequality of Ingham s type, see for instance Theorem 5 of [6]: Theorem 6 If the sequence (λ n n satisfies (37, then for all sequence (a n n Z (where Z = Z \ {}, the function f(t = n Z a n e n (t, satisfies the estimates (38 f(t dt n Z a n, for T > π γ Going back to the original functions e iλ kn t, the above equivalence (38 means that, for T > π γ, the function (from now on λ kn = λ kn f(t = n Z α n e iλ kn t, satisfies the estimates (39 f(t dt M k n C kn, B k= k n A k where means the Euclidean norm of the vector, for k n A k the vector C kn is given by C kn = (α n,, α n+k T, and the k k matrix B kn allows to pass from the coefficients a kn to α kn, namely C kn = B kn (a n,, a n+k T, and is given by B kn = (B kn, ij i, j k the matrix of size k k such that B kn, ij = n+j (λ kn+i λ kq if i j, (i, j (,, q = n q n + i if (i, j = (,, if i > j

21 More explicitly, we have B kn = λ kn λ kn+ (λ kn λ kn+ (λ kn λ kn+ (λ kn λ kn+ (λ kn λ kn+k λ kn+ λ kn (λ kn+ λ kn (λ kn+ λ kn+ (λ kn+ λ kn (λ kn+ λ kn+k (λ kn+ λ k n (λ k n+ λ kn+ (λ kn+ λ k n (λ k n+ λ kn+k (λ kn+k λ k n (λ k n+k λ kn+k We proceed similarly for n, the indices being decreasing from n to n k + Remark 6 If the standard gap condition (4 γ >, n, λ kn+ λ kn γ holds, then A = Z and B = and in that case f(t = n Z α n e iλ kn t satisfy Ingham s inequality (see [3]: (4 f(t dt n Z α n, for T > π γ Now, let U be a separable Hilbert space (in the sequel, U will be U For a c vector c = in U m, we set U, the norm in U m defined by c m m c U, = c l U l= Then we obtain the inequality of Ingham s type in U: Proposition 63 If we have the standard gap condition (4, then for all sequence (a n n in U, the function u(t = n Z a n e iλ kn t satisfies the estimates for T > π γ u(t U dt n Z a n U,

22 Proof As U is a separable Hilbert space, there exists a Hilbert basis (ψ k k of U Therefore, a n U can be written as a n = + k= a k nψ k We truncate a n as follows: for K N, let a (K n = K a k nψ k and set u K (t = k= K ( a k ne iλ kn t ψ k Since (ψ k k is a Hilbert basis, we have by Fubini s the- n Z orem k= k= K u K (t U = a k ne iλkn t n Z Thus, by applying Ingham s inequality (4, we have u K (t U dt = K T a k k= ne iλkn t dt n Z K Therefore k= n Z (a k n K (a k n n Z k= u K (t U dt n Z a (K n As u K u and a (K n a n when K +, we obtain the result In the same way, we obtain an Ingham s type inequality in a Hilbert space U in the case of the generalized gap condition (37 U Corollary 64 If the sequence (λ n n (α n n Z in U, the function satisfies (37, then for all sequence f(t = n Z α n e iλ kn t, satisfies the estimates (4 for T > π γ, where f(t dt M k n C kn U,, B k= k n A k C kn = (α n,, α n+k T U k

