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Applied Mathematics and Optimization

, Volume 18, Issue 1, pp 241–277 | Cite as

Exact boundary controllability onL2(Ω) ×H−1(Ω) of the wave equation with dirichlet boundary control acting on a portion of the boundaryΩ, and related problems

  • R. Triggiani
Article

Abstract

Consider the wave equation defined on a smooth bounded domain Ω ⊂R n with boundary Γ = Γ0 ∪ Γ1. The control action is exercised in the Dirichlet boundary conditions only on Γ1 and is of classL2(0,T: L21)); instead, homogeneous boundary conditions of Dirichlet (or Neumann) type are imposed on the complementary part Γ0. The main result of the paper is a theorem which, under general conditions on the triplet {Ω, Γ0, Γ1} with Γ0≠ ∅, guarantees exact controllability on the spaceL2(Ω) ×H−1(Ω) of maximal regularity forT greater than a computable timeT0>0, which depends on the triplet. This theorem generalizes prior results by Lasiecka and the author [L-T.3] (obtained via uniform stabilization) and by Lions [L.5], [L.6] (obtained by a direct approach, different from the one followed here). The key technical issue is a lower bound on theL2(∑1)-norm of the normal derivative of the solution to the corresponding homogeneous problem, which extends to a larger class of triplets {Ω, Γ0, Γ1} prior results by Lasiecka and the author [L-T.3] and by Ho [H.1].

Keywords

Wave Equation Bounded Domain Dirichlet Boundary Dirichlet Boundary Condition Technical Issue 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • R. Triggiani
    • 1
  1. 1.Department of Applied MathematicsUniversity of VirginiaCharlottesvilleUSA

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