A uniformly controllable and implicit scheme for the 1-D wave equation
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 2, p. 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 Δt. We introduce an implicit finite difference scheme which differs from the usual centered one by additional terms of order h 2 and Δt 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.

DOI : https://doi.org/10.1051/m2an:2005012
Classification:  35L05,  65M60,  93B05
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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {39},
     number = {2},
     year = {2005},
     pages = {377-418},
     doi = {10.1051/m2an:2005012},
     zbl = {1130.93016},
     mrnumber = {2143953},
     language = {en},
     url = {http://www.numdam.org/item/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 - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 2, pp. 377-418. doi : 10.1051/m2an:2005012. http://www.numdam.org/item/M2AN_2005__39_2_377_0/

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