Stability of nonlinear locally damped partial differential equations: the continuous and discretized problems. Part II

Transcription

1 . Stability of nonlinear locally damped partial differential equations: the continuous and discretized problems. Part II Fatiha Alabau-Boussouira 1 Emmanuel Trélat 2 1 Univ. de Lorraine, LMAM 2 Univ. Paris 6 (LJLL) and IUF 33th International Summer School of Automatic Control, Grenoble, september

2 Outline of the first lecture 1 Examples of dissipative infinite dimensional systems 2 Nonlocal dissipation: memory damping 3 Results for infinite dimensional systems: the indirect method 4 The main results 5 The proofs 6 Examples of PDE s 7 Lower estimates. Optimality 8 A methodology for lower energy estimates through comparison principles 9 Some results on non local dissipation

3 Examples of dissipative infinite dimensional systems We consider the following abstract equation u + Au + feedback operator[u] = 0 (u, u )(0) = (u 0, u 1 ) Here A stands for an unbounded linear operator in an Hilbert space H, which is closed, coercive self-adjoint with dense domain in H.

4 Examples of dissipative infinite dimensional systems Dissipative systems: infinite dimensions The feedback operator can be bounded, in case of locally distributed feedbacks for instance unbounded in H, in case of boundary feedback for instance linear or nonlinear frictional, i.e. local meaning that at time t and position in space x, it depends only on x and t nonlocal, i.e. as for the memory damping case involved for viscoelastic materials or with delay. The same questions than for the finite dimensional case arise, except now that there is a space dependence and the problem is set-up in an infinite dimensional framework (Sobolev spaces)

5 Examples of dissipative infinite dimensional systems Dissipative systems: infinite dimensions The feedback operator can be bounded, in case of locally distributed feedbacks for instance unbounded in H, in case of boundary feedback for instance linear or nonlinear frictional, i.e. local meaning that at time t and position in space x, it depends only on x and t nonlocal, i.e. as for the memory damping case involved for viscoelastic materials or with delay. The same questions than for the finite dimensional case arise, except now that there is a space dependence and the problem is set-up in an infinite dimensional framework (Sobolev spaces)

6 Examples of dissipative infinite dimensional systems Model examples We consider the wave equation. Case of frictional feedbacks: the feedback depends locally on the velocity and on the space variable. Two examples : locally distributed feedbacks: u tt u + ρ(., u t ) = 0 u = 0 in [0, ) Γ (u, u t )(0,.) = (u 0, u 1 ) in Ω, where Γ = Ω.

7 Examples of dissipative infinite dimensional systems locally distributed the feedback is "nonvanishing" in a subset ω of Ω. dissipative ρ monotone nondecreasing with respect to the second variable. In this case, formally multiply by u t and integrate over Ω. This gives, after an integration by parts and since u = 0 on the boundary ) (u tt u t + u u t dx = u t ρ(., u t ) dx 0 Ω Ω Hence the natural energy is E(t) = 1 ( u t (t) 2 + u(t) 2 dx ) 2 Ω The dissipation relation (for strong solutions) is: E (t) = u t ρ(., u t ) dx 0, t 0. Ω

8 Examples of dissipative infinite dimensional systems Using semi-group theory, one can show well-posedness in H 1 0 (Ω) L2 (Ω) for initial data in H 1 0 (Ω) L2 (Ω). If the initial data are in the domain of the associated infinitesimal generator, then one can show that the solution is a strong solution, in particular it takes values in H 2 (Ω) H 1 0 (Ω) H1 0 (Ω).

9 Examples of dissipative infinite dimensional systems Model examples boundary distributed feedbacks: u tt u = 0 u = 0 in [0, ) Γ 1 u ν + η(.)u + ρ(., u t) = 0 in [0, ) Γ 0 (u, u t )(0,.) = (u 0, u 1 ) in Ω, where {Γ 0, Γ 1 } is a partition of Γ η is a nonnegative function.

10 Examples of dissipative infinite dimensional systems Model examples Once again we multiply formally the equation by u t and integrate over Ω. This gives using Green s formula Ω ) ) (u tt u t + u u t dx + (ηuu t + ρ(., u t )u t dσ Γ 0 Thus the natural energy is given by E(t) = 1 ( u t (t) 2 + u(t) 2 dx + 2 Ω Γ 0 η u 2 dσ ) The dissipation relation (for strong solutions) is: E (t) = u t ρ(., u t ) dσ 0, t 0. Γ 0

11 Nonlocal dissipation: memory damping Memory type feedbacks/ non local feedback operators:: the feedback depends on the memory of the "material", this is the case for viscoelastic materials and is nonlocal with respect to time. u tt u + k u = 0 u = 0 in [0, ) Γ (u, u t )(0,.) = (u 0, u 1 ) in Ω, where t (k v)(t) = k(t s)v(s) ds. 0 and the kernel k is positive, differentiable and decaying at infinity.

