Vanishing viscosity in mean-field optimal control

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.

Automated summary

We prove the existence of Lipschitz-in-space optimal controls for a class of mean-field control problems with non-local continuity equation dynamics. The proof is based on a vanishing viscosity method, where we show the convergence of the problem with the addition of a diffusion term and a small viscosity parameter. By employing stochastic optimal control, we demonstrate the existence of an optimal control sequence for the problem with diffusion. We then construct the optimizer of the original problem by taking the viscosity parameter to zero.