23 6 A first observability inequality Proposition 65 Assume that the generalized gap condition (37 holds and that U is separable Let φ be the solution of ( with (ω, ω V H Then there exists a time T > and a constant C > (depending on T such that (5 holds if and only if (43 γ >, k =,, M, k n A k, ξ R Ln, B k n Φ kn ξ U, γ ξ, where the matrix Φ kn with coefficients in U and size k L n, where L n = k l n+i, is given as follow: for all i =,, k, we set i= where (44 { B (Φ kn ij = ϕ kn+i +j L n,i if L n,i < j L n,i, else, L n,i = L n, =, i l n+i i i = Proof We first show that (43 (5 Writting and ω = i ω = i l i a ki+jϕ ki+j j= l i b ki +jϕ ki +j j= where (λ ki a ki+j i, j, (b ki+j i, j l (N, then the solution φ of system without damping ( is given by Consequently φ(, t = l i ( a ki +j cos(λ ki t + b k i+j sin(λ ki t ϕ ki +j λ i j= ki (B φ (t = i l i ( a ki+jλ ki sin(λ ki t + b ki+j cos(λ ki t Bϕ ki+j j= By grouping the terms corresponding to the same eigenvalue, we get (Bφ (t = l i a ki +jbϕ ki +j λ ki sin(λ ki t + l j b ki +jbϕ ki +j cos(λ ki t i j= i j= = n Z α n e iλ kn t, 3

24 where α n = l n l n b kn +jb ϕ kn +j + i a kn +jbϕ kn +j λ kn n, j= α n = l n l n b kn+jb ϕ kn+j i a kn+jbϕ kn+j λ kn n j= Integrating the square of the norm of this identity between and T > and using Ingham s inequality (4 in U, for T large enough, we get (B φ (t U dt C j= M j= k= k n A k where C kn = (α n,, α n+k T is a vector of U k But for all k n A k, setting B k n C kn U,, à kn = ( λ kn a kn,, λ kn a kn +l n, λ kn+ a kn+,, λ kn+ a kn+ +l n+,, λ kn+k a kn+k,, λ kn+k a kn+k +l n+k T, B kn = ( b kn,, b kn+l n, b kn+,, b kn++l n+,, b kn+k,, we readily check that (B φ (t U dt C b kn+k +l n+k T, M k= k n A k Hence the assumption (43 yields M (Bφ (t U dt C ( B k n Φ kn à kn U, + ( à kn + k= ( k A n A k = C ω + ω H H B k n Φ kn Bkn B kn because (ϕ kn +i n,i is an orthonormal basis associated with the operator A It remains to show that (5 (43 Let k =,, M and k n A k be fixed Take ω = n+k li i=n j= a k i+jϕ ki+j and ω = n+k li i=n j= b k i+jϕ ki+j Then the solution φ of system ( is given by φ(, t = n+k i=n l i j= ( a ki+j cos(λ ki t + b k i +j sin(λ ki t ϕ ki+j, λ ki U, 4

25 and then (B φ (t = n+k i=n l i ( a ki +jλ ki sin(λ ki t + b ki +j cos(λ ki t Bϕ ki +j j= Applying again Ingham s inequality, we get for T large enough and Ãk n, Bkn define above By (5, we obtain (B φ (t U dt B k n Φ kn à kn U, + B k n Φ kn Bkn U, (45 B k n Φ kn à kn U, + B k n Φ kn Bkn U, = C C A ω + ω H H n+k i=n l i (a k i +jλ k i + b k i +j, j= for some C > Hence we conclude that B k n Φ kn ξ U γ ξ, This ends the proof Remark 66 If the standard gap condition (4 holds, then A = N and B = In this case, the assumption (43 becomes γ >, k n, ξ R ln, Φ kn ξ U γ ξ Moreover, if the standard gap condition (4 holds and if the eigenvalues are simple, the assumption (43 becomes γ >, k, B ϕ k U γ Remark 67 The above Proposition 65 yields a time T > and a constant C > depending on T such that (5 holds but the time T and the constant C do not depend on the delay τ Hence if the minimal time T is not strictly greater than τ, by choosing T > max{t, τ}, we still have A ω H + ω H C (B φ (t U dt, with the same constant C as before and that does not depend on τ This means that under the generalized gap condition (37, the condition (43 is equivalent to the sufficient condition of Theorem 44 5

