  In this paper, we study the $extended$ mean field control problem, which is a
class of McKean-Vlasov stochastic control problem where the state dynamics and
the reward functions depend upon the joint (conditional) distribution of the
controlled state and the control process. By considering an appropriate
controlled Fokker-Planck equation, we can formulate an optimization problem
over a space of measure-valued processes and, under suitable assumptions, prove
the equivalence between this optimization problem and the $extended$ mean-field
control problem. Moreover, with the help of this new optimization problem, we
establish the associated limit theory i.e. the $extended$ mean field control
problem is the limit of a large population control problem where the
interactions are achieved via the empirical distribution of state and control
processes.
