  We derive a Maximum Principle for optimal control problems with constraints
given by the coupling of a system of ODEs and a PDE of Vlasov-type. Such
problems arise naturally as ${\Gamma}$-limits of optimal control problems
subject to ODE constraints, modeling, for instance, external interventions on
crowd dynamics. We obtain these first-order optimality conditions in the form
of Hamiltonian flows in the Wasserstein space of probability measures with
forward-backward boundary conditions with respect to the first and second
marginals, respectively. In particular, we recover the equations and their
solutions by means of a constructive procedure, which can be seen as the
mean-field limit of the Pontryagin Maximum Principle applied to the discrete
optimal control problems, under a suitable scaling of the adjoint variables.

Optimization and Control;Analysis of PDEs;Dynamical Systems