  We study a family of McKean-Vlasov (mean-field) type ergodic optimal control
problems with linear control, and quadratic dependence on control of the cost
function. For this class of problems we establish existence and uniqueness of
an optimal control. We propose an $N$-particles Markovian optimal control
problem approximating the McKean-Vlasov one and we prove the convergence in
relative entropy, total variation and Wasserstein distance of the law of the
former to the law of the latter when $N$ goes to infinity. Some McKean-Vlasov
optimal control problems with singular cost function and the relation of these
problems with the mathematical theory of Bose-Einstein condensation is also
established.
