29 Pages
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Exponential stability of the wave equation with boundary time varying delay

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29 Pages
English

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Niveau: Supérieur, Doctorat, Bac+8
Exponential stability of the wave equation with boundary time-varying delay Serge Nicaise?, Cristina Pignotti†, Julie Valein‡ March 24, 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 = ? and ?D 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(x, t)

  • problems respectively

  • constant ? exists

  • ?universite de valenciennes et du hainaut cambresis

  • time-varying delay

  • delay effects

  • hilbert space

  • exponential stability result

  • stability estimate


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Expon
ential stability of the wave equation boundary time-varying delay Serge Nicaise, Cristina Pignotti,Julie Valein
March 24, 2009
with
Abstract We consider the wave equation with a time - varying delay term in the boundary condition in a bounded and smooth domain ΩIRn.suitable assumptions, we prove exponential stabilityUnder 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 ΩIRnbe an open bounded set with a boundary Γ of classC2.We assume that Γ is divided into two parts ΓDand ΓN, i.e. Γ = ΓDΓNwith ΓDΓN=and ΓD6=. In this domain Ω, we consider the initial boundary value problem
utt(x t)Δu(x t in Ω) = 0×(0+) (1.1) u(x t on Γ) = 0D×(0+) (1.2) uν(x t) =µ1ut(x t)µ2ut(x tτ(t)) on ΓN×(0+) (1.3) u(x0) =u0(x) andut(x0) =u1(x) in Ω (1.4) ut(x tτ(0)) =f0(x tτ(0)) in ΓN×(0 τ(0))(1.5) whereν(x) denotes the outer unit normal vector to the pointxΓ anduνis the normal derivative. Moreover,τ(t)>0 is the time-varying delay,µ1andµ2are positive real numbers and the initial datum (u0 u1 f0) belongs to a suitable space. On the functionτwe assume that there exists a positive constantτsuch that
0τ(t)τ t >0.
(1.6)
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1
Moreover, we assume τ0(t)<1t >0(1.7) and τW2,([0 T])T >0.(1.8) We are interested in giving an exponential stability result for such a problem. Let us denote byhv wior, equivalently, byvwthe euclidean inner product between two vectorsv wIRn . We assume that there existsx0IRnsuch that denoting bymthe standard multiplier m(x) :=xx0
we have m(x)ν(x)0 on ΓD(1.9) and, for some positive constantδ m(x)ν(x)δon ΓN.(1.10) It is well–known that ifµ2= 0that is in absence of delay, the energy of problem (1.1)(1.5) 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, ifµ1= 0that is if we have only the delay part in the boundary condition on ΓNsystem (1.1)(1.5) becomes unstable. See, for instance Datko, Lagnese and Polis [7]. The above problem, with bothµ1 µ2>0 and a constant delayτhas been studied in one space dimension by Xu, Yung and Li [30] and on networks by Nicaise and Valein [26] and in higher space dimension by Nicaise and Pignotti [25]. Assuming that
µ2< µ1(1.11) in [25], a stabilization result in general space dimension is given, by using a suitable observability estimate. This is done by applying inequalities obtained from Carleman estimates for the wave equation by Lasiecka, Triggiani and Yao in [19] and by using 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 µ2<1d µ1(1.12) wheredis a constant such that τ0(t)d <1t >0.(1.13)
Here, we extend this result to general space dimension. Moreover, we remove the hypothesis
τ(t)τ0>0t >0assumed in [27], that is the delay may degenerate. We will study also a nonlinear version of the above model. Consider the system
utt(x t)Δu(x t) = 0 in Ω×(0+) u(x t) = 0 on ΓD×(0+) uν(x t) =β1(ut(x t))β2(ut(x tτ(t))) on ΓN×(0+) u(x0) =u0(x) andut(x0) =u1(x) in Ω ut(x tτ(0)) =g0(x tτ Γ(0)) inN×(0 τ(0))whereβjI:RIR j= 12satisfy suitable growth assumptions. In particular we assume
|βj(s)| ≤cj|s|sIR j= 12
2
(1.14)
(1.15) (1.16) (1.17)
(1.18) (1.19)
(1.20)