Optimal feedback control, linear first-order PDE systems, and obstacle problems

We introduce an alternative approach for the analysis and numerical approximation of the optimal feedback control mapping. It consists in looking at a typical optimal control problem in such a way that feasible controls are mappings depending both in time and space. In this way, the feedback form of the problem is built-in from the very beginning. Optimality conditions are derived for one such optimal mapping, which by construction is the optimal feedback mapping of the problem. In formulating optimality conditions, costates in feedback form are solutions of linear, first-order transport systems, while optimal descent directions are solutions of appropriate obstacle problems. We treat situations with no constraint-sets for control and state, as well as the more general case where a constraint-set is considered for the control variable.