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A remark on observability of the wave equation with moving boundary

  • Kaïs Ammari , Ahmed Bchatnia EMAIL logo and Karim El Mufti
Published/Copyright: May 25, 2017
Journal of Applied Analysis
From the journal Journal of Applied Analysis Volume 23 Issue 1

Abstract

We deal with the wave equation with assigned moving boundary (0<x<a⁢(t)) upon which Dirichlet or mixed boundary conditions are specified. Here a⁢(t) is assumed to move slower than light and periodically. Moreover, a is continuous, piecewise linear with two independent parameters. Our major concern will be an observation problem which is based measuring, at each t>0, of the transverse velocity at a⁢(t). The key to the results is the use of a reduction theorem by Yoccoz [14].

Keywords: Strings with moving ends; observability; rotation number
MSC 2010: 35L05; 34K35; 93B07; 95B05

A Appendix

In this section, we treat the Dirichlet observability. We assume the following.

Assumption A.1.

The function

b⁢(t):=H′⁢(a⁢(t)+t)-H′⁢(-a⁢(t)+t)H′⁢(a⁢(t)+t)+H′⁢(-a⁢(t)+t)

satisfies

c1≤b⁢(t)≤c2 for all ⁢t∈ℝ,c1,c2>0.

Note that for a⁢(t) given by (1.7), we have

b(t)=a⁢(t)t+l2l1-l2⋅

This function satisfies

(A.1)2⁢amin⁢α-β1-β≤2⁢a⁢(t)⁢α-β1-β≤b⁢(t)≤2⁢a⁢(t)⁢α-β1+β≤2⁢amax⁢α-β1+β for all t∈ℝ,

and Assumption A.1 is satisfied.

Theorem A.2 (Dirichlet observability).

Under Assumptions 1.4 and A.1, we have that for all T≥ρ⁢(F), there exists a constant C*>0 such that for all solution u of system (1.1) with the mixed boundary condition (1.3) and initial data (ϕ,ψ)∈Hl1⁢(0,a⁢(0))×L2⁢(0,a⁢(0)), we have

(A.2)∫0T|ut(a(t),t)|2dt≥C*(∥ϕ∥Hl1⁢(0,a⁢(0))2+∥ψ∥L2⁢(0,a⁢(0))2).

Remark A.3.

The observability time is optimal here for the same reason as in Remark 1.6 and Theorem 1.5.

Remark A.4.

Using Φ given by (2.1), we transform system (1.1), (1.3) into

(A.3){∂τ2⁡V-∂ξ2⁡V=0for ⁢0<ξ<ρ⁢(F)/2,τ∈ℝ,V⁢(0,τ)=0,Vξ⁢(ρ⁢(F)/2,τ)+b⁢(t⁢(τ))⁢Vτ⁢(ρ⁢(F)/2,τ)=0for ⁢τ∈ℝ,V⁢(ξ,0)=ϕ2⁢(ξ),Vτ⁢(ξ,0)=ψ2⁢(ξ)for ⁢ξ∈(0,ω/2).

For the proof of Theorem A.2, we need the following lemmas.

Lemma A.5.

There exist positive constants C and ω such that

(A.4)EV⁢(τ)≤C⁢e-ω⁢τ⁢EV⁢(0).

Proof.

Define the Lyapunov function

E1⁢(τ)=12⁢∫0ρ⁢(F)[Vξ2⁢(ξ,τ)+Vτ2⁢(ξ,τ)]⁢𝑑ξ+δ⁢∫0ρ⁢(F)ξ⁢Vξ⁢(ξ,τ)⁢Vτ⁢(ξ,τ)⁢𝑑ξ.

For δ<1ρ⁢(F), we obtain

(A.5)0<(1-δ⁢ρ⁢(F))⁢EV⁢(τ)≤E1⁢(τ)≤(1+δ⁢ρ⁢(F))⁢EV⁢(τ).