12 Nonlocal dissipation: memory damping Model examples In this case, the natural energy is E u (t) = 1 2 u t(t) 2 L 2 (Ω) dx ( t 1 t 0 0 ) k(s) ds u(t) 2 L 2 (Ω) k(t s) u(t) u(s) 2 L 2 (Ω) ds and the dissipation relation is: E u(t) = 1 2 k(t) u(t) t 0 k (t s) u(s) u(t) 2 ds 0

13 Nonlocal dissipation: memory damping Model examples But also : higher order systems: Petrowsky equation u tt + 2 u + ρ(x, u t ) = 0 in Ω R, u = 0 = u on Σ = Γ R, (u, t u)(0) = (u 0, u 1 ) on Ω. We define the energy of a solution u by E(t) = 1 2 ( u t 2 + u 2 ) dx. The dissipation relation is E (t) = u t ρ(., u t ) dx 0, t 0. Ω Ω

14 Nonlocal dissipation: memory damping Model examples Coupled systems such as Timoshenko beams, 1-D : { ρ 1 ϕ tt k(ϕ x + ψ) x = 0 t > 0, 0 < x < L, ρ 2 ψ tt bψ xx + k(ϕ x + ψ) + ρ(ψ t ) = 0 t > 0, 0 < x < L, where Remark the functions ϕ and ψ denote respectively the transverse displacement of the beam and the rotation angle of the filament. The parameters ρ 1, ρ 2, k and b denote positive constants characterizing physical properties of the beam and the filament. Note that there is an additional difficulty here, since only the second equation is damped.

15 Nonlocal dissipation: memory damping Model examples Usually one considers two types of boundary conditions for this system, namely ϕ = ψ x = 0, t > 0, x = 0, x = L, ϕ = ψ = 0, t > 0, x = 0, x = L. The energy of solutions for both sets of boundary conditions is defined by E(t) = 1 2 L 0 ( ρ 1 ϕ 2 t + ρ 2 ψ 2 t + bψ 2 x + k ϕ x + ψ 2) dx. The dissipation relation for strong solutions reads E (t) = L 0 ψ t ρ(ψ t ) dx.

16 Results for infinite dimensional systems: the indirect method Results for infinite dimensional systems: the indirect method We have seen how to derive sharp upper energy estimates thanks to the optimal-weight convexity for finite dimensional systems method There exist a prolific literature on nonlinear stabilization of hyperbolic reversible PDE s in infinite dimensions: References Zuazua, Nakao, Komornik, Haraux in the 90 s for polynomially growing feedbacks and to Lasiecka and Tataru 93, Martinez 99, Liu and Zuazua 99, Eller et al. 2002, A.-B.-2005, 2010, Daoulatli Lasiecka et al 2008, A.-B. and Ammari and for arbitrary growing feedback close to 0

17 Results for infinite dimensional systems: the indirect method Our goals are here to present here a unified approach between the finite dimensional and infinite dimensional cases to distinguish between what is linked to geometric hypotheses and what is linked to nonlinearity of the feedback to have a general way to obtain simple, expected-optimal decay energy decay rates that relies on a methodology which adapts as well to other types of feedbacks and can also be adapted for discretized equations

18 Results for infinite dimensional systems: the indirect method The upper estimates for the finite dimensional case can be extended to the infinite dimensional case under geometric conditions on the damping region by either a direct method or an indirect method. In the direct method the results are proved user multiplier geometric conditions. These conditions are not sharp geometrically in general. One needs to understand that geometric arguments can be separated from the arguments to deal with the nonlinearity of the feedbacks. If we want to combine the sharp conditions of the geometric optics (Bardos Lebeau Rauch 1992, Burq Gérard 1997, Burq 1997 based on microlocal analysis, or issued from other techniques, such frequency domains approach or non harmonic analysis) with the optimal-weight convexity method, we have to use an indirect method, deriving the energy decay rates from an observability for the (linear) undamped corresponding evolution PDE.

19 Results for infinite dimensional systems: the indirect method Let H be a Hilbert space with norm, A be bounded linear self-ajoint coercive operator on H and B be a linear bounded self-adjoint nonnegative operator on H. We consider the following two problems The conservative problem φ + Aφ = 0, The linearly damped problem y + Ay + By = 0.

20 Results for infinite dimensional systems: the indirect method Then one can prove that the linearly damped system is exponentially stable, that is there exists δ > 0 and c > 0 such that y (t) 2 + A 1/2 y(t) 2 ce δt( y A 1/2 y 0 2), (y 0, y 1 )(initial data) D(A 1/2 ) H, t 0. if and only if all the solutions of the conservative system satisfy the following observabilty inequality T φ (0) 2 + A 1/2 φ(0) 2 C B 1/2 φ 2 dt. 0 for T > 0 (sufficiently large) and C 0 (not depending on the initial data). This result is due to Haraux 1989 (see Emmanuel s lecture for a proof).

21 Results for infinite dimensional systems: the indirect method This result allows to deduce the exponential stabilization of a linearly damped wave-like equation from an observability estimate for the conservative corresponding equation. Extension of this result to the case of linear unbounded damping operators by Ammari Tucsnak Such observability results are available under more general geometric assumptions than the multiplier ones (used for the direct method). With the indirect method, the geometric aspects are "hidden" in the assumption of the observability inequality. Can we combine this approach to the optimal approach to treat nonlinear stabilization thanks to the optimal-weight convexity in the case of nonlinear dampings B?

