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

Application to LQR problems

Published online by Cambridge University Press:  05 June 2007

Karim Ramdani ,
Takéo Takahashi  and
Marius Tucsnak
Karim Ramdani
Affiliation:
Institut Elie Cartan University of Nancy-I, POB 239, Vandœuvre-les-Nancy 54506, France; [email protected] INRIA Lorraine, Projet CORIDA.
Takéo Takahashi
Affiliation:
Institut Elie Cartan University of Nancy-I, POB 239, Vandœuvre-les-Nancy 54506, France; [email protected] INRIA Lorraine, Projet CORIDA.
Marius Tucsnak
Affiliation:
Institut Elie Cartan University of Nancy-I, POB 239, Vandœuvre-les-Nancy 54506, France; [email protected] INRIA Lorraine, Projet CORIDA.
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Abstract

We consider the approximation of a class ofexponentially stable infinite dimensional linear systems modellingthe damped vibrations of one dimensional vibrating systems or ofsquare plates. It is by now well known that the approximatingsystems obtained by usual finite element or finite difference arenot, in general, uniformly stable with respect to the discretizationparameter. Our main result shows that, by adding a suitablenumerical viscosity term in the numerical scheme, our approximationsare uniformly exponentially stable. This result is then applied toobtain strongly convergent approximations of the solutions of thealgebraic Riccati equations associated to an LQR optimal controlproblem. We next give an application to a non-homogeneous stringequation. Finally we apply similar techniques for approximating theequations of a damped square plate.

Keywords

Uniform exponential stabilityLQR optimal control problemwave equationplate equationfinite elementfinite difference
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 3 , July 2007 , pp. 503 - 527
Copyright
© EDP Sciences, SMAI, 2007

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