Weak and strong minima: from calculus of variation toward PDE optimization

This note summarizes some recent advances on the theory of optimality conditions for PDE optimization. We focus our attention on the concept of strong minima for optimal control problems governed by semi-linear elliptic and parabolic equations. Whereas in the field of calculus of variations this notion has been deeply investigated, the study of strong solutions for optimal control problems of partial differential equations (PDEs) has been addressed recently. We first revisit some well-known results coming from the calculus of variations that will highlight the subsequent results. We then present a characterization of strong minima satisfying quadratic growth for optimal control problems of semi-linear elliptic and parabolic equations and we end by describing some current investigations