26 63 A second observability inequality Proposition 68 Assume that the generalized gap condition (37 holds and that U is separable Let φ be the solution of ( with (ω, ω V H Then for a fixed real number m >, there exist a time T > and a constant C > such that (9 holds if and only if (46 γ >, k =,, M, k n A k, ξ R Ln, B k n Φ kn ξ U, Proof The proof is similar to the one of Proposition 65 because l n λ m kn n (a k n+j λ k n + b k = n+j j= n The details are therefore omitted l n (a k n+j λ( m k n j= γ λ m k n ξ + b k n+j λ m k n ω D(A m + ω D(A m Remark 69 If the standard gap condition (4 holds, the assumption (46 becomes γ >, k n, ξ R l n, Φ kn ξ U γ λ m k n ξ Moreover, if the standard gap condition (4 holds and if the eigenvalues are simple, the assumption (46 becomes γ >, k, B ϕ k U γ λ m k n A remark similar to Remark 67 can be made for polynomial stability 7 Examples We end up this paper by considering different examples for which our abstract framework can be applied To our knowledge, all the examples, with the exception of the first one, are new 7 A wave equation on -d networks with nodal feedbacks In this section we show that the result obtained in [7] enter in the framework of this paper Obviously we will use the same notations for a network that in [7] that we briefly recall (for more details see [7] We denote by E = {e j ; j N} the set of edges e j of length l j > of a given network R and V the set of vertices of R For a function u : R R, we set u j = u ej the restriction of u to e j For a fixed vertex v, we set E v = {j {,, N} ; v ē j } 6

27 If card(e v, v is an interior node Let V int be the set of interior nodes If card(e v =, v is an exterior node Let V ext be the set of exterior nodes For v V ext, we set E v = {j v } We now fix a partition of V ext : V ext = D N V c ext, where D = We actually will impose Dirichlet boundary condition at the nodes of D, Neumann boundary condition at the nodes of N and finally a feedback boundary condition at the nodes of V c ext We further fix a subset V c int of V int where a feedback transmission condition will be imposed By shortness, we denote by V c the set of controlled nodes, namely V c = V c int V c ext We here consider the following initial and boundary system: (47 u j t (x, t u j x (x, t = < x < l j, t >, j {,, N} u j (v, t = u l (v, t = u(v, t t >, j, l E v, v V int u j n j (v, t = α (v u u t (v, t α(v t (v, t τ t >, v V c j E v u j n j (v, t = t >, v V int \Vint c j E v u jv (v, t = u j v n jv (v, t = u(t = = u (, u (t = = u( u t t >, v D t >, v N t (v, t τ = f v (t τ v V c, < t < τ, where α (v i are fixed non-negative real numbers and the delay τ is positive To rewrite this system in the form (, we introduce H = L (R = {u : R R; u j L (, l j, j =,, N} and the operator { D(A H (48 A : (ϕ j j ( d dx ϕ j j where (49 D(A = {ϕ V N H (, l j ; ϕ j (v =, v V int ; n j j E v j= ϕ jv n jv (v =, v N V c ext}, 7

28 and V := {ϕ N H (, l j : ϕ j (v = ϕ k (v j, k E v, v V int ; ϕ jv (v =, v D} j= The operator A is self-adjoint and positive with a compact inverse in H Moreover D(A = V We now define U = U = U = R Vc, where V c is the cardinal of V c, with norm U = and the operators B i for i =, as U D(A (5 B i : (k v v Vc α (v i k v δ v v V c It is easy to verify that Bi (ϕ = ( α (v i ϕ(v T v V c for ϕ D(A and thus B i Bi (ϕ = v V c α (v i ϕ(vδ v for ϕ D(A Hence the system (47 can be rewritten in the form ( We notice that (8 is here reduced to < α, ϕ V, v V c (α (v ϕ(v α v V c (α (v ϕ(v, and therefore, the system (47 is well posed for α (v α (v for all v V c by Theorem, and the energy is decreasing for α (v < α (v for all v V c by Proposition 3 By Proposition 6 of [7], the generalized gap condition (37 holds with M = N + We know, by [7], that the hypothesis ( is satisfied Moreover, the hypothesis (8 is verified because ω m+ V C ω m X ω m D(A C (ω, ω, z m D(A ω m, D(A N where X = V ( H (, l j, by using Corollary 64 of [7] j= Now we define Ψ kn (v the matrix of size k L n by: for all i =,, k, we set { (Ψ kn (v ij = ϕkn+i +j L n, i (v if L n, i < j < L n, i, else, where L n, = and L n, i = i (l n+i for i Then, for all ξ R Ln, i = 8

