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

Arnaud Münch

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • 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 Δt. We introduce an implicit finite difference scheme which differs from the usual centered one by additional terms of order h2 and Δt2. 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 L2(0,T) and in such a way that it converges to the HUM control of the continuous wave, i.e. the minimal L2-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 39.2 (2010): 377-418. <http://eudml.org/doc/194266>.

@article{Münch2010,
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 Δt. We introduce an implicit finite difference scheme which differs from the usual centered one by additional terms of order h2 and Δt2. 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 L2(0,T) and in such a way that it converges to the HUM control of the continuous wave, i.e. the minimal L2-norm control. The results are illustrated with several numerical experiments. },
author = {Münch, Arnaud},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Exact boundary controllability; wave system; finite difference.},
language = {eng},
month = {3},
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/194266},
volume = {39},
year = {2010},
}

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
DA - 2010/3//
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 Δt. We introduce an implicit finite difference scheme which differs from the usual centered one by additional terms of order h2 and Δt2. 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 L2(0,T) and in such a way that it converges to the HUM control of the continuous wave, i.e. the minimal L2-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/194266
ER -

References

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Citations in EuDML Documents

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  1. Nicolae Cîndea, Enrique Fernández-Cara, Arnaud Münch, Numerical controllability of the wave equation through primal methods and Carleman estimates
  2. Arnaud Münch, Ademir Fernando Pazoto, Uniform stabilization of a viscous numerical approximation for a locally damped wave equation
  3. Farah Abdallah, Serge Nicaise, Julie Valein, Ali Wehbe, Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications
  4. Sylvain Ervedoza, Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes
  5. Sylvain Ervedoza, Resolvent estimates in controllability theory and applications to the discrete wave equation

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