uniform polynomial stability of c0-semigroups · uniform polynomial stability application uniform...

Outline

Uniform polynomial stability ofC0-Semigroups

SALEM NAFIRI

Supervisor: Pr. Lahcen Maniar

LMDP - UMMISCO

Departement of Mathematics

Cadi Ayyad University

Faculty of Sciences Semlalia Marrakech

14 February 2012

SALEM NAFIRI Supervisor: Pr. Lahcen Maniar

OutlineIntroductionUniform Polynomial StabilityApplication

Outline

1 Introduction

2 Uniform polynomial stability

3 Application



I- Introduction



Stability

(S)

x ′(t) = Ax(t) + Bu(t) , t > 0

x(0) = x0

(u(t)=Fx(t))=⇒ (S)

x ′(t) = Ax(t) , t > 0

x(0) = x0=⇒ x(t) = T (t)x0

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.SALEM NAFIRI Supervisor: Pr. Lahcen Maniar


Exponential Stability of (S)

Gearhart. Spectral theory for contraction semigroups on

Hilbert spaces. Trans. Amer. Math. Soc. 236 : 385-394, (1978).

Prüss. On the spectrum of C0-semigroups, Trans. Amer. Math.

Soc. 284 (1984), 847-857.

Huang. Characteristic conditions for exponential stability of

linear dynamical systems in Hilbert spaces, Ann. Di. Eq., 1

(1985), 43-56.

Theorem. (Gearhart 1978, Prüss 1984, Huang 1985)

Let A be the innitesimal generator of a C0-semigroup (T (t))t>0on the Hilbert space H. Then (T (t))t>0 is exponentially stable i

(.I − A)−1 ∈ H∞(L(X )).



Uniform Exponential Stability of (Sn) ! !

(Sn)

x ′n(t) = Anxn(t) , t > 0

xn(0) = x0n=⇒ xn(t) = Tn(t)x0n

(Sn) exp. stable i ∀n, ∃Mn, αn : ‖Tn(t)x0n‖ ≤ Mne−αnt‖x0n‖

=⇒‖Tn(t)x0n‖ ≤Me−αt‖x0n‖, ∀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. 407-438.

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) :503-527, 2007.



Uniform Exponential Stability of (Sn) ! !

S. Ervedoza and E. Zuazua. Uniformly exponentially stable

approximations for a class of damped systems, 2008.

S. Ervedoza, Ch. Zheng, E. Zuazua. On the observability of

time-discrete conserva- tive linear systems, 2008.

Sylvain Ervedoza and Enrique Zuazua. Uniform exponential

decay for viscous damped systems, 2009.



Uniform Exponential Stability of c0-semigroups :

Zhuangyi Liu and Songmu Zheng.(1993)

Let Tn(t) (n = 1, . . .) be a sequence of C0-semigroups pf operators

on the Hilbert spaces Hn and let An be the corresponding

innitesimal generators. Then Tn(t) are uniformly exp. stable i the

following three conditions hold :

1

supn∈NReλ; λ ∈ σ(An) = σ0 < 0;

2 ∃σ ∈ (σ0, 0) such that

supReλ>σ,n∈N

(λI − An)−1 = M0 <∞;

3 ∃M1 > 0 such that

‖Tn(t)‖L(Hn) 6 M1 <∞ ∀t > 0, n ∈ N



Uniform Exponential Stability of a thermoelasticsystem

Zhuangyi Liu and Songmu Zheng.(1993) : Uniform Exponential

Stability and Approximation in Control of a Thermoelastic

System.

(S)

utt − c2uxx + c2γθx = 0 , (0, π)× (0,+∞),θt + γuxt − θxx = 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 ! !



Polynomial Stability of (S)

(S)

x ′(t) = Ax(t) , t > 0

x(0) = x0=⇒ x(t) = T (t)x0

Denition

(S) is polynomially stable i

‖T (t)A−αx‖ ≤ Ct−1‖x‖, ∀t > 0.

where α > 0, x ∈ D(Aα).



II- Uniform polynomialstability



Polynomial Stability of (S)

Theorem. (Borichev and Tomilov, 2009)

Let T (t) be a bounded C0-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−1α ), t →∞.



Bátkai, A., Engel, K.-J., Prüss, J., Schnaubelt, R. :

Polynomial stability of operator semigroups. Math. Nachr. 279,

1425-1440 (2006).

Borichev Alex, Tomilov Yu, :Optimal polynomial decay of

functions and operator semigroups (2009).



Uniform polynomial stability of (Sn)

(Sn)

x ′n(t) = Anxn(t) , t > 0

xn(0) = x0n=⇒ xn(t) = Tn(t)x0n

‖tTn(t)A−αn ‖ ≤ Cn, ∀t > 0, ∀n.

=⇒ Cn = C??



Main result

Theorem (*)

Let Tn(t) (n = 1, . . .) be a uniformly bounded sequence of

C0-semigroups on the Hilbert spaces Hn and let An be the

corresponding innitesimal generators, such that iR ⊂ ρ(An). Thenfor a xed α > 0 the following conditions are equivalent :

1 sups, n∈N

|s|−α‖R(is,An)‖ <∞.

2 supt>0, n∈N

‖tTn(t)A−αn ‖ <∞.

3 supt>0, n∈N

‖t1αTn(t)A−1n ‖ <∞.



