Approximation of the controls for the wave equation with a potential

Abstract

This article deals with the approximation of the boundary controls of a 1-D linear wave equation with a variable potential by using a finite difference space semi-discrete scheme. Due to the high frequency numerical spurious oscillations, the semi-discrete model is not uniformly controllable with respect to the mesh-size and the convergence of the approximate controls cannot be guaranteed. In this paper we analyze how do the initial data to be controlled and their discretization affect the approximation of the controls. Under certain conditions on the potential, we prove that the convergence of the scheme is ensured if the highest frequencies of the discrete initial data have been previously filtered out. Several filtration procedures are proposed and analyzed. Moreover, we identify a class of (regular) continuous initial data which can be controlled uniformly without any special treatment.