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 end, introduce new notion localised metric subdifferential compactly supported measures, and investigate intrinsic linearised Cauchy problems associated to non-local continuity equations. particular, show that when velocity perturbations belong tangent cone convexification set admissible velocities, solutions these are solution corresponding inclusion. We then make use novel concepts provide synthetic geometric proof celebrated Pontryagin Maximum Principle an with inequality final-point constraints. addition, propose sufficient ensuring normality maximum principle.