Uniformly exponentially stable approximations for a class of second order evolution equations

Karim Ramdani; Takéo Takahashi; Marius Tucsnak

ESAIM: Control, Optimisation and Calculus of Variations (2007)

  • Volume: 13, Issue: 3, page 503-527
  • ISSN: 1292-8119

Abstract

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We consider the approximation of a class of exponentially stable infinite dimensional linear systems modelling the damped vibrations of one dimensional vibrating systems or of square plates. It is by now well known that the approximating systems obtained by usual finite element or finite difference are not, in general, uniformly stable with respect to the discretization parameter. Our main result shows that, by adding a suitable numerical viscosity term in the numerical scheme, our approximations are uniformly exponentially stable. This result is then applied to obtain strongly convergent approximations of the solutions of the algebraic Riccati equations associated to an LQR optimal control problem. We next give an application to a non-homogeneous string equation. Finally we apply similar techniques for approximating the equations of a damped square plate.

How to cite

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Ramdani, Karim, Takahashi, Takéo, and Tucsnak, Marius. "Uniformly exponentially stable approximations for a class of second order evolution equations." ESAIM: Control, Optimisation and Calculus of Variations 13.3 (2007): 503-527. <http://eudml.org/doc/250005>.

@article{Ramdani2007,
abstract = { We consider the approximation of a class of exponentially stable infinite dimensional linear systems modelling the damped vibrations of one dimensional vibrating systems or of square plates. It is by now well known that the approximating systems obtained by usual finite element or finite difference are not, in general, uniformly stable with respect to the discretization parameter. Our main result shows that, by adding a suitable numerical viscosity term in the numerical scheme, our approximations are uniformly exponentially stable. This result is then applied to obtain strongly convergent approximations of the solutions of the algebraic Riccati equations associated to an LQR optimal control problem. We next give an application to a non-homogeneous string equation. Finally we apply similar techniques for approximating the equations of a damped square plate. },
author = {Ramdani, Karim, Takahashi, Takéo, Tucsnak, Marius},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Uniform exponential stability; LQR optimal control problem; wave equation; plate equation; finite element; finite difference; uniform exponential stability; LQR optimal control problem; finite element},
language = {eng},
month = {6},
number = {3},
pages = {503-527},
publisher = {EDP Sciences},
title = {Uniformly exponentially stable approximations for a class of second order evolution equations},
url = {http://eudml.org/doc/250005},
volume = {13},
year = {2007},
}

TY - JOUR
AU - Ramdani, Karim
AU - Takahashi, Takéo
AU - Tucsnak, Marius
TI - Uniformly exponentially stable approximations for a class of second order evolution equations
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2007/6//
PB - EDP Sciences
VL - 13
IS - 3
SP - 503
EP - 527
AB - We consider the approximation of a class of exponentially stable infinite dimensional linear systems modelling the damped vibrations of one dimensional vibrating systems or of square plates. It is by now well known that the approximating systems obtained by usual finite element or finite difference are not, in general, uniformly stable with respect to the discretization parameter. Our main result shows that, by adding a suitable numerical viscosity term in the numerical scheme, our approximations are uniformly exponentially stable. This result is then applied to obtain strongly convergent approximations of the solutions of the algebraic Riccati equations associated to an LQR optimal control problem. We next give an application to a non-homogeneous string equation. Finally we apply similar techniques for approximating the equations of a damped square plate.
LA - eng
KW - Uniform exponential stability; LQR optimal control problem; wave equation; plate equation; finite element; finite difference; uniform exponential stability; LQR optimal control problem; finite element
UR - http://eudml.org/doc/250005
ER -

References

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