  We show the existence of Lipschitz-in-space optimal controls for a class of
mean-field control problems with dynamics given by a non-local continuity
equation. The proof relies on a vanishing viscosity method: we prove the
convergence of the same problem where a diffusion term is added, with a small
viscosity parameter. By using stochastic optimal control, we first show the
existence of a sequence of optimal controls for the problem with diffusion. We
then build the optimizer of the original problem by letting the viscosity
parameter go to zero.
