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

Application to LQR problems

¹ Institut Elie Cartan University of Nancy-I, POB 239, Vandœuvre-les-Nancy 54506, France; marius.tucsnak@loria.fr
² INRIA Lorraine, Projet CORIDA.

Received: 21 September 2005
Revised: 19 January 2006

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.