A uniformly controllable and implicit scheme for the 1-D wave equation
ESAIM: Mathematical Modelling and Numerical Analysis - 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 ${L}^{2}\left(0,T\right)$ 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.

DOI : https://doi.org/10.1051/m2an:2005012
Classification : 35L05,  65M60,  93B05
Mots clés : exact boundary controllability, wave system, finite difference
@article{M2AN_2005__39_2_377_0,
author = {M\"unch, Arnaud},
title = {A uniformly controllable and implicit scheme for the 1-D wave equation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {377--418},
publisher = {EDP-Sciences},
volume = {39},
number = {2},
year = {2005},
doi = {10.1051/m2an:2005012},
zbl = {1130.93016},
mrnumber = {2143953},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an:2005012/}
}
Münch, Arnaud. A uniformly controllable and implicit scheme for the 1-D wave equation. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 39 (2005) no. 2, pp. 377-418. doi : 10.1051/m2an:2005012. http://archive.numdam.org/articles/10.1051/m2an:2005012/

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