A uniformly controllable and implicit scheme for the 1-D wave equation

Abstract

This paper studies the exact controllability of a finite dimensional system obtained by discretizing in space and time the linear 1-D wave system with a boundary control at one extreme. It is known that usual schemes obtained with finite difference or finite element methods are not uniformly controllable with respect to the discretization parameters h and Δt. We introduce an implicit finite difference scheme which differs from the usual centered one by additional terms of order h2 and Δt2. Using a discrete version of Ingham's inequality for nonharmonic Fourier series and spectral properties of the scheme, we show that the associated control can be chosen uniformly bounded in L2(0,T) and in such a way that it converges to the HUM control of the continuous wave, i.e. the minimal L2-norm control. The results are illustrated with several numerical experiments.