Stability of an abstract wave equation with delay and a Kelvin Voigt damping

Transcription

1 Stability of an abstract wave equation with delay and a Kelvin Voigt damping University of Monastir/UPSAY/LMV-UVSQ Joint work with Serge Nicaise and Cristina Pignotti

2 Outline 1 Problem The idea Stability 2 3 The discrete spectrum The continuous spectrum Application to the stabilization of the wave equation with delay and a Kelvin Voigt damping 7 8

3 Outline 1 Problem The idea Stability 2 3 The discrete spectrum The continuous spectrum Application to the stabilization of the wave equation with delay and a Kelvin Voigt damping 7 8

4 Outline 1 Problem The idea Stability 2 3 The discrete spectrum The continuous spectrum Application to the stabilization of the wave equation with delay and a Kelvin Voigt damping 7 8

5 Outline 1 Problem The idea Stability 2 3 The discrete spectrum The continuous spectrum Application to the stabilization of the wave equation with delay and a Kelvin Voigt damping 7 8

6 Outline 1 Problem The idea Stability 2 3 The discrete spectrum The continuous spectrum Application to the stabilization of the wave equation with delay and a Kelvin Voigt damping 7 8

7 Outline 1 Problem The idea Stability 2 3 The discrete spectrum The continuous spectrum Application to the stabilization of the wave equation with delay and a Kelvin Voigt damping 7 8

8 Outline 1 Problem The idea Stability 2 3 The discrete spectrum The continuous spectrum Application to the stabilization of the wave equation with delay and a Kelvin Voigt damping 7 8

9 Outline 1 Problem The idea Stability 2 3 The discrete spectrum The continuous spectrum Application to the stabilization of the wave equation with delay and a Kelvin Voigt damping 7 8

10 Problem Outline Problem The idea Stability It is well-known that delay equations like the simplest one of parabolic type, u t (t; x) = u(t τ; x); with a delay parameter τ > 0, or of hyperbolic type, are not well-posed. u tt (t; x) = u(t τ; x); Their instability is given in the sense that there is a sequence of initial data remaining bounded, while the corresponding solutions, at a fixed time, go to infinity in an exponential manner, see Jordan, Dai & Mickens and Dreher, Quintanilla & Racke.

11 Problem The idea Stability The same phenomenon of instability is given for a general class of problems of the type d n u (t) = Au(t τ); dtn n N; fixed, whenever ( A) is linear operator in a Banach space having a sequence of real eigenvalues (λ k ) k such that 0 < λ k as k.

12 Problem The idea Stability The so-called α β-system with delay, { utt (t) + aau(t τ) ba β θ(t) = 0, θ t (t) + da α θ(t) + ba β u t (t) = 0 for functions u, θ : [0; + ) H, with A being a self-adjoint operator in the Hilbert space H, having a countable complete orthonormal system of eigenfunctions (φ j ) j with corresponding eigenvalues 0 < λ j as j. The thermoelastic plate equations appear with α = β = 1 2 and A = ( D ) 2.

13 A := {(β, α) α β, α 2β 1/2}, (τ = 0). (1.20) Outline eigenvalues 0 < Problem λj as j. The thermoelastic plate equations appear with α = β = 1 2 The idea and A = ( D) 2, where D denotes the Laplace operator realized in L Stability 2 (G) on some bounded domain Application G in R n withtodirichlet the wave boundary equation conditions. This original α-β-system without delay (τ = 0) was introduced by Muñoz Rivera & Racke [26] and, independently, by Ammar Khodja & Benabdallah [1], and investigated with respect to exponential stability and analyticity of the associated semigroup, the latter also for the Cauchy problem, where Ω = R n, and, more general, A = ( ) η for η > 0 arbitrary, and in arbitrary L p -spaces for 1 < p <, see Denk & Racke [6]. It was shown that we have a strong smoothing property for parameters (β, α) in the region Asm (see Figure 1.1), where It was shown that we have a strong smoothing property for parameters (α; β) in the region A sm := {(β, Asm α); := {(β, 1 α) 12β 2β < α α < 2β, < α 2β, > α 1}, > 2β (τ = 0), 1}, (τ(1.19) = 0). and that the analyticity (in L p (R n )) is given in the region Aan (see Figure 1.2), where α β Figure 1.1: Area of smoothing Asm (without delay)

