  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.
