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

Münch, Arnaud

ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 2, pp. 377-418

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 $\Delta t$. We introduce an implicit finite difference scheme which differs from the usual centered one by additional terms of order $h^2$ and $\Delta t^2$. 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 $L^2(0,T)$ and in such a way that it converges to the HUM control of the continuous wave, i.e. the minimal $L^2$-norm control. The results are illustrated with several numerical experiments.

Classification : 35L05, 65M60, 93B05
Keywords: exact boundary controllability, wave system, finite difference

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