22 Results for infinite dimensional systems: the indirect method This result allows to deduce the exponential stabilization of a linearly damped wave-like equation from an observability estimate for the conservative corresponding equation. Extension of this result to the case of linear unbounded damping operators by Ammari Tucsnak Such observability results are available under more general geometric assumptions than the multiplier ones (used for the direct method). With the indirect method, the geometric aspects are "hidden" in the assumption of the observability inequality. Can we combine this approach to the optimal approach to treat nonlinear stabilization thanks to the optimal-weight convexity in the case of nonlinear dampings B?

23 The main results So we will now consider the case of nonlinear damping operators B. This is a joint work with Kaïs Ammari.

24 The main results We consider the following second order differential equation {ẅ(t) + Aw(t) + a(.)ρ(., ẇ) = 0, t (0, ), x Ω w(0) = w 0, ẇ(0) = w 1. where Ω is a bounded open set in R N, with a boundary Γ sufficiently smooth. we set H = L 2 (Ω), with its usual scalar product denoted by, H and the associated norm H A : D(A) H H is a densely defined self-adjoint linear operator satisfying for some C > 0. Au, u H C u 2 H u D(A)

25 The main results We define the energy of a solution by E w (t) = 1 2 ( ) A 1/2 w(t) 2 H + ẇ(t)) 2 H We make the following assumptions on the feedback ρ and on a. ρ C(Ω R; R) is a continuous monotone nondecreasing function with respect to the second variable on Ω such that ρ(., 0) = 0 on Ω. There exists a continuous strictly increasing odd function g in C([ 1, 1]; R) continuously differentiable in a neighbourhood of 0 and satisfying g(0) = g (0) = 0 with c 1 g( v ) ρ(., v) c 2 g 1 ( v ), v 1, a.e. on Ω, c 1 v ρ(., v) c 2 v, v 1, a.e. on Ω, where c i > 0 for i = 1, 2.

26 The main results a L (Ω), with a 0 on Ω a > 0 such that a a on ω. where ω stands for the subregion of Ω on which the feedback ρ is active U = L 2 (ω).

27 The main results Define exactly as in the finite dimensional case H(x) = xg( x), x [0, r 2 0 ] Thanks to the above assumption, H is of class C 1 and is strictly convex on [0, r 2 0 ], where r 0 > 0 is a sufficiently small number. Ĥ the extension of H to R where Ĥ(x) = + for x R\[0, r 2 0 ]. Ĥ stands for the convex conjugate function of Ĥ, i.e.: Ĥ (y) = sup x R {xy Ĥ(x)}. L is defined by Ĥ (y), if y (0, + ), L(y) = y 0, if y = 0,

28 The main results Define Λ H on (0, r 2 0 ] by Λ H (x) = H(x) xh (x). Recall that we proved in part I that L is strictly increasing continuous and onto from [0, + ) on [0, r 2 0 ). We also define a function K r from (0, r] on [0, + ) by: K r (τ) = r τ dy yl 1 (y), a function ψ r as the (implicitly) strictly increasing onto function 1 defined from [ L 1 (r), + ) on [ 1, + ) by: L 1 (r) ψ r (z) = z + K r (L( 1 z )) z, z 1 L 1 (r). Hence ψ 1 r ( t T 0 ) goes to as t goes to.

29 The main results We set H 1/2 = D(A 1/2 ). Then, we can state the following result.

30 The main results Theorem (A.-B. Ammari JFA 2011) Assume that ρ and a satisfy the above assumptions, that there exists r 0 > 0 sufficiently small so that the function H is strictly convex on [0, r0 2] and that H (x) lim x 0 + Λ H (x) = 0 Moreover assume that there exists T > 0 such that the following observability inequality is satisfied for the linear conservative system { φ(t) + Aφ(t) = 0, φ(0) = φ 0, φ(0) = φ 1. c T E φ (0) T 0 a φ 2 H dt, (φ 0, φ 1 ) H 1/2 H. with a certain c T > 0. Then, the energy of the solution of the damped equation satisfies

31 The main results Theorem (continued) ( 1 ) E w (t) βtl ψr 1 ( t T T 0 ), for t sufficiently large. If further, lim sup x 0 + Λ H (x) < 1 then we have the simplified decay rate E w (t) βt (H ) 1( DT 0 ), t T for t sufficiently large. Here D is a positive constant which is independent of E w (0) and T, whereas T 0 depends on T, β is a positive constant chosen so that ( 2αT E w (0) β > max, E ) w(0) C T L(H (r0 2)),, δ where the constants C T > 0, α and δ > 0 are explicit positive constants.

