Weak KAM approach to first-order Mean Field Games with state constraints

Piermarco Cannarsa (1) , Wei Cheng (2) , Cristian Mendico (3, 4) , Kaizhi Wang (5)

1 DIPMAT - Dipartimento di Matematica [Roma II]
2 Zhejiang University
3 Université Paris Dauphine-PSL
4 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
5 Shanghai Jiao Tong University [Shanghai]

We study the asymptotic behavior of solutions to the constrained MFG system as the time horizon $T$ goes to infinity. For this purpose, we analyze first Hamilton-Jacobi equations with state constraints from the viewpoint of weak KAM theory, constructing a Mather measure for the associated variational problem. Using these results, we show that a solution to the constrained ergodic mean field games system exists and the ergodic constant is unique. Finally, we prove that any solution of the first-order constrained MFG problem on $[0,T]$ converges to the solution of the ergodic system as $T \to +\infty$.

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Piermarco Cannarsa, Wei Cheng, Cristian Mendico, Kaizhi Wang. Weak KAM approach to first-order Mean Field Games with state constraints. 2020. ⟨hal-02886570⟩