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 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.As per journal style, reference citation should be expanded form in abstract. So kindly check and confirm the reference citation present in the abstract is correct.Please change "Gomes in" below by "Gomes et al. in "

我们考虑确定性平均场博弈，其中典型主体的动态相对于状态变量是非线性的，而相对于控制变量是仿射的。这里考虑的问题的具体实例是控制加速度的平均场博弈（参见 Achdou 等人，NoDEA Nonlinear Differ Equ Appl 27(3):33, 2020；Cannarsa 和 Mendico，Minimax Theory Appl 5(2):221 -250, 2020；Cardaliaguet 和 Mendico 在《非线性肛门》203: 112185, 2021）。我们将注意力集中在离散时间和有限状态空间中类似问题的平均场博弈的近似上，这些问题属于（Gomes in J Math Pures Appl (9) 93(3):308-328, 2010）的框架。对于这些近似，我们证明了解的存在性以及在附加单调性假设下的唯一性。在我们的主要结果中，我们建立了离散平均场博弈问题的均衡向连续平均场博弈问题的均衡的收敛。最后，我们提供了两个 MFG 问题的一些数值结果。在第一个中，典型玩家的动态相对于状态而言是非线性的，而在第二个中，典型玩家控制其加速度。根据期刊风格，参考引用应该以抽象形式扩展。因此，请检查并确认摘要中的参考引文是否正确。请将下面的“Gomes in”改为“Gomes et al. in”