Exponential stability of the wave equation with boundary time varying delay

Description

Niveau: Supérieur, Doctorat, Bac+8
Exponential stability of the wave equation with boundary time-varying delay Serge Nicaise?, Cristina Pignotti†, Julie Valein‡ September 16, 2009 Abstract We consider the wave equation with a time - varying delay term in the boundary condition in a bounded and smooth domain ? ? IRn. Under suitable assumptions, we prove exponential stability of the solution. These results are obtained by introducing suitable energies and suitable Lyapounov functionals. Such analysis is also extended to a nonlinear version of the model. 2000 Mathematics Subject Classification: 35L05, 93D15 Keywords and Phrases: wave equation, delay feedbacks, stabilization 1 Introduction We are interested in the effect of a time–varying delay in boundary stabilization of the wave equation in domains of IRn. Delay effects arise in many pratical problems and it is well known that they can induce some unstabilities, see [5, 6, 7, 25, 30]. Let ? ? IRn be an open bounded set with a boundary ? of class C2. We assume that ? is divided into two parts ?D and ?N , i.e. ? = ?D ? ?N , with ?D ? ?N = ?, ?D 6= ? and ?N 6= ?. In this domain ?, we consider the initial boundary value problem utt(x, t)?∆u(x, t) = 0 in ?? (0,+∞) (1.1) u(x, t) = 0 on ?D ? (0,+∞) (1.2) ∂u ∂? (x, t) = ?µ1ut

• constant such


• smallest constants


• ?universite de valenciennes et du hainaut cambresis


• time-varying delay


• delay effects


• domain ?


• exponential stability result


• stability estimate

Sujets

• Saint Nicaise

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 23
• Langue: English

Extrait

Expon

ential stability of the wave equation boundary time-varying delay Serge Nicaise ∗ , Cristina Pignotti, † Julie Valein ‡

September 16, 2009

with

Abstract We consider the wave equation with a time - varying delay term in the boundary condition in a bounded and smooth domain Ω⊂IR n .Under suitable assumptions, we prove exponential stability of the solution. These results are obtained by introducing suitable energies and suitable Lyapounov functionals. Such analysis is also extended to a nonlinear version of the model.

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

1 Introduction

We are interested in the effect of a time–varying delay in boundary stabilization of the wave equation in domains of IR n .Delay effects arise in many pratical problems and it is well known that they can induce some unstabilities, see [5, 6, 7, 25, 30]. 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 =∅, Γ D 6=∅and Γ N 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) = 0 on Γ D ×(0+∞) (1.2) ∂ ∂ u ν (x t) =−µ 1 u t (x t)−µ 2 u t (x t−τ(t)) 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(0)) =f 0 (x t−τ(0)) in Γ N ×(0 τ(0))(1.5) whereν(x) denotes the outer unit normal vector to the pointx∈Γ and ν u ∂ ∂ is the normal derivative. Moreover,τ(t)>0 is the time-varying delay,µ 1 andµ 2 are positive real numbers and the initial datum (u 0  u 1  f 0 ) belongs to a suitable space. On the functionτwe assume that there exists a positive constantτsuch that

0≤τ(t)≤τ ∀t >0.

(1.6)

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1

and

Moreover, we assume

τ 0 (t)<1∀t >0 τ∈W 2,∞ ([0 T])∀T >0.

(1.7)

(1.8)

Note that, ift < τ(t), thenu t (x t−τ(tneed an initial value in the past.)) is in the past and we Moreover, by (1.7) and the mean value theorem, we have τ(t)−τ(0)< t

which implies t−τ(t)>−τ(0). We thus obtain the initial condition (1.5). We are interested in giving an exponential stability result for such a problem. n Let us denote byhv wior, equivalently, byv∙wthe euclidean inner product between two vectorsv w∈IR. We assume that there existsx 0 ∈IR n such that denoting bymthe standard multiplier

m(x) :=x−x 0 (1.9) we have m(x)∙ν(x)≤0 on Γ D (1.10) and, for some positive constantδ m(x)∙ν(x)≥δon Γ N .(1.11) It is well–known that ifµ 2 = 0that is in absence of delay, the energy of problem (1.1)−(1.4) is exponentially decaying to zero. See for instance Chen [3], Lagnese [16, 17], Lasiecka and Triggiani [18], Komornik and Zuazua [15], Komornik [13, 14]. On the contrary, ifµ 1 = 0that is if we have only the delay part in the boundary condition on Γ N system (1.1)−(1.4) becomes unstable. See, for instance Datko, Lagnese and Polis [7]. The above problem, with bothµ 1  µ 2 >0 and a constant delayτhas been studied in one space dimension by Xu, Yung and Li [30], on networks by Nicaise and Valein [26] and in higher space dimension by Nicaise and Pignotti [25]. Assuming that

µ 2 < µ 1 (1.12) in [25] a stabilization result in general space dimension is given, by using a suitable observability inequality obtained combining Carleman estimates for the wave equation by Lasiecka, Triggiani and Yao [19] and compactness-uniqueness arguments. The case of time–varying delay has been studied by Nicaise, Valein and Fridman [27] in one space dimension. In [27] an exponential stability result is given, under the condition µ 2 < √ 1−d µ 1 (1.13) wheredis a constant such that τ 0 (t)≤d <1∀t >0.(1.14) Here, under the same conditions (1.13)−(1.14)we extend this result to general space dimension. Moreover, we remove the hypothesis

τ(t)≥τ 0 >0∀t >0 assumed in [27], that is the delay may degenerate. We will study also a nonlinear version of the above model. Consider the system

2

(1.15)