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

Arnaud Münch

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2005)

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

Abstract

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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 Δ t . We introduce an implicit finite difference scheme which differs from the usual centered one by additional terms of order h 2 and Δ 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.

How to cite

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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 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 -

References

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