Exact boundary controllability on L2(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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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 classL 2(0,T: L 2(Γ1)); 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 spaceL 2(Ω) ×H −1(Ω) of maximal regularity forT greater than a computable timeT 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 theL 2(∑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].