Uniformly exponentially stable approximations for a class of second order evolution equations Application to LQR optimization problems

We consider the approximation of a class of exponentially stable infinite dimensional linear systems modelling the damped vibrations of one dimensional vibrating systems. It is by now well known that the approximating systems obtained by usual finite element or finite difference are not 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 finally give applications to systems governed by a non-homogeneous string equation and a plate equation.