Uniform Polynomial Stability of (Sn)

Lemma 1

Let Tn(t) (n = 1, . . .) be a sequence of C0-semigroups on the

Hilbert spaces Hn and let An be the corresponding innitesimal

generators and let C+ := z ∈ C : Re z > 0. Then Tn(t) is

uniformly bounded i

C+ ⊂ ρ(An), and

sup ξξ>0n∈N

∫R

(‖R(ξ + iη,An)x‖2 + ‖R(ξ + iη,A∗n

)x‖2)dη <∞

for all x ∈ Hn.



Sketch of the Proof of Lemma 1

=⇒We consider the rescaled semigroup T

−ξn (t) := e−ξtTn(t).

R(ξ + iη,An)x =∫∞0

e−iηtT−ξn (t)xdt.Plancherel's Theorem implies :

sup ξξ>0n∈N

∫R ‖R(ξ + iη,An)x‖2dη 6 πM2‖x‖2.

By symmetry we obtain the same estimate for the resolvent of A∗n.⇐=We use the inversion formula :⟨Tn(t)x , x∗

⟩= 1

2πit limω→∞

∫ τ+iωτ−iω eλt

⟨R2(τ + iβ,An)x , x∗

⟩dλ,

we choose τ = 1

t:

We deduce ‖Tn(t)‖ 6 eC4π ∀t > 0, n ∈ N, with C > 0 a constant

independent of n.



Lemma 2

Let Tn(t) (n = 1, . . .) be a sequence of uniformly bounded

C0-semigroups on the Hilbert spaces Hn and let An be the

corresponding innitesimal generators, such that iR ⊂ ρ(An). Thenfor a xed α > 0, the following assertions are equivalent :

1 supRe λ>0, n∈N

‖R(λ,An)‖1+|λ|α <∞.

2 supRe λ>0, n∈N

‖R(λ,An)A−αn ‖ <∞.

3 sups, n∈N

|s|−α‖R(is,An)‖ <∞.



Lemma 3 : Moment inequality

Let α < β < γ, then there exists a constant k(α, β, γ) > 0, such

that the following inequality hold :

‖Aβnx‖ ≤ k(α, β, γ)‖Aγnx‖β−αγ−α .‖Aαn x‖

γ−βγ−α , ∀x ∈ D(Aγn), ∀n ∈ N.



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 : 5.34.

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

C0-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. 341-363 Birkhäuser, Basel

(2001)




First case : γ = 0, α = −α1, β = −β1 (0 < β1 < α1) and

p ≤ α1 < p + 1.

A−β1n =1

2πi

∫Γa

λ−β1R(λ,An)dλ,

where we assume that ‖R(λ,An)‖ ≤ M1+|λ| .

We can show for λ = se±πi

‖AαnRp+1(−s,An)‖ ≤ c(1+s)n+1−α (p ≤ α < p + 1)

We show that ‖A−β1n x‖ ≤ k(α1, β1)‖x‖α1−β1

α1 .‖A−α1n x‖β1α1 ,

with k(α1, β1) = cβ1α1M

(p+1)(α1−β1)

α1

(1

α1−β1 + 1

β1

).

General case : α < β < γ and x ∈ D(Aγn).We apply the last inequality to the element Aγnx with α1 = γ − αand β1 = γ − β.




(1)⇔ (2)R(λ,An) is bounded on D = λ ∈ C/|λ| 6 ε ; S any subset of

ρ(An) s.t D ∩ S = ∅.We remark that : R(λ,An)

|λ|α = 1+|λ|α|λ|α

R(λ,An)1+|λ|α , we can show by

induction :

R(λ,An)A−pn =R(λ,An)

λp+

p−1∑k=0

(−1)pA−(p−k)n

λk+1.

For α = p ∈ N : ‖R(λ,An)A−α‖ ≤ dα‖R(λ,An)‖1+|λ|α + cα,

‖R(λ,An)‖1+|λ|α ≤ ‖R(λ,An)A−α‖,

with cα, dα > 0 independent of n.

For α = p /∈ N : we apply the moment inequality.




(1)⇔ (3)Apply the maximum principle to the function

F (λ) = R(λ,An)λ−α(1 + λ2

B2 ) on the domain

D := λ ∈ C : Re(λ) > 0, 1 6 |λ| 6 B for large B , and to use

the estimate ‖R(λ,An)‖ ≤ MRe(λ) .



III- Application



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)

utt − a∆u + αy = 0 in Ω,ytt − a∆y + αu = 0 in Ω,u = 0, on Γ0, a∂vu + γu + ut = 0 on Γ1y = 0 on Γ,

with Ω a bounded domain in R.Γ = Γ0 ∪ Γ1, Γ0 ∩ Γ1 = ∅.



Wave equation with a localized lineardissipation :

Kim Dang Phung.(2007) : Polynomial decay rate for the

dissipative wave equation.

(S)

utt −∆u + α(x)∂tu = 0 in Ω× R+,u = 0 on ∂Ω× R+,(u(., 0), ∂tu(., 0)) = (u0, u1) 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).



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, 380-438.

(S)

yt − yxx = 0 in (0,∞)× (0, 1),ztt − zxx = 0 in (0,∞)× (−1, 0),y(t, 1) = 0 = z(t,−1), t ∈ (0,∞),y(t, 0) = z(t, 0), yx(t, 0) = zx(t, 0) t ∈ (0,∞),y(0) = y0 in(0, 1),z(0) = z0, zt(0) = z1 in(−1, 0).



Thanks For Your Attention


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