14 Problem The idea Stability The α β-system with delay is not well-posed in the region A 1 in := {(β, α); 0 β α 1, α 12 }, (β, α) (1, 1). α β Figure 1.3: Area of instability A 1 in (with delay) The paper is organized as follows: In Section 2 we 20 shall October discuss 2017, the second-order LJLL-GdT thermoe- Contrôle

15 Problem The idea Stability A similar result will hold for the related system { utt (t) + aau(t) ba β θ(t) = 0, θ t (t) + da α θ(t τ) + ba β u t (t) = 0 in the region A 2 in := {(β, α); 0 β α 1, (β, α) (1, 1)}. α β Figure 1.4: Area of instability A 2 in (with delay) for t 0 and x {0, L}, and with initial conditions

16 The idea Outline Problem The idea Stability Datko: The effect of a small delay ẅ(t) + Aw(t) + BB ẇ(t τ) = 0, t 0, w(0) = w 0, ẇ(0) = w 1, ẇ(t) = f 0 (t), t ( τ, 0), τ > 0 is the time delay.

17 Problem The idea Stability ẅ(t) + Aw(t) + α 1 BB ẇ(t) + α 2 BB ẇ(t τ) = 0, t 0, w(0) = w 0, ẇ(0) = w 1, ẇ(t) = f 0 (t), t ( τ, 0), τ > 0 is the time delay. 0 < α 2 < α 1.

18 Problem The idea Stability u tt (x, t) u(x, t) + au t (x, t τ) = 0, x Ω, t > 0, (1) u(x, t) = 0, x Γ 0, t > 0 (2) u ν (x, t) = ku t(x, t), x Γ 1, t > 0 (3) u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), x Ω, (4) u t (x, t) = g(x, t), x Ω, t ( τ, 0), (5) where ν stands for the unit normal vector of Ω pointing towards the exterior of Ω and u ν is the normal derivative. Moreover, the constant τ > 0 is the time delay, a and k are two positive numbers and the initial data are taken in suitable spaces.

19 Problem The idea Stability Theorem (A-Nicaise-Pignotti) For any k > 0 there exist positive constants a 0, C 1, C 2 such that E(t) C 1 e C2t E(0), (6) for any regular solution of problem (29)-(31) with 0 a < a 0. The constants a 0, C 1, C 2 are independent of the initial data but they depend on k and on the geometry of Ω.

20 Problem The idea Stability The opposite problem, that is to contrast the effect of a time delay in the boundary condition with a velocity term in the wave equation, is still, as far as we know, open and it seems to be much harder to deal with. However, there is a positive answer by Datko, Lagnese and Polis [6] in the one dimensional case for the problem u tt (x, t) u xx (x, t) + 2au t (x, t) + a 2 u(x, t) = 0, 0 < x < 1, t > 0, (7) u(0, t) = 0, t > 0 (8) u x (1, t) = ku t (1, t τ), t > 0; (9) (10)

21 Problem The idea Stability with a, k positive real numbers. Indeed, through a careful spectral analysis, in [6] the authors have shown that, for any a > 0, if k satisfies 0 < k < 1 e 2a, (11) 1 + e 2a then the spectrum of the system (7) (9) lies in R ω β, where β is a positive constant depending on the delay τ.

22 Problem The idea Stability Stabilization by switching time-delay ẅ(t) + Aw(t) = 0, 0 t T 0, (12) ẅ(t) + Aw(t) + µ 1 BB ẇ(t) = 0, (2i + 1)T 0 t (2i + 2)T 0, (13) ẅ(t) + Aw(t) + µ 2 BB ẇ(t T 0 ) = 0, (2i + 2)T 0 t (2i + 3)T 0, (14) w(0) = w 0, ẇ(0) = w 1, (15) where T 0 > 0 is the time delay, µ 1, µ 2 are real numbers and the initial datum (w 0, w 1 ) belongs to a suitable space.

23 Examples Outline Problem The idea Stability Pointwise stabilization: u tt (x, t) u xx (x, t) = 0, (0, l) (0, 2l), (16) u tt (x, t) u xx (x, t) + a u t (ξ, t 2l) δ ξ = 0, (0, l) (2l, + ), (17) u(0, t) = 0, u x (l, t) = 0, (0, + ), (18) u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), (0, l), (19)

24 Problem The idea Stability Boundary stabilization: u tt (x, t) u xx (x, t) = 0 (0, l) (0, + ), (20) u(0, t) = 0, (0, + ), (21) u x (l, t) = 0, (0, 2l), (22) u x (l, t) = µ 1 u t (l, t), (2(2i + 1)l, 2(2i + 2)l), i N, (23) u x (l, t) = µ 2 u t (l, t 2l), (2(2i + 2)l, 2(2i + 3)l), i N, (24) u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), (0, l), (25) where l > 0, µ 1, µ 2, a and ξ (0, l) are constants.

