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 goes to infinity.
Cannarsa, P., Cheng, W., Mendico, C., Wang, K. (2023). Weak KAM Approach to First-Order Mean Field Games with State Constraints. JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS, 35(2), 1885-1916 [10.1007/s10884-021-10071-9].
Weak KAM Approach to First-Order Mean Field Games with State Constraints
Piermarco Cannarsa
;Cristian Mendico;
2023-01-01
Abstract
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 goes to infinity.| File | Dimensione | Formato | |
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