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.
Mots clés : uniform exponential stability, LQR optimal control problem, wave equation, plate equation, finite element, finite difference
@article{COCV_2007__13_3_503_0,
author = {Ramdani, Karim and Takahashi, Tak\'eo and Tucsnak, Marius},
title = {Uniformly exponentially stable approximations for a class of second order evolution equations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {503--527},
publisher = {EDP-Sciences},
volume = {13},
number = {3},
year = {2007},
doi = {10.1051/cocv:2007020},
mrnumber = {2329173},
zbl = {1126.93050},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2007020/}
}
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 PY - 2007 SP - 503 EP - 527 VL - 13 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2007020/ DO - 10.1051/cocv:2007020 LA - en ID - COCV_2007__13_3_503_0 ER -
%0 Journal Article %A Ramdani, Karim %A Takahashi, Takéo %A Tucsnak, Marius %T Uniformly exponentially stable approximations for a class of second order evolution equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2007 %P 503-527 %V 13 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2007020/ %R 10.1051/cocv:2007020 %G en %F COCV_2007__13_3_503_0
Ramdani, Karim; Takahashi, Takéo; Tucsnak, Marius. Uniformly exponentially stable approximations for a class of second order evolution equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 3, pp. 503-527. doi : 10.1051/cocv:2007020. http://www.numdam.org/articles/10.1051/cocv:2007020/
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