Relaxed Lagrangian Approach to First-Order Non-Convex Mean Field Type Control Problem

TL;DR

This paper introduces a relaxed Lagrangian method on Wasserstein space to establish the existence of equilibria in non-convex first-order Mean Field Control problems, extending classical results.

Contribution

It presents a novel relaxed Lagrangian approach that guarantees the existence of equilibria in non-convex Mean Field Control problems, broadening the scope of classical convexity-based results.

Findings

Existence of relaxed Nash equilibria in non-convex settings

The approach generalizes classical convex case results

Framework applicable to a broad class of control problems

Abstract

This paper addresses the existence of equilibria for Mean Field type Control problems of first-order with non-convex action functional. Introducing a relaxed Lagrangian approach on the Wasserstein space to handle the lack of convexity. we prove the existence of new relaxed Nash equilibria and we show that our existence result encompasses the classical Mean Field Control problem's existence result under convex data conditions.