Exact controllability of the wave equation with Neumann boundary control

AbstractWe consider the wave equation defined on a smooth bounded domainΩ⊂Rn 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=Z1×Z2 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 = L2(Ω)×[H1(Ω)]′ in the Neumann case Γ0=Ø in the absence of geometrical conditions on Ω, but with a special classVΣ of controls, larger thanL2(Σ). The key technical issues are, in all cases, lower bounds on theL2(Σ1)-norm of appropriate traces of the solution to the corresponding homogeneous problem. These are obtained by multiplier techniques.

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