25 Problem The idea Stability Theorem (A-Nicaise-Pignotti) We suppose that ξ = l 2. Then for any a (0, 2) C 1, C 2 > 0 s.t. for all initial data in H, the solution of (29)-(30) satisfies E p (t) C 1 e C2t. (26) The constant C 1 depends on the initial data, on l and on a, while C 2 depends only on l and on a.

26 Problem The idea Stability Where For any µ 1, µ 2 satisfying one of the following conditions 1 < µ 2 < µ 1, µ 1 < µ 2 < 1, (27) C 1, C 2 > 0 s.t. for all initial data in H, the solution of (20) (25) satisfies E b (t) C 1 e C2t. (28) E p (t) = E b (t) = 1 2 l and H = { u H 1 (0, l), u(0) = 0 } L 2 (0, l). 0 { u x (x, t) 2 + u t (x, t) 2 }dx,

27 Problem The idea Stability Stability of an abstract wave equation with delay and Kelvin-Voigt damping We consider a stabilization problem for an abstract wave equation with delay and a Kelvin Voigt damping. We prove an exponential stability result for appropriate damping coefficients by using a frequency domain approach.

28 Problem The idea Stability Our main goal is to study the internal stabilization of a delayed abstract wave equation with a Kelvin Voigt damping. More precisely, given a constant time delay τ > 0, we consider the system given by: u (t) + a BB u (t) + BB u(t τ) = 0, in (0, + ), (29) u(0) = u 0, u (0) = u 1, (30) B u(t τ) = f 0 (t τ), in (0, τ), (31)

29 Problem The idea Stability where a > 0 is a constant, B : D(B) H 1 H is a linear unbounded operator from a Hilbert space H 1 into another Hilbert space H equipped with the respective norms H1, H and inner products (, ) H1, (, ) H, and B : D(B ) H H 1 is the adjoint of B. The initial datum (u 0, u 1, f 0 ) belongs to a suitable space. We suppose that the operator B satisfies the following coercivity assumption: there exists C > 0 such that B v H1 C v H, v D(B ). (32) We set V = D(B ) and we assume that it is closed with the norm v V := B v H1 and that it is compactly embedded into H.

30 Problem The idea Stability To restitute the well-posedness character and its stability we propose to add the Kelvin Voigt damping term a BB u. Hence the stabilization of problem (29) (31) is performed using a frequency domain approach combined with a precise spectral analysis.

31 Outline We introduce the auxiliary variable z(ρ, t) = B u(t τρ), ρ (0, 1), t > 0. (33) Then, problem (29) (31) is equivalent to u (t) + a BB u (t) + Bz(1, t) = 0, in (0, + ), (34) τz t (ρ, t) + z ρ (ρ, t) = 0 in (0, 1) (0, + ), (35) u(0) = u 0, u (0) = u 1, (36) z(ρ, 0) = f 0 ( ρτ), in (0, 1), (37) z(0, t) = B u(t), t > 0. (38)

32 If we denote U := (u, u, z), then U := (u, u, z t ) = ( u, abb u Bz(1, t), τ 1 ) z ρ. Therefore, problem (34) (38) can be rewritten as { U = AU, U(0) = (u 0, u 1, f 0 ( τ)), (39)

33 where the operator A is defined by u v A v z := abb v Bz(, 1) τ 1 z ρ with domain { D(A) := (u, v, z) D(B ) D(B ) H 1 (0, 1; H 1 ) : } ab v + z(1) D(B), B u = z(0),, (40) in the Hilbert space H := D(B ) H L 2 (0, 1; H 1 ), (41) equipped with the standard inner product 1 ((u, v, z), (u 1, v 1, z 1 )) H = (B u, B u 1 ) H1 + (v, v 1 ) H + ξ (z, z 1 ) H1 dρ, 0 where ξ > 0 is a parameter fixed later on.