29 we have B k n Φ kn ξ U, = α l B k n Ψ kn (vξ v V c = αξ l T Ψ kn (v T B T k n B k n Ψ kn (vξ v V c ( = ξ T αψ l kn (v T B T k n B k n Ψ kn (v v V c ξ Therefore by setting M k n = Ψ l kn (v T B T k n B k n Ψ kn (v, v Vcα we see that the assumption (43 becomes γ >, k {,, M}, k n A k, λ min ( M k n γ, and the assumption (46 becomes m R +, γ >, k {,, M}, k n A k, λ min ( M k n γ λ m k n which corresponds respectively to the conditions (44 and (5 from the paper N [7], because λ k kπ L, where L = l j j= Note that if the standard gap condition (4 holds and if all eigenvalues are simple (ie, l k =, then the condition (43 becomes (5 γ >, k, v V c α (v ϕ k (v γ, while the conditions (46 becomes (5 m R +, γ >, k, v V c α (v ϕ k (v γ λ m k Consequently, we find again all the results from [7] (see for instance the examples treated in section 7 of [7], here we can even precise the dependence of the decay with respect to the delay τ 7 An Euler-Bernoulli beam with interior damping We consider an Euler-Bernoulli beam of length with interior damping and a delay term at ξ Two types of boundary conditions will be considered Without delay, these two problems were analyzed in [3, 4], where some decay rates similar to the ones proved below were obtained 9

30 7 Mixed boundary conditions We consider the following initial and boundary system: (53 ω t (x, t + 4 ω ω x (x, t + α 4 t (ξ, tδ ω ξ + α t (ξ, t τδ ξ = < x <, t > ω(, t = ω x (, t = ω x (, t = 3 ω x (, t = t > 3 ω(x, = ω (x, ω t (x, = ω (x < x < ω t (ξ, t τ = f (t τ < t < τ, where ξ (,, α, α > and τ > To enter into the framework of section, we rewrite this system in the form ( For that purpose, we introduce H = L (, and the operator (54 A : D(A H : ϕ d4 dx 4 ϕ where D(A = {ϕ H 4 (, ; ϕ( = ϕ x ( = ϕ x ( = 3 ϕ x ( = } The 3 operator A is self-adjoint and positive with a compact inverse in H We now define U = U = U = R and the operators B and B as (55 B i : U D(A : k α i k δ ξ, i =, It is easy to verify that B i (ϕ = α i ϕ(ξ for ϕ D(A and thus B i B i (ϕ = α i ϕ(ξδ ξ for ϕ D(A and i =, Then the system (53 can be rewritten in the form ( We notice that (8 is equivalent to < α, α αα, and consequently, this system is well posed for α α by Theorem, and the energy is decreasing for α < α by Proposition 3 Let us now state the next well known results about the spectral properties of A Proposition 7 The eigenvalues of the operator A defined in (54 are simple and are given by λ k = ( k+ π 4 of associated eigenvector ϕ k (x = sin( k+ πx, for all k N Consequently the standard gap condition (4 holds, ie, there exists a constant γ > such that λ k+ λ k γ > k, and moreover for all k, B ϕ k U = α sin((kπ + π ξ The hypothesis ( was verified in [3] Moreover, we have by Lemma 9 of [9] the Lemma 7 ξ is a rational number with an irreductible fraction ξ = p, where p is odd q if and only if there exists a constant γ > such that ( k, sin (kπ + π ξ > γ 3