32 The main results The proof of our main result relies on a result important in itself, since it allows to compare discrete energy inequalities to continuous ones. One can show that if then the following property holds: The function M defined by H (x) lim x 0 + Λ H (x) = 0 M(x) = xl 1 (x), x [0, r 2 0 ). is such that lim x 0 M (x) = 0, where L is defined as above. + Hence thanks to this assumption, the function x x κm(x) is strictly increasing in [0, δ] [0, r0 2 ] for a certain κ > 0 and δ > 0 sufficiently small. We shall assume this last property from now on. Under this last property, we have the result

33 The main results The proof of our main result relies on a result important in itself, since it allows to compare discrete energy inequalities to continuous ones. One can show that if then the following property holds: The function M defined by H (x) lim x 0 + Λ H (x) = 0 M(x) = xl 1 (x), x [0, r 2 0 ). is such that lim x 0 M (x) = 0, where L is defined as above. + Hence thanks to this assumption, the function x x κm(x) is strictly increasing in [0, δ] [0, r0 2 ] for a certain κ > 0 and δ > 0 sufficiently small. We shall assume this last property from now on. Under this last property, we have the result

34 The main results The proof of our main result relies on a result important in itself, since it allows to compare discrete energy inequalities to continuous ones. One can show that if then the following property holds: The function M defined by H (x) lim x 0 + Λ H (x) = 0 M(x) = xl 1 (x), x [0, r 2 0 ). is such that lim x 0 M (x) = 0, where L is defined as above. + Hence thanks to this assumption, the function x x κm(x) is strictly increasing in [0, δ] [0, r0 2 ] for a certain κ > 0 and δ > 0 sufficiently small. We shall assume this last property from now on. Under this last property, we have the result

35 The main results Theorem (time discrete inequlity, A.-B. Ammari JFA 2011) Assume that the above assumption holds and let T > 0 and ρ T > 0 be given. Let δ > 0 be such that the function defined by x x ρ T M(x) is strictly increasing on [0, δ]. Assume that Ê is a nonnegative, nonincreasing function defined on [0, ) with Ê(0) < δ and satisfying ( ) Ê((k + 1)T ) Ê(kT ) 1 ρ T L 1 (Ê(kT )), k N. Then Ê satisfies the upper estimate ( 1 ) Ê(t) TL, for t sufficiently large, ψr 1 ( (t T )ρ T T )

36 The main results Theorem (continued) If moreover lim sup x 0 + Λ H (x) < 1, then we have the simplified decay rate Ê(t) T (H ) 1( D T ), ρ T (t T ) for t sufficiently large and where D is a positive constant independent of Ê(0) and of T. Proof of the time discrete inequality: We start as follows.

37 The proofs We fix the initial data (w(0), ẇ(0)). We denote by E w the energy of solutions of the Cauchy problem in w. We choose a constant β such that β > max( αt E C T, w (0) Ew (0) L(H (r0 2 )), δ the constants C T > 0, α and δ > 0 are explicit and will be chosen later on. Let us associate to our damping, the weight function uniquely determined by the optimal-weight convexity method, that is f (s) = L 1( s ), s [0, βr0 2 ), β ) where where L is the continuous strictly increasing function from [0, + ) onto [0, r0 2 ), seen before. We set Ê w = E w β. We will prove that Êw satisfies the discrete inequalities

38 The proofs ( ) Ê w ((k + 1)T ) Êw(kT ) 1 ρ T L 1 (Êw(kT )), k N, for a suitable constant ρ T > 0. For this, several intermediate results will be needed. We start with the following technical Lemma, which proof relies on the optimal-weight convexity method.

39 The proofs Lemma (A.-B. Ammari JFA 2011) Assume that ρ and a satisfy the above assumptions and that there exists r 0 > 0 sufficiently small so that the function H is strictly convex on [0, r 2 0 ]. Let (w 0, w 1 ) D(A 1/2 ) H be given and (φ 0, φ 1 ) = (w 0, w 1 ) and w and φ be the respective solutions of the nonlinearly damped and of the conservative problems. Then the following inequality holds T ( f (E φ (0)) a(x) ẇ 2 + a(x) ρ(x, ẇ) 2) dx dt 0 Ω ) C 1 TH (f (E φ (0))) + C 2 (f T (E φ (0)) + 1 a(x)ρ(x, ẇ)ẇ dx dt, 0 Ω where C 1 = Ω (1 + c 2 2), C 2 = and Ω = dσ, with dσ = a(.)dx. Ω ( 1 c 1 + c 2 ),

40 The proofs The proof of this result relies on the convexity properties of H. Here we have to deal also with integration with respect to space and make use of Jensen s and Young s inequalities. The next steps will be based on a comparison between the localized kinetic energy of the linearly damped problem and the linear and nonlinear kinetic energies for the nonlinearly damped problem and on a comparison with conservative and linearly damped kinetic energies.

41 The proofs Let us now consider z solution of the linear locally damped problem { z + Az + a(x)ż = 0, z(0) = w 0, ż(0) = w 1. Then, we have the following comparison Lemma. Lemma T 0 Ω T ( a(x) ż 2 dx dt C a(x) ẇ 2 + a(x) ρ(x, ẇ 2) dx dt. 0 Ω where C > 0 is a universal constant.

