Uniform polynomial stability of C 0 -Semigroups

Transcription

1 Uniform polynomial stability of C 0 -Semigroups LMDP - UMMISCO Departement of Mathematics Cadi Ayyad University Faculty of Sciences Semlalia Marrakech 14 February 2012

2 Outline 1 2 Uniform polynomial stability 3

3 I-

4 Stability Outline { x (S) (t) = Ax(t) + Bu(t), t 0 x(0) = x 0 { (u(t)=fx(t)) x = (S) (t) = Ax(t), t 0 = x(t) = T (t)x x(0) = 0 x 0 Stability of (S) We distinguish several forms of stability : 1 exponential stability 2 polynomial stability 3... Aucun corps ne se met en mouvement ou revient au repos par lui-même Ibn SINA.

5 Exponential Stability of (S) Gearhart. Spectral theory for contraction semigroups on Hilbert spaces. Trans. Amer. Math. Soc. 236 : , (1978). Prüss. On the spectrum of C 0 -semigroups, Trans. Amer. Math. Soc. 284 (1984), Huang. Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Di. Eq., 1 (1985), Theorem. (Gearhart 1978, Prüss 1984, Huang 1985) Let A be the innitesimal generator of a C 0 -semigroup (T (t)) t 0 on the Hilbert space H. Then (T (t)) t 0 is exponentially stable i (.I A) 1 H (L(X )).

6 Uniform Exponential Stability of (S n )!! { x (S n ) n (t) = A n x n (t), t 0 x n (0) = xn 0 = x n (t) = T n (t)x 0 n (S n ) exp. stable i n, M n, α n : T n (t)x 0 n M n e αnt x 0 n = T n (t)x 0 n Me αt x 0 n, n?? Infante J. A. and E. Zuazua. Boundary observability for the space semi-discretization of the 1-D wave equation, M2AN, 33, 2 (1999), pp K. Ramdani, T. Takahashi, and M. Tucsnak. Uniformly exponentially stable approximations for a class of second order evolution equations-application to LQR problems. ESAIM Control Optim. Calc. Var., 13(3) : , 2007.

7 Uniform Exponential Stability of (S n )!! S. Ervedoza and E. Zuazua. Uniformly exponentially stable approximations for a class of damped systems, S. Ervedoza, Ch. Zheng, E. Zuazua. On the observability of time-discrete conserva- tive linear systems, Sylvain Ervedoza and Enrique Zuazua. Uniform exponential decay for viscous damped systems, 2009.

8 Uniform Exponential Stability of c 0 -semigroups : Zhuangyi Liu and Songmu Zheng.(1993) Let T n (t) (n = 1,...) be a sequence of C 0 -semigroups pf operators on the Hilbert spaces H n and let A n be the corresponding innitesimal generators. Then T n (t) are uniformly exp. stable i the following three conditions hold : 1 2 σ (σ 0, 0) such that sup{reλ; λ σ(a n )} = σ 0 < 0; n N 3 M 1 > 0 such that sup {(λi A n ) 1 } = M 0 < ; Reλ σ,n N T n (t) L(Hn) M 1 < t > 0, n N

9 Uniform Exponential Stability of a thermoelastic system Zhuangyi Liu and Songmu Zheng.(1993) : Uniform Exponential Stability and Approximation in Control of a Thermoelastic System. u tt c 2 u xx + c 2 γθ x = 0, (0, π) (0, + ), (S) θ t + γu xt θ xx = 0, (0, π) (0, + ), u x=0,π = θ x=0,π = 0, t > 0. For certain systems u(t) 0 polynomially and not exponentially!!

10 Polynomial Stability of (S) { x (S) (t) = Ax(t), t 0 = x(t) = T (t)x x(0) = 0 x 0 Denition (S) is polynomially stable i T (t)a α x Ct 1 x, t > 0. where α > 0, x D(A α ).

11 II- Uniform polynomial stability

12 Polynomial Stability of (S) Theorem. (Borichev and Tomilov, 2009) Let T (t) be a bounded C 0 -semigroup on a Hilbert space H with generator A such that ir ρ(a). Then for a xed α > 0 the following conditions are equivalent : (i) R(is, A) = O( s α ), s. (ii) T (t)( A) α = O(t 1 ), t. (iii) T (t)( A) 1 = O(t 1 α ), t.

13 Bátkai, A., Engel, K.-J., Prüss, J., Schnaubelt, R. : Polynomial stability of operator semigroups. Math. Nachr. 279, (2006). Borichev Alex, Tomilov Yu, :Optimal polynomial decay of functions and operator semigroups (2009).

14 Uniform polynomial stability of (S n ) { x (S n ) n (t) = A n x n (t), t 0 x n (0) = xn 0 = x n (t) = T n (t)x 0 n tt n (t)a α n C n, t > 0, n. = C n = C??

15 Main result Outline Theorem (*) Let T n (t) (n = 1,...) be a uniformly bounded sequence of C 0 -semigroups on the Hilbert spaces H n and let A n be the corresponding innitesimal generators, such that ir ρ(a n ). Then for a xed α > 0 the following conditions are equivalent : 1 sup s α R(is, A n ) <. s, n N 2 sup tt n (t)a α n <. t 0, n N 3 sup t 1 α T n (t)a 1 n <. t 0, n N

16 of (S n ) Lemma 1 Let T n (t) (n = 1,...) be a sequence of C 0 -semigroups on the Hilbert spaces H n and let A n be the corresponding innitesimal generators and let C + := {z C : Re z > 0}. Then T n (t) is uniformly bounded i C + ρ(a n ), and sup ξ ξ>0 R n N for all x H n. ( R(ξ + iη, A n )x 2 + R(ξ + iη, A n ) x 2) dη <