31 Therefore, by applying Proposition 34 and Theorem 44, we obtain the following results: Proposition 73 Assume that α < α Then (i The energy of system (53 decays to if and only if ξ m, m, k N k + (ii The energy of system (53 decays exponentially if ξ is a rational number with an irreductible fraction ξ = p, where p is odd q Remark 74 As mentioned before, in the case α = we recover the results from [3] 7 Other boundary conditions We here consider the following initial and boundary system: (56 ω t (x, t + 4 ω ω x (x, t + α 4 t (ξ, tδ ω ξ + α t (ξ, t τδ ξ = < x <, t > ω(, t = ω(, t = ω x (, t = ω x (, t = t > ω(x, = ω (x, ω t (x, = ω (x < x < ω t (ξ, t τ = f (t τ < t < τ, where ξ (,, α, α > and τ > This system (56 is not exponentially stable if α = as shown in [4] Hence we only consider the polynomial decay of system (56 As before we rewrite this system in the form ( by introducing H = L (, and the operator (57 A : D(A H : ϕ d4 dx 4 ϕ with D(A = {ϕ H 4 (, V ; ϕ x ( = ϕ x ( = }, V = H (, H (, The operator A is self-adjoint and positive with a compact inverse in H We then define U = U = U = R and the operators B, B by (55 Then the system (56 can be rewritten in the form ( and consequently, this system is well posed for α α by Theorem, and the energy is decreasing for α < α by Proposition 3 The spectral properties of A are well known and can be summarized as follows: Proposition 75 The eigenvalues of the operator A defined in (57 are simple and given by λ k = k4 π 4 of eigenvector ϕ k (x = sin(kπx, for all k N Therefore there exists a constant γ > such that the standard gap condition (4 is verified and moreover B ϕ k U = α sin(kπξ 3

32 The hypothesis ( was verified in [4] Let us prove that the condition (8 is satisfied { Lemma 76 Let m R + Then there exists C > such that for all ω X = u V : u (, ξ H 4 (, ξ, u } (ξ, H 4 (ξ,, u x ( = u x ( =, we have ω m+ V C ω m X ω D(A m, where the natural norm in X is given by u X = u H 4 (,ξ + u H 4 (ξ, Proof Let us fix a cut-off [ function ] η D(, [ such] that η = in a neighbourhood of ξ, η = on 3 + ξ 3, and η = on, ξ 3 Since ( ηω D(A, by Lemma 5, we have Since ( ηω m+ V ( ηω D(A m for some C > (depending on η we get (58 ( ηω m+ V In a second step, we set C ( ηω m X ( ηω D(A m = sup ϕ D(A m = sup ϕ D(A m (( ηω, ϕ ϕ m D(A (ω, ( ηϕ ϕ m D(A C ω D(A m, C ω m X ω D(A m ω (x = η(ξ xω (ξ x if < x < l := ξ ω (x = η(x + ξω (x + ξ if < x < l := ξ For any j =,, we introduce the following extension of ω j : (Eω j (x = ω j (x if x (, l j, (Eω j (x = n i= ν iω j ( i x if x ( (n l j,, where ω j is extended by zero outside its support and the real numbers ν i are the unique solution of the system n i= ν i = n i= i ν i = n i= i ν i = n i= 3i ν i = n i= ki ν i =, k =,, n 4, and finally n N is choosen large enough such that n m + 3 3

