Mean-Field First Order Optimality Condition under Low Regularity of Controls

We show that mean-field optimal controls satisfy a first order optimality condition (at a.e. time) without any a priori requirement on their spatial regularity. This principle is obtained by a careful limit procedure of the Pontryagin maximum principle for finite particle systems. In particular, our result applies to the case of mean-field selective optimal control problems for multipopulation and replicator dynamics.