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We provide a new version of the Tikhonov theorem for two-scale systems ofstochastic differential equations which also covers the McKean-Vlasov case.Differently from what is usually done in the literature, we prove a type ofconvergence for the "fast" variable, which allows the limiting process to bediscontinuous. This is relevant for the second part of the paper, where wepresent a new application of this theory to the approximation of the solutionof mean field control problems. Towards this aim, we construct a two-scalesystem whose "fast" component converges to the optimal control process, whilethe "slow" component converges to the optimal state process. The interest insuch a procedure is that it allows to approximate the solution of the controlproblem avoiding the usual step of the minimization of the Hamiltonian.

A Tikhonov theorem for McKean-Vlasov two-scale systems and a new application to mean field optimal control problems