Vanishing viscosity in mean-field optimal control

TL;DR

This paper demonstrates the existence of optimal controls in mean-field problems using a vanishing viscosity approach, combining stochastic control and non-local PDE analysis to handle the limit as diffusion vanishes.

Contribution

It introduces a novel vanishing viscosity method to establish Lipschitz-in-space optimal controls for mean-field control problems with non-local dynamics.

Findings

Proves convergence of diffusive control problems as viscosity approaches zero.

Establishes existence of optimal controls for non-local mean-field dynamics.

Utilizes stochastic control techniques to handle the vanishing viscosity limit.

Abstract

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.