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

TL;DR

This paper explores the structural properties of the value function in mean-field optimal control problems of Bolza type, including sensitivity relations, semiconcavity, and trajectory characterization.

Contribution

It provides new insights into the value function's structure, sensitivity, and optimal trajectory characterization in measure spaces for mean-field control problems.

Findings

Established sensitivity relations between costates and value function derivatives.

Proved semiconcavity properties of the value function in measure spaces.

Characterized optimal trajectories via set-valued feedback mappings.

Abstract

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.