33 We obtain an extension of ω to a function Eω, which belongs to D(à (due d4 to the four first properties of the ν i, where à is the positive operator dx on 4 the star shaped network S = j=, (, l j j=, ( (n l j,, with interior vertex ξ (identified to and Dirichlet boundary conditions at all other vertices Therefore, we can apply the interpolation lemma 5 to à and then write But we easily check that Consequently, we have Eω m+ D(à C Eω m D(à Eω D(à m ηω m+ D(A Eω m+ D(à ηω m+ D(A C Eω m D(à Eω D(à m Moreover, as E is an extension operator, we have and thus Eω D( à K ω X, ηω m+ D(A C ω m X Eω D(à m To estimate the last factor, we use a duality argument We write For ϕ D(à m, we have S Eωϕ = j=, Eω D( à m = sup ϕ D(à m lj ω j (xϕ j (x dx + By changes of variables, we obtain where S Eωϕ = j=, lj j=, (Eω, ϕ ϕ D( à m (n l j (Eω j (xϕ j (x dx ω j (x(f ϕ j (x dx, n (F ϕ j (x = ϕ j (x + χ j (x ν i i ϕ j ( i x x (, l j, i= the cut-off function χ j being fixed such that χ j on [, l j /3] and χ j on [5l j /6, l j ] (reminding that ω j (x for x > l j /3 Now we notice that the 33

34 conditions on ν i guarantees that F ϕ belongs to D(A m and by Leibniz s rule we have F ϕ m C ϕ D(A D(Ã m Therefore By duality, we conclude that S Eωϕ C ω D(A m ϕ D(Ã m Eω D( Ã m C ω D(A m Consequently, with the previous inequalities, we obtain (59 ηω m+ D(A C ω m X ω D(A m The conclusion follows from (58 and (59 This lemma leads to (8 Indeed for (ω, ω, z D(A, we have ω m+ V C ω m X ω m D(A C (ω, ω, z m D(A ω m D(A Now, we denote by S the set of all real numbers ρ such that ρ / Q and if [, a,, a n, ] is the expansion of ρ as a continued fraction, then the sequence (a n is bounded It is well known that S is uncountable and that its Lebesgue measure is zero Roughly speaking, the set S contains all irrational numbers which are badly approximated by rational numbers In particular, by the Euler- Lagrange theorem, S contains all irrational quadratic numbers (ie the roots of a second order equation with rational coefficients By a classical result, we have the Lemma 77 If s S, then there exists a positive constant γ such that sin(kπs γ k k Therefore, by applying Proposition 34 and Theorem 53 with m =, we obtain the next results: Proposition 78 Assume that α < α Then (i The energy of system (56 decays to if and only if ξ is irrational (ii The energy of system (56 decays polynomially like (+t if ξ belongs to S Remark 79 Again in the case α = we recover the results from [3, 4] 34

35 73 Examples with distributed damping terms 73 A non homogeneous string with distributed damping terms (-d We consider the following initial and boundary system: (6 ω t (x, t ω ω x (x, t + α t (x, tχ ω I + α t (x, t τχ I = in (, (, ω(, t = ω(, t = t > ω(x, = ω (x, ω t (x, = ω (x in (, ω t (x, t τ = f (x, t τ in I (, τ, where here and below χ I denotes the characteristic function of the set I In the remainder of this subsection we assume that α, α >, τ > and Later we will need that I I [, ] (6 δ [, ] and ɛ > : [δ, δ + ɛ] I We rewrite this system in the form ( For that purpose, we introduce H = L (, and the operator (6 A : D(A H : ϕ d dx ϕ where D(A = H (, H (, and V = D(A = H (, The operator A is self-adjoint and positive with a compact inverse in H We then define U i = L (I i and the operators B i as (63 B i : U i H V : k α i k χ Ii where k is the extension of k by zero outside I i (which defines an element of L (, It is easy to verify that B i (ϕ = α i ϕ Ii for ϕ V and thus B i B i (ϕ = α i ϕ Ii χ Ii = α i ϕχ Ii for ϕ V and i =, Then the system (6 can be rewritten in the form ( Moreover B i ϕ U i = α i I i ϕ dx Therefore, we notice that (8 is equivalent to < α, α I ϕ dx αα I ϕ dx, and consequently, this system is well posed for α α by Theorem, and the energy is decreasing for α < α by Proposition 3 35

