Semiconcavity and Sensitivity Analysis in Mean-Field Optimal Control and Applications

TL;DR

This paper explores the mathematical properties of the value function in mean-field optimal control, focusing on semiconcavity, sensitivity relations, and regularity propagation, with implications for optimal feedback design.

Contribution

It introduces new interpolation and linearisation formulas for non-local flows, establishing semiconcavity and sensitivity relations in the Wasserstein space.

Findings

Proved semiconcavity estimates for the value function.

Established variants of sensitivity relations linking superdifferential and adjoint curves.

Analyzed regularity propagation and optimal feedback conditions.

Abstract

In this article, we investigate some of the fine properties of the value function associated to an optimal control problem in the Wasserstein space of probability measures. Building on new interpolation and linearisation formulas for non-local flows, we prove semiconcavity estimates for the value function, and establish several variants of the so-called sensitivity relations which provide connections between its superdifferential and the adjoint curves stemming from the maximum principle. We subsequently make use of these results to study the propagation of regularity for the value function along optimal trajectories, as well as to investigate sufficient optimality conditions and optimal feedbacks for mean-field optimal control problems.