On the Properties of the Value Function Associated to a Mean-Field Optimal Control Problem of Bolza Type

In this paper, we obtain several structural results for the value function associated to a mean-field optimal control problem of Bolza type in the space of measures. After establishing the sensitivity relations bridging between the costates of the maximum principle and metric superdifferentials of the value function, we investigate semiconcavity properties of this latter with respect to both variables. We then characterise optimal trajectories using set-valued feedback mappings defined in terms of suitable directional derivatives of the value function.