42 The proofs Similarly, let φ be the solution of the conservative problem { φ(t) + Aφ(t) = 0, Then we can prove. φ(0) = w 0, φ(0) = w 1. Lemma (A.-B. Ammari JFA 2011) Let T > 0 be given, then there exists k T > 0 such that for all (w 0, w 1 ) D(A 1/2 ) H T 0 Ω T a φ 2 dx dt k T 0 Ω a ż 2 dx dt where z is the solution of the above linearly damped equation with Cauchy data (w 0, w 1 ).

43 The proofs Theorem (A.-B. Ammari JFA 2011) Under the above hypotheses and setting Êw = E w /β, we have ( ) Ê w (T ) Êw(0) 1 ρ T L 1 (Êw(0)). where ρ T = c T 4k T (C 2 H (r 2 0 ) + 1). Proof. Thanks to our observability hypothesis, we have c T E φ (0) On the other hand we have T a φ 2 dx dt 0 Ω

44 The proofs continued. Hence we have T 0 Ω T c T E φ (0) k T 0 T a φ 2 dx dt k T Ω We choose β such that a ż 2 dx dt 0 Ω a ż 2 dx dt. T ( k T C a(x) ẇ 2 + a(x) ρ(x, ẇ 2) dx dt. 0 Ω f (E φ (0)) H (r 2 0 ). This together with the inequality of our technical Lemma, that is

45 The proofs continued. T ( f (E φ (0)) a(x) ẇ 2 + a(x) ρ(x, ẇ) 2) dx dt 0 Ω ) C 1 TH (f (E φ (0))) + C 2 (f T (E φ (0)) + 1 a(x)ρ(x, ẇ)ẇ dx dt, and the definition of the weight function f (s) = L 1 (s/β), so that 0 Ω imply H (f (s)) = sf (s)/β C T E φ (0)f (E φ (0)) αt β E φ(0)f (E φ (0)) + T 0 Ω a(x)ρ(x, ẇ)ẇ dx dt, where C T depends on c T, k T and α is independent of T.

46 The proofs continued. On the other hand, the dissipation relation for w gives T 0 Since E φ (0) = E w (0), we obtain Ω a(x)ρ(x, ẇ)ẇ dx dt = E w (0) E w (T ). [ ( E w (T ) E w (0) 1 C T αt β ) ] f (E w (0)). Thanks to our choice of β, we have C T αt β > C T 2 = ρ T > 0. Recalling the definition of the weight function f and setting Ê w (t) = E w (t)/β, we proved the desired result.

47 The proofs continued. We set E k = Êw(kT ). Then using the invariance by translation in time of the problem in w, we deduce that E k satisfies E k+1 E k + ρ T M(E k ) 0, k N, with E 0 = Êw(0). Hence we proved the time discrete inequality for all k N. We shall now "go" from this time discrete inequality to a continuous one.

48 The proofs We are now able to give a sketch of the proof of the upper estimate. Theorem (A.-B. Ammari JFA 2011) Assume that the above assumption holds and let T > 0 and ρ T > 0 be given. Let δ > 0 be such that the function defined by x x ρ T M(x) is strictly increasing on [0, δ]. Assume that Ê is a nonnegative, nonincreasing function defined on [0, ) with Ê(0) < δ and satisfying ( ) Ê((k + 1)T ) Ê(kT ) 1 ρ T L 1 (Ê(kT )), k N. Then Ê satisfies the upper estimate ( 1 ) Ê(t) TL, for t sufficiently large, ψr 1 ( (t T )ρ T T )

49 The proofs We are now able to give a sketch of the proof of the upper estimate. Theorem (A.-B. Ammari JFA 2011) Assume that the above assumption holds and let T > 0 and ρ T > 0 be given. Let δ > 0 be such that the function defined by x x ρ T M(x) is strictly increasing on [0, δ]. Assume that Ê is a nonnegative, nonincreasing function defined on [0, ) with Ê(0) < δ and satisfying ( ) Ê((k + 1)T ) Ê(kT ) 1 ρ T L 1 (Ê(kT )), k N. Then Ê satisfies the upper estimate ( 1 ) Ê(t) TL, for t sufficiently large, ψr 1 ( (t T )ρ T T )

50 The proofs Theorem (continued) If moreover lim sup x 0 + Λ H (x) < 1, then we have the simplified decay rate Ê(t) T (H ) 1( D T ), ρ T (t T ) for t sufficiently large and where D is a positive constant independent of Ê(0) and of T.

51 The proofs Proof. Let l N be an arbitrary fixed integer and set T 0 = T ρ T, r = Ê(0). We have E k+1+i E k+i + ρ T M(E k+i ) 0, for i = 0..., i = l. We sum these inequalities from i = 0 to i = l,. Since (E k ) k is a nonincreasing sequence whereas M is a nondecreasing function, we obtain E k+l+1 E k + 1 T 0 (l + 1)TM(E k+l ) 0

52 The proofs continued. so that, we have for any arbitrary p N M(E p ) T 0 T Thanks to our hypothesis: ( Ep l ) inf. l {0,...,p} l + 1 H (x) lim x 0 + Λ H (x) = 0 we deduce that for δ > 0, the function ψ : x x ρ T M(x) is strictly increasing on [0, δ). Now, set K r (τ] = r τ 1 M(v)

53 The proofs continued. Then since ψ is increasing, we can compare E k with the corresponding discretization point in a suitable Euler scheme, and prove that we have E k Kr 1 ( kt ), k N. T 0 Using this inequality in the above one, we get M(E p ) T 0 T inf l {0,...,p} (K r 1 ( (p l)t l + 1 T 0 ) ). For t T given, let p N be the unique integer so that t [pt, (p + 1)T ). Let θ (0, t T ] be arbitrary and l N be the unique integer so that θ [lt, (l + 1)T ).

