A numerical method for the controls of the heat equation

This work is devoted to analyze a numerical scheme for the approximation of the linear heat equation's controls. It is known that, due to the regularizing effect, the efficient computation of the null controls for parabolic type equations is a difficult problem. A possible cure for the bad numerical behavior of the approximating controls consists of adding a singular perturbation depending on a small parameter ε which transforms the heat equation into a wave equation. A space discretization of step h leads us to a system of ordinary differential equations. The aim of this paper is to show that there exists a sequence of exact controls of the corresponding perturbed semi-discrete systems which converges to a control of the original heat equation when both h (the mesh size) and ε (the perturbation parameter) tend to zero.