Exact controllability of the wave equation with Neumann boundary control

Abstract

We consider the wave equation defined on a smooth bounded domainΩ⊂R ⁿ with boundary Γ=Γ₀⋃Γ₁, with Γ₀ possibly empty and Γ₁ nonempty and relatively open in Γ. The control action is exercised in the Neumann boundary conditions only on Γ₁, while homogeneous boundary conditions of Dirichlet type are imposed on the complementary part Γ₀. We study by a direct method (i.e., without passing through “uniform stabilization”) the problem of exact controllability on some finite time interval [0,T] for initial data on some preassigned spaceZ=Z ₁ ×Z ₂ based on Ω and with control functions in some preassigned space$V_{\Sigma _1 }$ based on Γ₁ and [0,T]. We consider several choices of pairs [Z,$V_{\Sigma _1 }$] of spaces, and others may be likewise studied by similar methods. Our main results are exact controllability results in the following cases: (i)$Z = H_{\Gamma _0 }^1 (\Omega ) \times L^2 (\Omega )$ and$V_{\Sigma _1 } = L^2 (\Sigma _1 ); (ii) Z = L^2 (\Omega ) \times [H_{\Gamma _0 }^1 (\Omega )]\prime$ and$V_{\Sigma _1 } = [H^1 (0,T;L^2 (\Gamma _1 ))]\prime$, both under suitable geometrical conditions on the triplet {Ω, Γ₀, Γ₁} expressed in terms of a general vector field; (iii)Z = L ² (Ω)×[H ¹ (Ω)]′ in the Neumann case Γ₀=Ø in the absence of geometrical conditions on Ω, but with a special classV _Σ of controls, larger thanL ² (Σ). The key technical issues are, in all cases, lower bounds on theL ² (Σ ₁)-norm of appropriate traces of the solution to the corresponding homogeneous problem. These are obtained by multiplier techniques.