54 The proofs continued. Then, thanks to the above inequality, we have so that we finally get M(Ê(t)) M(E p) T 0 T Ê(t) TM 1( inf θ (0,(t T )] inf l {0,...,p} (K r 1 ( (p l)t T 0 ) l + 1 ), (1 θ K r 1 ((t T θ) )) ). T 0 Using the generalized weighted Gronwall Theorem (A.-B. 2005), we deduce that ( 1 ) Ê(t) TL, t T. ψ 1 r ( t T ) T 0 If we further assume that lim sup x 0 + Λ H (x) < 1, then we obtain the claimed simplified estimate.

55 The proofs The proof of the main result follows easily, since we proved that E k satisfies the discrete inequality, and so that Ê satisfies the claimed decay rate. Remark one can show that the condition can be violated only when H (x) lim x 0 + Λ H (x) = 0 lim inf x 0 + Λ H(x) = 0, and lim x 0 + H (x) Λ H (x) does not exist. Is it possible? We could not prove it nor exhibit counterexamples.

56 Examples of PDE s We can give some examples of applications: the nonlinearly locally damped wave equation: u tt u + a(x)ρ(x, u t ) = 0,, (x, t) Ω (0, + ) u = 0, on Ω (0, + ), u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), on Ω, Hence under some smoothness assumptions on a and Ω, and assuming the Geometric Control Condition, that is (G.C.C) : every ray of geometric optics of Ω with length greater than T 0 hits the open set ω for a sufficiently large T 0 > 0. we deduce a sharp upper energy estimate as seen before.

57 Examples of PDE s It also extends to the case where (Ω, g) is a smooth and connex Riemannian manifold, and A the Laplacian on Ω for the metrics g. Once again, it suffices to assume (GCC) to conclude. It also extends to nonlinear Euler-Bernouilli plate equation, that is to u tt + 2 u + a(x)ρ(x, u t ) = 0, Ω (0, + ), u = 0, u = 0, Ω (0, + ), u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), Ω, Also holds for general second order uniformly elliptic operators with smooth coefficients. Extends to boundary damped cases.

58 Examples of PDE s Examples of decay rates It works as for the finite dimensional case Example 1 (polynomial case): vskip 2mm let g be given by g(x) = x p where p > 1 on (0, r 0 ]. Then E(t) Cβ E(0) t 2 p 1, (1) for t sufficiently large and for all (u 0, u 1 ) in R 2. Example 2 (exponential case): let g be given by g(x) = e 1 x 2 on (0, r 0 ]. Then E(t) Cβ E(0) (ln(t)) 1, for large t.

59 Examples of PDE s Example 3 (polynomial-logarithmic, close to linear): let g be given by g(x) = x(ln( 1 x )) p where p > 0. Then E(t) C β E(0) e 2( p t DT 0 ) 1/(p+1) t 1/(p+1) (2) for t sufficiently large. Example 4 (faster than any polynomial less than exponential) : let g be given by g(x) = e (ln( 1 x ))p, 1 < p < 2, x [0, r 0 ]. Then E(t) C β E(0) e 2(ln(t))1/p

60 Examples of PDE s So for upper estimates, this gives a unified approach between finite and infinite dimensional systems. The upper estimates are of the same order for a same feedback for the infinite dimensional system and the corresponding finite dimensional system Uniform upper estimates for semi-discretized and fully discretized PDE s? Results with Emmanuel Trélat and Yannick Privat to prove uniform upper estimates for semi-discretized and fully discretized PDE s.

61 Lower estimates. Optimality Lower estimates, optimality What about optimality in the infinite dimensional setting? Indeed the proof of the lower energy estimate of the finite dimensional case no longer works. Recall the proof for the scalar equation u + νu + f (u) + g(u ) = 0 We wrote

62 Lower estimates. Optimality E (t) = u g(u ) H( u 2 ) with E(t) = 1 2 ( u 2 + ν u 2 + F (u)) 1/2 u 2 For the wave equation with a distributed damping g(u ), we have seen that E (t) = u g(u ) dx, but here E(t) = 1 ( u t 2 + u(t) 2 ) dx Ω 2 Ω No direct comparison between u t 2 (t,.) and the energy E(t). There is an additional variable (the space one), and one has to consider integration with respect to the space.

63 Lower estimates. Optimality As far as we know, in the infinite dimensional estimate, very few lower estimates and/or optimality results are available. Haraux 1995, in the specific case : 1-D wave, globally distributed feedback, polynomial lim sup(t 3/(p 1) E(t)) > 0. t for initial data in W 2, (0, 1) W 1, (0, 1) It requires smoothness of the solutions. The proof does not work for the boundary damped one-dimensional wave equation, for systems such as Timoshenko beams or for higher order equations such as Petrowsky (plate) equation. Haraux proof is based on a weak estimate of the L -norm of the velocity. It requires an upper estimate of the energy. The lower estimate for the energy is obtained indirectly.

