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.

Keywords: 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 }, 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://www.numdam.org/articles/10.1051/m2an:2005012/} }

TY - JOUR AU - Münch, Arnaud TI - A uniformly controllable and implicit scheme for the 1-D wave equation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2005 SP - 377 EP - 418 VL - 39 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2005012/ DO - 10.1051/m2an:2005012 LA - en ID - M2AN_2005__39_2_377_0 ER -

%0 Journal Article %A Münch, Arnaud %T A uniformly controllable and implicit scheme for the 1-D wave equation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2005 %P 377-418 %V 39 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2005012/ %R 10.1051/m2an:2005012 %G en %F M2AN_2005__39_2_377_0

Münch, Arnaud. A uniformly controllable and implicit scheme for the 1-D wave equation. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 (2005) no. 2, pp. 377-418. doi : 10.1051/m2an:2005012. http://www.numdam.org/articles/10.1051/m2an:2005012/

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