Regularity of the Optimal Feedback and the Value Function in Convex Problems of Optimal Control

An optimal feedback mapping, leading to necessary and sufficient conditions for optimality in terms of a closed-loop differential inclusion, is derived in the setting of fully convex generalized problems of Bolza. Results are translated to the format of control problems with linear dynamics and convex costs. Properties of the feedback mapping, with focus on single-valuedness and continuity, are analyzed through those of the value function and of the Hamiltonian. Conditions guaranteeing differentiability of the value function are obtained through the analysis of its subdifferential as a maximal monotone operator and of the generalized Hamiltonian dynamics.