Approximation of exact controls for semilinear wave and heat equations through space-time methods

Abstract

We consider from the algorithmic and numerical viewpoints the exact controllability problems for linear and semilinear heat and wave equations. We notably report on some recent iterative approaches yielding to strongly convergent approximations of controlled solutions for semilinear equations. From the numerical perspective, we focus on the control-then-discretize strategy where the optimality system associated with each problem is solved within a space-time framework leading to strongly convergent approximations with respect to the parameters of discretization. The role of global Carleman type estimates is emphasized in the robustness of the approaches. Numerical experiments in the one and two dimensional case illustrate our results of convergence.