Interior feedback stabilization of wave equations with time dependent delay

Description

Interior feedback stabilization of wave equations with time dependent delay Serge Nicaise? and Cristina Pignotti† Abstract We study the stabilization problem by interior (weak/strong) damping of the wave equation with boundary or internal time–varying delay feedback in a bounded and smooth domain ? ? IRn. By introducing suitable Lyapunov functionals exponential stability estimates are obtained if the delay effect is appropriately compensated by the internal damping. 2000 Mathematics Subject Classification: 35L05, 93D15 Keywords and Phrases: wave equation, delay feedbacks, stabilization 1 Introduction Let ? ? IRn be an open bounded set with a boundary ? of class C2. We assume that ? is divided into two parts ?0 and ?1, i.e. ? = ?0 ? ?1, with ?0 ? ?1 = ? and meas ?0 6= ?. Moreover, we assume that there exists x0 ? IR n such that denoting by m the standard multiplier m(x) := x? x0, we have m(x) · ?(x) ≤ 0 on ?0 (1.1) and, for some positive constant ?, m(x) · ?(x) ≥ ? on ?1. (1.2) We consider the problem utt(x, t)?∆u(x, t)? a∆ut(x, t) = 0 in ?? (0,+∞) (1.3) u(x, t) = 0 on ?0 ? (0,+∞) (1.4) µutt(x, t) = ? ∂(u

• parts ?0


• problems also


• av ?


• let ?


• constant delay


• also ∫


• delay effect


• general dimension


• function ?

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• Published by: pefav
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• Language: English

Exrait

Interior feedback stabilization of wave equations with time dependent delay Serge Nicaise ∗ and Cristina Pignotti †

Abstract We study the stabilization problem by interior (weak/strong) damping of the wave equation with boundary or internal time–varying delay feedback in a bounded and smooth domain Ω ⊂ IR n . By introducing suitable Lyapunov functionals exponential stability estimates are obtained if the delay effect is appropriately compensated by the internal damping.

2000 Mathematics Subject Classification: 35L05, 93D15 Keywords and Phrases: wave equation, delay feedbacks, stabilization 1 Introduction

Let Ω ⊂ IR n be an open bounded set with a boundary Γ of class C 2 . We assume that Γ is divided into two parts Γ 0 and Γ 1 , i.e. Γ = Γ 0 ∪ Γ 1 , with Γ 0 ∩ Γ 1 = ∅ and meas Γ 0 6 = ∅ . Moreover, we assume that there exists x 0 ∈ IR n such that denoting by m the standard multiplier m ( x ) := x − x 0 , we have m ( x ) ∙ ν ( x ) ≤ 0 on Γ 0 (1.1) and, for some positive constant δ, m ( x ) ∙ ν ( x ) ≥ δ on Γ 1 . (1.2) We consider the problem

u tt ( x, t ) − Δ u ( x, t ) − a Δ u t ( x, t ) = 0 in Ω × (0 , + ∞ ) (1.3) u ( x, t ) = 0 on Γ 0 × (0 , + ∞ ) (1.4) µu tt ( x, t ) = − ∂ ( u ∂ + ν au t ) ( x, t ) − ku t ( x, t − τ ( t )) on Γ 1 × (0 , + ∞ ) (1.5) u ( x, 0) = u 0 ( x ) and u t ( x, 0) = u 1 ( x ) in Ω (1.6) u t ( x, t ) = f 0 ( x, t ) in Γ 1 × ( − τ (0) , 0) , (1.7) where ν ( x ) denotes the outer unit normal vector to the point x ∈ Γ and ∂ ∂ u ν is the normal derivative. Moreover, τ = τ ( t ) is the time delay, µ, a, k are real numbers, with µ ≥ 0 , a > 0 , and the initial datum ( u 0 , u 1 , f 0 ) belongs to a suitable space. Note that for µ > 0 , (1.5) is a so–called dynamic boundary condition. ∗ Universite´deValenciennesetduHainautCambre´sis,MACS,ISTV,59313ValenciennesCedex9,France † DipartimentodiMatematicaPuraeApplicata,Universita`diL’Aquila,ViaVetoio,Loc.Coppito,67010L’AquilaItaly 1