Interior feedback stabilization of wave equations with time dependent delay

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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

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Saint Nicaise Delay (audio effect)

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∂ + νau 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 ∂∂uν 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

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