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

- Volume: 39, Issue: 2, page 377-418
- ISSN: 0764-583X

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topMünch, Arnaud. "A uniformly controllable and implicit scheme for the 1-D wave equation." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 39.2 (2005): 377-418. <http://eudml.org/doc/245995>.

@article{Münch2005,

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 $\Delta t$. 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.},

author = {Münch, Arnaud},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {exact boundary controllability; wave system; finite difference},

language = {eng},

number = {2},

pages = {377-418},

publisher = {EDP-Sciences},

title = {A uniformly controllable and implicit scheme for the 1-D wave equation},

url = {http://eudml.org/doc/245995},

volume = {39},

year = {2005},

}

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 - Modélisation Mathématique et Analyse Numérique

PY - 2005

PB - EDP-Sciences

VL - 39

IS - 2

SP - 377

EP - 418

AB - 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.

LA - eng

KW - exact boundary controllability; wave system; finite difference

UR - http://eudml.org/doc/245995

ER -

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