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.
