This was part of

**Distributed Solutions to Complex Societal Problems Reunion Workshop**## On the long time behavior of mean field control problems

**
Alekos Cecchin, Università di Padova
**

**Wednesday, February 22, 2023**

**Abstract**: We study the long time behavior of controlled McKean-Vlasov dynamics with a non-degenerate idiosyncratic noise, without assuming convexity, nor monotonicity, of the cost coefficients. Instead, we consider a drift component which is uniformly convex, but just outside an arbitrarily large ball. Using coupling by reflection arguments, we establish uniform in time estimates for the Lipschitz constant of the value function of the mean field control problem and of its measure derivative. Thus we show existence and uniqueness of a solution to the ergodic problem and then the turnpike property, that is related to exponential convergence of the optimal trajectories to the equilibrium, as the time horizon grows. We also prove uniform in time propagation of chaos for the related symmetric control problem with N agents.