Journal of Mathematical Sciences

, Volume 83, Issue 2, pp 165–174 | Cite as

Controllability in a filled domain for the multidimensional wave equation with a singular boundary control

  • S. A. Avdonin
  • M. I. Belishev
  • S. A. Ivanov
Article

Abstract

In accordance with the demands of the so-called local approach to inverse problems, the set of “waves” uf (·, T) is studied, where uf (x,t) is the solution of the initial boundary-value problem utt−Δu=0 in Ω×(0,T), u|t<0=0, u|∂Ω×(0,T)=f, and the (singular) control f runs over the class L2((0,T); H−m (∂Ω)) (m>0). The following result is established. Let ΩT={x ∈ Ω : dist(x, ∂Ω)<T)} be a subdomain of Ω ⊂ ℝn (diam Ω<∞) filled with waves by a final instant of time t=T, let T*=inf{T : ΩT=Ω} be the time of filling the whole domain Ω. We introduce the notation Dm=Dom((−Δ)m/2), where (−Δ) is the Laplace operator, Dom(−Δ)=H2(Ω)∩H 0 1 (Ω);D−m=(Dm)′;D−mT)={y∈D−m:supp y ⋐ ΩT. If T<T., then the reachable set R m T ={ut(·, T): f ∈ L2((0,T), H−m (∂Ω))} (∀m>0), which is dense in D−mT), does not contain the class C 0 T). Examples of a ∈ C 0 , a ∈ R m T , are presented.

Keywords

Inverse Problem Wave Equation Laplace Operator Boundary Control Local Approach 

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Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • S. A. Avdonin
  • M. I. Belishev
  • S. A. Ivanov

There are no affiliations available

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