Extended mean field control problem: a propagation of chaos result

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.