Exact internal controllability of vibrations on a thin body

We study by the Hilbert uniqueness method, (HUM), of Lions, the exact controllability of a 3-dimensional body of very small thickness. Afterwards, we let the thickness parameter go to zero and we prove that the limit of the sequence of exact controls, is the exact control given by HUM, of the 2-dimensional limit system. We prove in particular strong convergence results for the sequence of controls.