Mean-field limit for a class of ergodic stochastic control problems

We establish existence and uniqueness of an optimal control for a family of McKean- Vlasov (mean-field) type ergodic optimal control problems with linear control, and quadratic depen dence on control of the cost function. We propose an N-particles Markovian optimal control problem approximating the McKean-Vlasov one and prove the convergence in relative entropy, total variation of the law of the former to the law of the latter when N goes to infinity.