Stabilization of the wave equation with a delay term in the boundary or internal feedbacks

Serge Nicaise Universite

Description

Niveau: Supérieur, Doctorat, Bac+8
Stabilization of the wave equation with a delay term in the boundary or internal feedbacks Serge Nicaise Universite de Valenciennes et du Hainaut Cambresis MACS, Institut des Sciences et Techniques de Valenciennes 59313 Valenciennes Cedex 9 France Cristina Pignotti Dipartimento di Matematica Pura e Applicata Universita di L'Aquila Via Vetoio, Loc. Coppito, 67010 L'Aquila Italy Abstract In this paper we consider, in a bounded and smooth domain, the wave equation with a delay term in the boundary condition. We also consider the wave equation with a delayed velocity term and mixed Dirichlet-Neumann boundary condition. In both cases, under suitable assumptions, we prove exponential stability of the solution. These results are obtained by introducing suitable energies and by using some observability inequalities. Some unstability examples are also given. 2000 Mathematics Subject Classification: 35L05, 93D15 Keywords and Phrases: wave equation, delay feedbacks, stabilization 1 Introduction We investigate the effect of time delay in boundary or internal stabilization of the wave equation in domains of IRn. Such effects arise in many pratical problems and it is well known, at least in one–dimension, that they can induce some unstabilities, see [3, 4, 14]. To our knowledge, the analysis in higher dimension is not yet done. In this paper, we give some stability results under a sufficient condition and further we show that if this condition is not satisfied, then there exist some delays for which the system is destabilized.

• feedback


• datum u0 ?


• unstability examples


• let ? ?


• domain ?


• ?µ2 ≤


• boundary feedback


• µ1 ?


• instance chen


• internal stabilization

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• Publié par: profil-zyak-2012
• Nombre de lectures: 27
• Langue: English

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Stabilization of the wave equation with a delay term in the boundary or internal feedbacks

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Serge Nicaise Universit´edeValenciennesetduHainautCambre´sis MACS, Institut des Sciences et Techniques de Valenciennes 59313 Valenciennes Cedex 9 France

Cristina Pignotti Dipartimento di Matematica Pura e Applicata Universita`diL’Aquila Via Vetoio, Loc. Coppito, 67010 L’Aquila Italy

Abstract

In this paper we consider, in a bounded and smooth domain, the wave equation with a delay term in the boundary condition. We also consider the wave equation with a delayed velocity term and mixed Dirichlet-Neumann boundary condition. In both cases, under suitable assumptions, we prove exponential stability of the solution. These results are obtained by introducing suitable energies and by using some observability inequalities. Some unstability examples are also given.

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

Introduction

We investigate the effect of time delay in boundary or internal stabilization of the wave equation in domains of IR n .Such effects arise in many pratical problems and it is well known, at least in one–dimension, that they can induce some unstabilities, see [3, 4, 14]. To our knowledge, the analysis in higher dimension is not yet done. In this paper, we give some stability results under a sufficient condition and further we show that if this condition is not satisfied, then there exist some delays for which the system is destabilized. So, in a certain sense, our sufficient condition is also necessary in order to have a general stability result.

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Let Ω⊂IR n be an open bounded set with a boundary Γ of classC 2 .We assume that Γ is divided into two parts Γ D and Γ N , i.e. Γ = Γ D ∪Γ N ,with Γ D ∩Γ N =∅and Γ D 6=∅. In this domain Ω, we consider the initial boundary value problem

u tt (x, t)−Δu(x, t in Ω) = 0×(0,+∞) (1.1) u(x, t on Γ) = 0 D ×(0,+∞) (1.2) ∂ ∂ u ν (x, t) =−µ 1 u t (x, t)−µ 2 u t (x, t−τ) on Γ N ×(0,+∞) (1.3) u(x,0) =u 0 (x) andu t (x,0) =u 1 (x) in Ω (1.4) u t (x, t−τ) =f 0 (x, t−τ) in Γ N ×(0, τ),(1.5) whereν(x) denotes the unit normal vector to the pointx∈Γ and u∂ ν∂ is the normal derivative. Moreover,τ >0 is the time delay,µ 1 andµ 2 are positive real numbers and the initial datum (u 0 , u 1 , f 0 ) belongs to a suitable space. We are interested in giving an exponential stability result for such a problem. Let us denote byhv, wior, equivalently, byv∙wthe euclidean inner product between two vectorsv, w∈IR n . We assume that there exists a scalar functionv∈C 2 (Ω) such that (i)vis strictly convex in Ω,that is there existsα >0 such that hD 2 (v)(x)ξ, ξi ≥2α|ξ| 2 ,∀x∈Ω,∀ξ∈IR n ,(1.6)

whereD 2 (v) denotes the Hessian matrix ofv; (ii)the vector fieldH:=rvverifies

H(x)∙ν(x)≤0,∀x∈Γ D .(1.7) For the above assumptions see [11] where some observability estimates for second– order hyperbolic equations are given. It is well–known that ifµ 2 = 0,that is in absence of delay, the energy of problem (1.1)–(1.5) is exponentially decaying to zero. See for instance Chen [2], Lasiecka and Triggiani [10], Lagnese [9], Komornik and Zuazua [8], Komornik [6, 7]. On the contrary, ifµ 1 = 0,delay part in the boundary condition on Γthat is if we have only the N ,system (1.1)–(1.5) becomes unstable. See, for instance Datko, Lagnese and Polis [4]. Although these examples involve only one space dimension, we can expect that a similar phenomenon occurs in higher space dimension. So, it is interesting to seek a sta- bilization result whenµ 1 andµ 2 both nonzero. In this case, the boundary feedback isare composed of two parts and only one of them has a delay. This problem has been studied in one space dimension by Xu, Yung and Li [14]. After a spectral analysis the authors have proved a stability result ifµ 2 < µ 1 .In their paper it is also shown that ifµ 2 > µ 1 the system is unstable and ifµ 1 =µ 2 some unstabilities may occur. Here, coherently with [14], assuming that µ 2 < µ 1 (1.8)

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