34 We will show that A generates a C 0 semigroup on H by proving that A cid is maximal dissipative for an appropriate choice of c in function of ξ, τ and a. We prove the next result. Lemma If ξ > 2τ a, then there exists a = maximal dissipative in H. ( 1 a + ξ 2τ ) 1 > 0 such that A a 1 Id is

35 We have then the following result. Proposition The system (29) (31) is well posed. More precisely, for every (u 0, u 1, f 0 ) H, there exists a unique solution (u, v, z) C(0, +, H) of (39). Moreover, if (u 0, u 1, f 0 ) D(A) then (u, v, z) C(0, +, D(A)) C 1 (0, +, H) with v = u and u is indeed a solution of (29) (31).

36 The discrete spectrum The continuous spectrum As D(B ) is compactly embedded into H, the operator BB : D(BB ) H H has a compact resolvent. Hence let (λ k ) k N be the set of eigenvalues of BB repeated according to their multiplicity (that are positive real numbers and are such that λ k + as k + ) and denote by (ϕ k ) k N the corresponding eigenvectors that form an orthonormal basis of H.

37 The discrete spectrum The continuous spectrum Lemma If τ a, then any eigenvalue λ of A satisfies R λ < 0.

38 The discrete spectrum The continuous spectrum If a < τ, we show that there exist some pairs of (a, τ) for which the system (29) (31) becomes unstable. Hence the condition τ a is optimal for the stability of this system. Lemma There exist pairs of (a, τ) such that 0 < a < τ and for which the associated operator A has a pure imaginary eigenvalue.

39 The discrete spectrum The continuous spectrum Recall that an operator T from a Hilbert space X into itself is called singular if there exists a sequence u n D(T ) with no convergent subsequence such that u n X = 1 and Tu n 0 in X. T is singular if and only if its kernel is infinite dimensional or its range is not closed.

40 The discrete spectrum The continuous spectrum Let Σ := { λ C; aλ + e λτ = 0 }. The following results hold: Theorem 1 If λ Σ, then λi A is singular. 2 If λ Σ, then λi A is a Fredholm operator of index zero.

41 The discrete spectrum The continuous spectrum Lemma If τ a, then Σ {λ C : R λ < 0}.

42 The discrete spectrum The continuous spectrum Corollary It holds and therefore if τ a σ(a) = σ pp (A) Σ, σ(a) {λ C : R λ < 0}.

43 We show that if τ a and ξ > 2τ a, the semigroup eta decays to the null steady state with an exponential decay rate. Theorem (A-Nicaise-Pignotti) If ξ > 2τ a and τ a, then there exist constants C, ω > 0 such that the semigroup e ta satisfies the following estimate e ta L(H) C e ωt, t > 0. (42)

44 To obtain this, our technique is based on a frequency domain approach and combines a contradiction argument to carry out a special analysis of the resolvent. We will employ the following frequency domain theorem for uniform stability of a C 0 semigroup on a Hilbert space:

45 Lemma A C 0 semigroup e tl on a Hilbert space H satisfies e tl L(H) C e ωt, for some constant C > 0 and for ω > 0 if and only if and Rλ < 0, λ σ(l), (43) sup (λi L) 1 L(H) <. (44) R λ 0 where σ(l) denotes the spectrum of the operator L.

46 According to Corollary 9 the spectrum of A is fully included into R λ < 0, which clearly implies (43). Then the proof of Theorem 10 is based on the following lemma that shows that (44) holds with L = A. Lemma The resolvent operator of A satisfies condition sup (λi L) 1 L(H) <. (45) R λ 0

47 Suppose that condition (45) is false. By the Banach-Steinhaus Theorem, there exists a sequence of complex numbers λ n such that R λ n 0, λ n + and a sequence of vectors Z n = (u n, v n, z n ) t D(A) with Z n H = 1 (46) such that i.e., (λ n I A)Z n H 0 as n, (47) λ n u n v n f n 0 in D(B ), (48) λ n v n + a B(B v n + z n (1)) g n 0 in H, (49) λ n z n + τ 1 ρ z n h n 0 in L 2 ((0, 1); H 1 ). (50)

48 Our goal is to derive from (47) that Z n H converges to zero, that furnishes a contradiction. We notice that from (48) we have (λ n I A)Z n H R ((λ n I A)Z n, Z n ) H ( R λ n a 1 B u n 2 ξ H 1 + 2τ 1 ) z n (1) 2 H a 1 + a 2 B v n 2 H 1 = R λ n a 1 B v n + B 2 ( f n ξ λ n + H 1 2τ 1 ) z n (1) 2 H a 1 + a 2 B v n 2 H 1, ( ) 1 where a 1 = a + ξ 2τ.