We derive E1 with respect to τ and get

E1′⁢(τ)=[Vξ⁢Vτ]ξ=0ξ=ρ⁢(F)-δ2⁢∫0ρ⁢(F)[Vξ2⁢(ξ,τ)+Vτ2⁢(ξ,τ)]⁢𝑑ξ+δ2⁢[ξ⁢(Vξ2+Vτ2)]ξ=0ξ=ρ⁢(F)
=[δ2⁢(1+b⁢(t⁢(τ))2)-b⁢(t⁢(τ))]⁢Vτ2⁢(ρ⁢(F),τ)-δ2⁢∫0ρ⁢(F)[Vξ2⁢(ξ,τ)+Vτ2⁢(ξ,τ)]⁢𝑑ξ.

We choose δ small enough and, taking into account (A.1) and (A.5), we get E1′⁢(τ)≤-ω⁢E1⁢(τ). The proof is complete. ∎

Lemma A.6.

If T≥ρ⁢(F), then there exists C⁢(T)>0 such that for all (ϕ2,ψ2)∈Hl1⁢(0,ρ⁢(F)/2)×L2⁢(0,ρ⁢(F)/2), we have

(A.6)C(T)∫0T|Vξ(ρ(F)/2,τ)|2dτ≥∥ϕ2∥Hl1⁢(0,ρ⁢(F)/2)2+∥ψ2∥L2⁢(0,ρ⁢(F)/2)2

and

(A.7)C(T)∫0T|Vτ(ρ(F)/2,τ)|2dτ≥∥ϕ2∥Hl1⁢(0,ρ⁢(F)/2)2+∥ψ2∥L2⁢(0,ρ⁢(F)/2)2.

Proof.

The energy identity for system (A.3) gives

EV⁢(T)-EV⁢(0)=-∫oTb⁢(t⁢(τ))⁢|Vτ⁢(ρ⁢(F)/2,τ)|2⁢𝑑τ.

Using (A.1) and (A.4), we obtain

∫0T|Vτ(ρ(F)/2,τ)|2dτ≥C∫0Tb(t(τ))|Vτ(ρ(F)/2,τ)|2dτ≥C(EV(0)-EV(T))≥CEV(0)(1-e-ω⁢T).

This permit to conclude the second inequality in Lemma A.6.

For the first inequality, it suffices to use (A.3) and (A.1). ∎

Proof of Theorem A.2.

For the proof of (A.2), we state as above

∂t⁡u=∂ξ⁡V⁢∂t⁡ξ+∂τ⁡V⁢∂t⁡τ.

Next we have

|∂tu(a(t),t)|2=14{|∂ξV(ρ(F)/2,τ)[H′(x+t)-H′(-x+t)]+∂τV(ρ(F)/2,τ)[H′(x+t)+H′(-x+t)]}2.

Making use of Young’s inequalities, Lemma 2.4, (A.6), (A.7) and (1.8), we obtain the desired result. ∎

Acknowledgements

The authors are very grateful to the anonymous referees for their helpful comments and suggestions, that improved the manuscript.

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Received: 2015-7-23
Revised: 2016-9-6
Accepted: 2017-3-23
Published Online: 2017-5-25
Published in Print: 2017-6-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

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  2. A note on properties of the restriction operator on Sobolev spaces
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  4. On a new proof and an extension of Jack’s lemma
  5. On the conjecture of Hayami and Owa concerning the class ℛ(α)
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  7. The Weyl–von Neumann theorem in von Neumann factors
  8. A remark on observability of the wave equation with moving boundary
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https://doi.org/10.1515/jaa-2017-0007
Keywords for this article
Strings with moving ends; observability; rotation number

Articles in the same Issue

  1. Frontmatter
  2. A note on properties of the restriction operator on Sobolev spaces
  3. Approximation of entire transcendental functions of several complex variables in some Banach spaces for slow growth
  4. On a new proof and an extension of Jack’s lemma
  5. On the conjecture of Hayami and Owa concerning the class ℛ(α)
  6. The order of starlikeness of uniformly convex functions
  7. The Weyl–von Neumann theorem in von Neumann factors
  8. A remark on observability of the wave equation with moving boundary
  9. Best approximation in quotient probabilistic normed space
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