Recent results in Shape Optimization and Optimal Control for PDEs

In this paper we will present some recent advances in the numerical approximation of two classical problems: shape optimization and optimal control for evolutive partial differential equations. For shape optimization we present two novel techniques which have shown to be rather efficient on some applications. The first technique is based on multigrid methods whereas the second relies on an adaptive sequential quadratic programming. With respect to the optimal control of evolutive problems, the approximation is based on the coupling between a POD representation of the dynamical system and the classical Dynamic Programming approach. We look for an approximation of the value function characterized as the weak solution (in the viscosity sense) of the corresponding Hamilton-Jacobi equation. Several tests illustrate the main features of the above methods.