Necessary Optimality Conditions for Optimal Control Problems in Wasserstein Spaces

TL;DR

This paper establishes first-order necessary optimality conditions for control problems in Wasserstein spaces, introducing new concepts like localised metric subdifferential and tangent cones, and providing a geometric proof of Pontryagin's Maximum Principle.

Contribution

It introduces a novel notion of localised metric subdifferential and geometric methods to derive optimality conditions in Wasserstein spaces, extending control theory.

Findings

Derived first-order necessary optimality conditions in Wasserstein spaces.

Introduced a new notion of localised metric subdifferential for probability measures.

Provided a geometric proof of Pontryagin Maximum Principle in this setting.

Abstract

In this article, we derive first-order necessary optimality conditions for a constrained optimal control problem formulated in the Wasserstein space of probability measures. To this end, we introduce a new notion of localised metric subdifferential for compactly supported probability measures, and investigate the intrinsic linearised Cauchy problems associated to non-local continuity equations. In particular, we show that when the velocity perturbations belong to the tangent cone to the convexification of the set of admissible velocities, the solutions of these linearised problems are tangent to the solution set of the corresponding continuity inclusion. We then make use of these novel concepts to provide a synthetic and geometric proof of the celebrated Pontryagin Maximum Principle for an optimal control problem with inequality final-point constraints. In addition, we propose sufficient conditions ensuring the normality of the maximum principle.