Some remarks on homogenization and exact boundary controllability for the one-dimensional wave equation

Abstract

This paper contains three results concerning the homogenization and exact controllability for the one-dimensional wave equation. First, we give sufficient conditions on the initial data to ensure the convergence of the conormal derivatives associated with the wave equation with a rapidly oscillating coefficient and zero Dirichlet boundary conditions. Secondly, we apply this result to prove the existence of a class of initial data whose associated boundary controls are uniformly bounded and obtain some information (in particular, its limit behavior) on this class of data. Finally, we prove that all initial data in L 2 × H − 1 L^2\times H^{-1} may be uniformly controlled but at the price of adding an internal feedback control in our system. The main advantage of this last procedure is that we have explicit formulae for both states and controls.