36 Proposition 7 The eigenvalues of the operator A defined in (6 are simple and given by λ k = (kπ of associated eigenvector ϕ k (x = sin(kπx, for all k N Hence there exists a constant γ > such that the standard gap condition (4 holds Moreover if (6 holds, then there exists γ > such that k, B ϕ k U γ Proof It is well known that the eigenvectors of the operator A are ϕ k (x = sin(kπx of eigenvalue (kπ, k of multiplicity Hence, the standard gap condition (4 is verified From the definition of B we have Bϕ k U = α I ϕ k (x dx [ δ+ɛ α sin(kπx dx = α δ x sin(kπx kπ ( α ɛ kπ ɛ α, ] δ+ɛ for k E( ɛπ + =: k ɛ, where E(x is the entire part of x This leads to the conclusion because for k {,, k ɛ }, we have δ+ɛ δ sin(kπx dx > δ Lemma 7 The operators A and B = (B B L(U, V where U = U U satisfy assumption ( Proof Let i {, } and ϕ L (I i It can be easily checked that v = (λ + A B i ϕ satisfies (64 { λ v d v dx = α i ϕχ Ii v( = v( = As v L (,, v can be written as v = c k ϕ k By replacing v in (64, c k must satisfy k= and therefore Moreover v = αi c k = λ + λ ( ϕ, ϕ k, k k= αi λ + λ ( ϕ, ϕ k ϕ k k λv L (, = λ λ + λ α i ( ϕ, ϕ k k k= 36

37 Now, we set z = λ λ +λ k z = and, if λ = β + iy, with y R, then β + y (β y + λ k + 4β y C(β, where C(β is a positive constant depending only on β Indeed, if y β +λ k, then z β + y ( β +λ k β + β +λ k ( β +λ k which is bounded uniformly in k, and if y β +λ k, then z β + y 4β y, which is a decreasing function with respect to y and thus z β + β +λ k ( 4β β, +λ k which is again uniformly bounded in k Therefore, we have λv L (, C(βα i ( ϕ, ϕ k C(βα i ϕ L (I i, k= which leads to λb j v α L (I j j λv L (, C(βα iα j ϕ L (I i, j {, } Consequently, the operator λ λb j (λi + A B i is bounded on C β and the lemma is proved Therefore, by applying Theorem 44, we obtain the Proposition 7 If α < α and (6 holds, then the energy of system (6 decays exponentially 73 The wave equation with distributed damping terms Let Ω be an open bounded domain of R n, n, with a boundary Γ of class C We assume that Γ is divided into two parts Γ D and Γ N, ie Γ = Γ D Γ N, with Γ D Γ N = and Γ D Let O O Ω such that O is an open neighborhood of Γ N (ie Γ N O Moreover, we assume that x R n is such that (65 (x x ν(x x Γ D 37

38 We consider the following initial and boundary system: (66 ω ω t (x, t ω(x, t + α t (x, tχ ω O + α t (x, t τχ O = in Ω (,, ω(x, t = on Γ D (,, ω ν (x, t = on Γ N (,, ω(x, = ω (x, ω t (x, = ω (x in Ω, ω t (x, t τ = f (x, t τ in O (, τ, where ω ν is the normal derivative of ω and α, α >, τ > In order to reformulate this system in the form (, we introduce H = L (Ω and the operator (67 A : D(A H : ϕ ϕ where D(A = {ϕ V H ϕ (Ω : ν (x = on Γ N} and V = D(A = {ϕ H (Ω : ϕ = on Γ D } The operator A is self-adjoint and positive with a compact inverse in H We then define U i = L (O i and the operators B i as (68 B i : U i H V : k α i k χ Oi where k is the extension of k by zero outside O i (which defines an element of L (Ω It is easy to verify that Bi (ϕ = α i ϕ Oi for ϕ V and thus B i Bi (ϕ = α i ϕ Oi χ Oi for ϕ V and i =, Then the system (66 can be rewritten in the form ( Moreover B i ϕ U i = α i O i ϕ dx, and therefore (8 is equivalent to < α, α O ϕ dx αα O ϕ dx Consequently, this system is well posed for α α by Theorem, and the energy is decreasing for α < α by Proposition 3 To obtain the exponential decay of system (66, we simply check the observability inequality (5 and the hypothesis (, which are the aim of the two following propositions Proposition 73 There exists a time T such that for all times T > T there exists a positive constant C (depending on T for which the observability inequality (5 holds for any regular solution of system ( Proof This proposition is proved in [6] and [4] Proposition 74 The operators A and B = (B B L(U, V where U = U U satisfy the assumption ( 38