64 Lower estimates. Optimality Optimality results only in peculiar cases: for 1-D wave equations with boundary feedbacks Vancostenoble (1999), Martinez-Vancostenoble (2000) using the explicit form of the solution through D Alembert s formula. Set Ω = (0, 1) and consider u tt u xx = 0 on (0, + ) Ω, u = 0 on (0, + ) {0}, u x + ν ρ(u t) = 0 on (0, + ) {1}, u(0,.) = u 0 (.), u t(0,.) = u 1 (.) on Ω,

65 Lower estimates. Optimality Theorem (Vancostenoble and Martinez SICON 2000) Assume that g is a strictly increasing and odd C 1 function on R such that g(0) = g (0) = 0. Assume that ρ is a continuous nondecreasing function on R such that ρ = or ρ = g 1 in a neighbourhood of 0. Then for all initial data of the form (u 0, u 1 )(x) = (2A 0 x, 0), with A 0 0, then n 0 N, n n 0, E(2n) = 1 2 V 2 (2t n ) where (t n ) n n0 is a real positive increasing sequence such that t n n as n, and where V : [0, ) [0, ) is the solution of the differential equation, we have seen before, that is V + g(v ) = 0, V (0) = 2 E(2n 0 )/2. Remark They also derive some lower estimates for E(2n) under additional hypotheses on the feedback function.

66 Lower estimates. Optimality Optimality is a hard open question in the general case. Is it possible to obtain lower estimates for general dampings? less regular solutions? boundary damping cases? multi-dimensional situations? other PDE s?

67 A methodology for lower energy estimates through comparison principles Let us consider the damped wave equation, with a nonlinear locally distributed damping where u tt (t, x) u xx (t, x) + a(x)g(u t (t, x)) = 0, 0 < t, x Ω, u(t, c) = u(t, d) = 0, for 0 < t, u(0, x) = u 0 (x), u t (0, x) = u 1 (x), x Ω, Ω = (c, d) R, with < c < d <, a L (Ω) and a 0 a.e. on Ω with a > 0 on an open subset ω of Ω. The energy of a solution is defined by E(t) = 1 (ut 2 + ux 2 ) dx. 2 Ω

68 A methodology for lower energy estimates through comparison principles Theorem (A-B 2011) Assume that (u 0, u 1 ) (H0 1(Ω) H2 (Ω)) H0 1 (Ω) and that g satisfies (H1). We assume that H is strictly convex close to 0 and either 0 < lim inf x 0 or that there exists µ > 0 such that ( H(µ x) z1 0 < lim inf x 0 µ x x Λ H(x) lim sup Λ H (x) < 1, x 0 1 ) H(y) dy, and lim sup Λ H (x) < 1, x 0 for a certain z 1 (0, z 0 ] (arbitrary). Then the energy satisfies the lower estimate ( (H C(E 1 (0)) ) ( 1 1 )) 2 E(t), t T1 + T 0, t T 0 where C(E 1 (0)) depends explicitly on E 1 (0).

69 A methodology for lower energy estimates through comparison principles Let us now state the general result for the 1 D locally damped wave equation for general dampings and more regular solutions. We recall the regularity result proved by Haraux 1995 : If (u 0, u 1 ) W 2, (Ω) W 1, (Ω), the solution of the 1 D locally damped wave equation is such that u t L ([0, ); W 1, (Ω)). Thanks to this regularity result, we prove

70 A methodology for lower energy estimates through comparison principles Theorem (A.-B. JDE 2010) Assume the above hypotheses on g and H. Assume that (u 0, u 1 ) W 2, (Ω) W 1, (Ω) and E(0) > 0. Then, the energy satisfies the lower estimate ( (H C ) 1 1 ) 3/2 1 ( ) E(t), sufficiently large t, t T 0 where C 1 > 0 depends on u tx L ([0, ) Ω). In the peculiar case g(s) = s p 1 s close to the origin, H(x) = x (p+1)/2, so that the resulting lower extimate we get is E(t) C 1 (t T 0 ) 3/(p 1), for sufficiently large t. We can remark that this estimate is strictly better than the one obtained for strong solutions.

71 A methodology for lower energy estimates through comparison principles But in both cases, if we compare to the upper estimate we obtained before there is a gap between the lower and upper estimate so that contrarily to what happens to the finite dimensional case, we cannot deduce optimality. This result can be generalized to general dampings radial solutions of wave equations in annulus type domains in 2-D and 3-D less regular solutions, with weaker estimates Plate equation in multi-d Timoshenko beams

72 A methodology for lower energy estimates through comparison principles It also generalizes formally to the multi-dimensional locally damped wave equation Theorem (A.-B. JDE 2010) Assume that Ω and g are such that there exist solutions u such that u t L ([0, ); W 1, (Ω)). Then the energy satisfies the lower estimate ( (H C ) 1 1 ) (N+2)/2 1 ( ) E(t), t T1, t T 0 where C 1 > 0 and T 1 depend on u t L ([0, ) Ω). The proofs of these results rely on interpolation theory (Gagliardo-Nirenberg type inequalities relying on some regularity of the solutions), or on estimates involving the energy of higher order.

