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

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.