Necessary Optimality Conditions for Optimal Control Problems in Wasserstein Spaces

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.

Automated summary

This article derives first-order necessary optimality conditions for a constrained optimal control problem in the Wasserstein space of probability measures, introduces a new notion of localised metric subdifferential, and investigates intrinsic linearised Cauchy problems associated with non-local continuity equations. By using these concepts, the Pontryagin Maximum Principle for optimal control problems with inequality final-point constraints is proven synthetically and geometrically, and sufficient conditions for the normality of the maximum principle are proposed.