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

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