39 Proof Let i {, } and ϕ L (O i It can be easily checked that v = (λ + A B i ϕ satisfies λ v v = α i ϕχ Oi in Ω, (69 v = on Γ D, v ν = on Γ N As v L (Ω, it can be written as v = By replacing v in (69, c k must satisfy and therefore v = c k ϕ k k= αi c k = λ + λ ( ϕ, ϕ k, k k= αi λ + λ ( ϕ, ϕ k ϕ k k Moreover, by Parseval s identity, we have λv L (Ω = λ λ + λ α i ( ϕ, ϕ k k k= Now, for λ = β + iy, with y R, we have checked in the previous subsection that λ λ + λ C(β, k where C(β is a positive constant depending only on β Therefore, we have λv L (Ω C(βα i ( ϕ, ϕ k C(βα i ϕ L (O, i k= which leads to λbj v α L (O j λv j L (Ω C(βα iα j ϕ L (O i, j {, } Consequently, the operator λ λb j (λi + A B i is bounded on C β and the lemma is proved Therefore, by applying Theorem 44, we obtain the Proposition 75 If α < α, then the energy of system (66 decays exponentially Remark 76 This result is a generalization of [6] because in [6] the authors supposed that O = O 39

40 733 A beam with distributed damping terms We consider the following initial and boundary system: (7 ω t (x, t + 4 ω ω x (x, t + α 4 t (x, tχ ω I + α t (x, t τχ I = in (, (,, ω(, t = ω(, t = ω x (, t = ω x (, t = t >, ω(x, = ω (x, ω t (x, = ω (x in Ω, ω t (x, t τ = f (x, t τ in I (, τ, where α, α >, τ > and where I I [, ] We rewrite this system in the form ( by introducing H = L (,, the operator A by (57 and the operators B and B by (63 Hence this system is well posed for α α by Theorem, and the energy is decreasing for α < α by Proposition 3 By the results of the previous subsections, we know that the standard gap condition holds and if (6 holds that the eigenvector ϕ k (x = sin(kπx of A associated with λ k = k4 π 4 satisfies for some γ > B ϕ k U γ, Lemma 77 The operators A and B = (B B L(U, V where U = U U satisfy assumption ( Proof The proof is the same than in the previous subsection Therefore, by applying Proposition 34 and Theorem 44, we obtain the Proposition 78 If α < α and (6 holds, then the energy of system (7 decays exponentially 734 A wave equation on -d networks with internal damping terms In this last subsection we consider again a wave equation on a given network R but we suppose that the feedbacks are located in the edges Namely with the notations from section 7 we consider the system: (7 u j t (x, t u j x (x, t + α (j u j t (x, tχ I (j +α (j u j t (x, t τχ = < x < l I (j j, t >, j {,, N}, u j (v, t = u l (v, t = u(v, t t >, j, l E v, v V int, u j n j (v, t = t >, v V\D, j E v u jv (v, t = t >, v D, u(t = = u (, u t (t = = u(, u j t (x, t τ = f j (x, t τ in I(j (, τ, j {,, N}, 4