  We study Mean Field stochastic control problems where the cost function and
the state dynamics depend upon the joint distribution of the controlled state
and the control process. We prove suitable versions of the Pontryagin
stochastic maximum principle, both in necessary and in sufficient form, which
extend the known conditions to this general framework. Furthermore, we suggest
a variational approach to study a weak formulation of these control problems.
We show a natural connection between this weak formulation and optimal
transport on path space, which inspires a novel discretization scheme.
