# Intrinsic Lipschitz Regularity of Mean-Field Optimal Controls

## TL;DR

This paper establishes conditions for Lipschitz regularity in space for solutions to mean-field optimal control problems, combining mean-field approximations with Wasserstein calculus to ensure regularity of feedback controls.

## Contribution

It introduces a novel approach that combines mean-field approximations and Wasserstein calculus to prove Lipschitz regularity of optimal controls in infinite-dimensional settings.

## Key findings

- Lipschitz regularity of controlled vector fields is achieved under new sufficient conditions.
- A reformulation of coercivity estimates in Wasserstein calculus is key to the analysis.
- Uniform Lipschitz bounds are obtained along empirical measure approximations.

## Abstract

In this article, we provide sufficient conditions under which the controlled vector fields solution of optimal control problems formulated on continuity equations are Lipschitz regular in space. Our approach involves a novel combination of mean-field approximations for infinite-dimensional multi-agent optimal control problems, along with a careful extension of an existence result of locally optimal Lipschitz feedbacks. The latter is based on the reformulation of a coercivity estimate in the language of Wasserstein calculus, which is used to obtain uniform Lipschitz bounds along sequences of approximations by empirical measures.