49 Hence using the inequality we obtain that B v n + B f n 2 H 1 2 B v n 2 H B f n 2 H 1, (λ n I A)Z n H R λ n 2a 1 λ n 2 B f n 2 H 1 + ( ξ 2τ 1 ) z n (1) 2 H a 1 ( a + 2 2a 1 λ n 2) B v n 2 H 1.

50 Hence for n large enough, say n n, we can suppose that and therefore for all n n, we get a 2 2a 1 λ n 2 a 4. (λ n I A)Z n H R λ n 2a 1 λ n 2 B f n 2 H 1 + ( ξ 2τ 1 ) z n (1) 2 H a 1 + a 4 B v n 2 H 1. (51)

51 By this estimate, (47) and (48), we deduce that z n (1) 0, B v n 0, in H 1, as n, (52) and in particular, from the coercivity (32), that v n 0, in H, as n. This implies according to (48) that u n = 1 v n + 1 f n 0, in D(B ), as n, (53) λ n λ n as well as z n (0) = B u n 0, in H 1, as n. (54)

52 By integration of the identity (50), we have z n (ρ) = z n (0) e τλnρ + τ Hence recalling that Rλ n 0 ρ 0 e τλn(ρ γ) h n (γ) dγ. (55) 2τ ρ 0 z n (ρ) 2 H 1 dρ 2 z n (0) 2 H 1 + h n (γ) 2 H 1 dγρ dρ 0, as n. All together we have shown that Z n H converges to zero, that clearly contradicts Z n H = 1.

53 We study the internal stabilization of a delayed wave equation. More precisely, we consider the system given by : u tt (x, t) a u t (x, t) u(x, t τ) = 0, in Ω (0, + ), (56) u = 0, on Ω (0, + ), (57) u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), in Ω, (58) u(x, t τ) = f 0 (t τ), in Ω (0, τ), (59) where Ω is a smooth open bounded domain of R n and a, τ > 0 are constants.

54 This problem enters in our abstract framework with H = L 2 (Ω), B = div : D(B) = H 1 (Ω) n L 2 (Ω), B = : D(B ) = H0 1 (Ω) H 1 := L 2 (Ω) n, the assumption (32) being satisfied owing to Poincaré s inequality.

55 The operator A is then given by A u v z := v a v + div z(, 1) τ 1 z ρ, with domain { D(A) := (u, v, z) H0 1 (Ω) H0 1 (Ω) L 2 (Ω; H 1 (0, 1)) : } a v + z(, 1) H 1 (Ω), u = z(, 0) in Ω, (60) in the Hilbert space H := H 1 0 (Ω) L2 (Ω) L 2 (Ω (0, 1)).

56 Corollary If τ a, the system (56) (59) is exponentially stable in H, namely for ξ > 2τ a, the energy E(t) = 1 ( ( u(x, t) 2 + u t (x, t) 2 ) dx+ 2 Ω satisfies ξ Ω 1 0 ) u(x, t τρ) 2 dxdρ, E(t) Me ωt E(0), t > 0, (u 0, u 1, f 0 ) D(A), for some positive constants M and ω.

57 Outline By a careful spectral analysis combined with a frequency domain approach, we have shown that the system (29) (31) is exponentially stable if τ a and that this condition is optimal. But from the general form of (29), we can only consider interior Kelvin-Voigt dampings. Hence an interesting perspective is to consider the wave equation with dynamical Ventcel boundary conditions with a delayed term and a Kelvin-Voigt damping.

58 Outline E. M. Ait Ben Hassi, K. Ammari, S. Boulite and L. Maniar, Feedback stabilization of a class of evolution equations with delay, Journal of Evolution Equations, 1 (2009), K. Ammari, S. Nicaise and C. Pignotti, Stability of abstract wave equation with delay and a Kelvin Voigt damping, Asymptotic Analysis, 95 (2015), K. Ammari and S. Nicaise, Stabilization of elastic systems by collocated feedback, Lecture Notes in Mathematics, 2124, Springer, Cham, K. Ammari, S. Nicaise and C. Pignotti, Stabilization by switching time-delay, Asymptotic Analysis, 83 (2013),

59 R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26 (1988), R. Datko, J. Lagnese and P. Polis, An exemple on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1985), S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), R. Racke, Instability of coupled systems with delay, Commun. Pure Appl. Anal., 11 (2012),

60 Thank you for your attention