Exact boundary controllability of the wave equation as the limit of internal controllability

This paper presents the study of the following problem of exact controllability concerning the wave equation with Dirichlet boundary conditions. Using Lions’s Hilbert uniqueness method (HUM), Zuazua has given a positive answer to the problem of exact controllability when the control is distributed and acts on an $\varepsilon $-neighborhood of a part $\Gamma _0 $ of the boundary satisfying some geometrical conditions. The main interest is in the passage to the limit when $\varepsilon $ goes to 0, which means when the neighborhood of $\Gamma _0 $ shrinks to $\Gamma _0 $ itself.