INTRINSIC LIPSCHITZ REGULARITY OF MEAN-FIELD OPTIMAL CONTROLS

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.

Automated summary

In this article, sufficient conditions for the Lipschitz regularity of the controlled vector fields solution of optimal control problems formulated on continuity equations are provided. The approach involves mean-field approximations and a careful extension of the existence result of locally optimal Lipschitz feedbacks.