73 A methodology for lower energy estimates through comparison principles It is a deep question since it probably requires to understand which properties of the initial data lead to solutions that "have" or have not the expected asymptotic behavior. How to select such initial data? It is not known at present If one aims to obtain lower estimate, then how far the fact that the lower estimates approaches the upper one is linked to regularity properties of the solutions? Therefore a better understanding of how certain qualitative properties of the solution depend on the initial data is a challenging question. Such a question raises also for feedbacks which are close to linear around the origin.

74 Some results on non local dissipation The case of memory damping We have explored links between finite and infinite dimensional nonlinear dissipative systems. We gave applications to the wave model with locally distributed feedback. It applies as well to higher order PDE s such as Petrowsky equation..., to systems such as Timoshenko beams with a single control so that it is also robust to more constraining feedbacks design... This sharp analysis and in particular the optimal-weight convexity method can also be extended to other types of dampings such as memory dampings. These dampings are non local, the resulting systems are not autonomous, so that the invariance by translation in time is not preserved.

75 Some results on non local dissipation I will just state some of the results we have obtained. We consider u tt u + k u = 0 u = 0 in [0, ) Γ (u, u t )(0,.) = (u 0, u 1 ) in Ω, (k v)(t) = t 0 k(t s)v(s) ds. and the kernel k is positive, nonincreasing, and decaying at infinity. Moreover, for well-posedness, one considers the condition 0 k(t) dt < 1

76 Some results on non local dissipation I will just state some of the results we have obtained. We consider u tt u + k u = 0 u = 0 in [0, ) Γ (u, u t )(0,.) = (u 0, u 1 ) in Ω, (k v)(t) = t 0 k(t s)v(s) ds. and the kernel k is positive, nonincreasing, and decaying at infinity. Moreover, for well-posedness, one considers the condition 0 k(t) dt < 1

77 Some results on non local dissipation Model examples In this case, the natural energy is E u (t) = 1 2 u t(t) 2 L 2 (Ω) dx ( t 1 t 0 0 ) k(s) ds u(t) 2 L 2 (Ω) k(t s) u(t) u(s) 2 L 2 (Ω) ds and the dissipation relation is: E u(t) = 1 2 k(t) u(t) t 0 k (t s) u(s) u(t) 2 ds 0

78 Some results on non local dissipation The optimal-weight convexity method can be extended, in an involved way, to treat nonlocal dissipation. The dissipation terms are very different from the frictional case the problem is not invariant by translation in time.

79 Some results on non local dissipation The case of memory damping Assume and k (t) χ ( k(t)) for a.e. t 0 χ is a nonnegative measurable function on [0, k 0 ], for some k 0 > 0, strictly increasing and of class C 1 on [0, k 1 ], for some k 1 (0, k 0 ], such that χ(0) = χ (0) = 0 χ 0 > 0 such that χ χ 0 on [k 1, k 0 ] k0 0 dx χ(x) =, k0 0 x χ(x) dx < 1

80 Some results on non local dissipation Theorem (A.-B.-Cannarsa 2009) Assume that the convolution kernel k : [0, ) [0, ) is a locally absolutely continuous function as above and that χ is strictly convex on an interval of the form (0, δ] with δ > 0 and where lim inf x 0 + Λ(x) > 1 2, Λ(x) = χ(x)/x χ (x), x (0, δ]. Then lim sup Λ(x) < 1 and k (t) = χ(k(t)) for a.e. t 0. x 0 + E u (t) κ(e u (0))k(t), t T 1 where T 1 > 0 and κ(e u (0)) are explicit positive constants.

81 Some results on non local dissipation We can also give an explicit estimate if the lim sup = 1 but it has a different expression. The approach works as well for abstract PDE s and applies to elasticity, higher order PDE s. for semi linear terms (with a growth condition... It requires a sharp analysis of the damping mechanisms and of the intrinsic properties of such dissipative systems. Remark The exponential or power-like case was known (see e.g. Munoz-Rivera and Salvaterria 2001, A.-B.-Cannarsa-Sforza 2008), it is much more difficult to get a sharp estimate for generally decaying kernels k (also easily computable for all examples) satisfying the condition (to guarantee well-posedness): 0 k(s) ds < 1

82 Some results on non local dissipation This shows that there is a unified approach between the finite dimensional and infinite dimensional setting between the continuous stabilization problem and the discretized problem, and between frictional and memory dampings. Next lectures on how to derive uniform estimates for discretized PDE s.

83 Some results on non local dissipation This shows that there is a unified approach between the finite dimensional and infinite dimensional setting between the continuous stabilization problem and the discretized problem, and between frictional and memory dampings. Next lectures on how to derive uniform estimates for discretized PDE s.

84 Some results on non local dissipation Thanks for your attention