Wasserstein regularity in mean field control problems

This work deals with a class of mean field control problems that are obtained as limits of optimal control problems for large particle systems. Developing on [Cardaliaguet, P. & Souganidis, P. E.(2023). Regularity of the value function and quantitative propagation of chaos for mean field control problems, Nonlinear Differ. Equ. Appl.], we analyse the value function U in Wasserstein metric and we prove its smoothness in an open and dense set of the space, time and probability measures using the strategy of the linearized system. The definition of this set exploits the concept of strong stability. Then, we focus on chaos propagation: we study the properties of the optimal solutions of the interacting particle system starting from the aforementioned open and dense set. We also show some classical results on flows of probability measures via simple analytical tools.