Approximation of Deterministic Mean Field Games with Control-Affine Dynamics

Abstract

We consider deterministic mean field games where the dynamics of a typical agent is non-linear with respect to the state variable and affine with respect to the control variable. Particular instances of the problem considered here are mean field games with control on the acceleration (see Achdou et al. in NoDEA Nonlinear Differ Equ Appl 27(3):33, 2020; Cannarsa and Mendico in Minimax Theory Appl 5(2):221-250, 2020; Cardaliaguet and Mendico in Nonlinear Anal 203: 112185, 2021). We focus our attention on the approximation of such mean field games by analogous problems in discrete time and finite state space which fall in the framework of (Gomes et al. in J Math Pures Appl (9) 93(3):308-328, 2010). For these approximations, we show the existence and, under an additional monotonicity assumption, uniqueness of solutions. In our main result, we establish the convergence of equilibria of the discrete mean field games problems towards equilibria of the continuous one. Finally, we provide some numerical results for two MFG problems. In the first one, the dynamics of a typical player is nonlinear with respect to the state and, in the second one, a typical player controls its acceleration.