State Constraints in Optimal Control: A Case Study in Proximal Normal Analysis

We consider an optimal control problem on a given interval $[0,T]$ whose trajectories must satisfy the state constraint $g(t,x(t)) \leqq 0$ a.e. Infinite-dimensional perturbations of this constraint give rise to a value function V, whose epigraph is a closed set containing sensitivity information, controllability and penalization results and even necessary conditions for optimality. Studying epi V when the domain of V is $L^2 [0,T]$ and $AC^2 [0,T]$ allows the derivation of a variety of necessary conditions, each with its own merits, and provides a concrete illustration of the intricacies of infinite-dimensional proximal normal analysis.