17 Sketch of the Proof of Lemma 1 = We consider the rescaled semigroup T ξ n (t) := e ξt T n (t). R(ξ + iη, A n )x = e iηt T ξ 0 n (t)xdt. Plancherel's Theorem implies : sup ξ R R(ξ + iη, A n)x 2 dη πm 2 x 2. ξ>0 n N By symmetry we obtain the same estimate for the resolvent of A n. = We use the inversion formula : Tn (t)x, x = 1 2πit lim ω we choose τ = 1 t : We deduce T n (t) ec 4π independent of n. τ+iω τ iω eλt R 2 (τ + iβ, A n )x, x dλ, t > 0, n N, with C > 0 a constant

18 Lemma 2 Let T n (t) (n = 1,...) be a sequence of uniformly bounded C 0 -semigroups on the Hilbert spaces H n and let A n be the corresponding innitesimal generators, such that ir ρ(a n ). Then for a xed α > 0, the following assertions are equivalent : R(λ,A 1 sup n) 1+ λ <. Re λ>0, n N α 2 sup R(λ, A n )A α n <. Re λ>0, n N 3 sup s α R(is, A n ) <. s, n N

19 Lemma 3 : Moment inequality Let α < β < γ, then there exists a constant k(α, β, γ) > 0, such that the following inequality hold : A β n x k(α, β, γ) A γ nx β α γ α. A α γ β n x γ α, x D(A γ n), n N.

20 Lemma 3 has been established in the particular case n = 1. K. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, p : 141, Th : Similarly (1) (2) was established for n = 1 by : Huang, S.-Z., van Neerven, J.M.A.M. : B-convexity, the analytic Radon-Nikodym property and individual stability of C 0 -semigroups. J. Math. Anal. Appl. 231, 1-20 (1999) Latushkin, Yu., Shvydkoy, R. : Hyperbolicity of semigroups and Fourier multipliers, In : Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000), Oper. Theory Adv. Appl., vol. 129, pp Birkhäuser, Basel (2001)

21 Sketch of the Proof of Lemma 3 First case : γ = 0, α = α 1, β = β 1 (0 < β 1 < α 1 ) and p α 1 < p + 1. A β 1 n = 1 λ β1 R(λ, A n )dλ, 2πi Γ a where we assume that R(λ, A n ) We can show for λ = se ±πi A α n Rp+1 ( s, A n ) M 1+ λ. c (1+s) n+1 α (p α < p + 1) We show that A β 1 n x k(α 1, β 1 ) x α 1 β 1 α 1. A α 1 n x β 1 β 1 ( α 1 M (p+1) (α 1 β 1 ) α 1 with k(α 1, β 1 ) = c α 1 β β 1 ). General case : α < β < γ and x D(A γ n). We apply the last inequality to the element A γ nx with α 1 = γ α and β 1 = γ β. 1 α 1,

22 Sketch of the Proof of Lemma 2 (1) (2) R(λ, A n ) is bounded on D = {λ C/ λ ε} ; S any subset of ρ(a n ) s.t D S =. We remark that : R(λ,An) λ α induction : = 1+ λ α λ α R(λ,A n) 1+ λ α, we can show by R(λ, A n )A p n = R(λ, A p 1 n) + λ p k=0 ( 1) p A (p k) n λ k+1. For α = p N : R(λ, A n )A α R(λ,A d n) α 1+ λ + α c α, R(λ, A n )A α, R(λ,A n) 1+ λ α with c α, d α > 0 independent of n. For α = p / N : we apply the moment inequality.

23 Sketch of the Proof of Lemma 2 (1) (3) Apply the maximum principle to the function F (λ) = R(λ, A n )λ α (1 + λ2 ) B 2 on the domain D := {λ C : Re(λ) 0, 1 λ B} for large B, and to use the estimate R(λ, A n ) M Re(λ).

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25 System of partially damped wave equations : Zhuangyi Liu and Bopeng Rao.(2006) : Frequency domain approach for the polynomial stability of a system of partially damped wave equations. (S) u tt a u + αy = 0 in Ω, y tt a y + αu = 0 in Ω, u = 0, on Γ 0, a v u + γu + u t = 0 on Γ 1 y = 0 on Γ, with Ω a bounded domain in R. Γ = Γ 0 Γ 1, Γ 0 Γ 1 =.

26 Wave equation with a localized linear dissipation : Kim Dang Phung.(2007) : Polynomial decay rate for the dissipative wave equation. u tt u + α(x) t u = 0 in Ω R +, (S) u = 0 on Ω R +, (u(., 0), t u(., 0)) = (u 0, u 1 ) in Ω, where Ω is a bounded domain in R with a boundary Ω at least Lipschitz. Here, α is a nonnegative function in L (Ω) and depends on a non-empty proper subset ω of Ω on which 1 α L (ω) (in particular, {x Ω; α(x) > 0} is a non-empty open set).

27 Hyperbolic-parabolic coupled system : J. Rauch, X. Zhang. Zuazua E.((2005)) : Polynomial decay for a hyperbolic-parabolic coupled system. Zhang X., Zuazua E., Polynomial decay and control of a 1-d hyperbolic-parabolic coupled system, J. Dierential Equations 204 (2004), 2, y t y xx = 0 in (0, ) (0, 1), z tt z xx = 0 in (0, ) ( 1, 0), y(t, 1) = 0 = z(t, 1), t (0, ), (S) y(t, 0) = z(t, 0), y x (t, 0) = z x (t, 0) t (0, ), y(0) = y 0 in(0, 1), z(0) = z 0, z t (0) = z 1 in( 1, 0).

28 Thanks For Your Attention