On the approximation of the boundary control of the wave equation with numerical viscosity

This article deals with the approximation of the boundary control of the 1-D linear wave equation. Due to the spurious high frequencies, the semi-discrete models obtained with finite difference or classical finite element methods are not uniformly controllable as the discretization parameter h goes to zero (see [8]). We propose a new strategy for the approximation of the boundary control based on the addition of a numerical vanishing viscous term. This will damp out the spurious high frequencies and will ensure the existence of a convergent sequence of approximate controls. We present an approximation algorithm and some numerical experiments.