# Exact boundary controllability on*L*_{2}(Ω) ×*H*^{−1}(Ω) of the wave equation with dirichlet boundary control acting on a portion of the boundary*∂*Ω, and related problems

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## Abstract

Consider the wave equation defined on a smooth bounded domain Ω ⊂*R*^{ n } with boundary Γ = Γ_{0} ∪ Γ_{1}. The control action is exercised in the Dirichlet boundary conditions only on Γ_{1} and is of class*L*_{2}(0,*T: L*_{2}(Γ_{1})); instead, homogeneous boundary conditions of Dirichlet (or Neumann) type are imposed on the complementary part Γ_{0}. The main result of the paper is a theorem which, under general conditions on the triplet {Ω, Γ_{0}, Γ_{1}} with Γ_{0}≠ ∅, guarantees exact controllability on the space*L*_{2}(Ω) ×*H*^{−1}(Ω) of maximal regularity for*T* greater than a computable time*T*_{0}>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 the*L*_{2}(∑_{1})-norm of the normal derivative of the solution to the corresponding homogeneous problem, which extends to a larger class of triplets {Ω, Γ_{0}, Γ_{1}} 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## Preview

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