The goal of this paper is to study the long time behavior of solutions of the first-order mean field game (MFG) systems with a control on the acceleration. The main issue for this is the lack of small time controllability of the problem, which prevents to define the associated ergodic mean field game problem in the standard way. To overcome this issue, we first study the long-time average of optimal control problems with control on the acceleration: we prove that the time average of the value function converges to an ergodic constant and represent this ergodic constant as a minimum of a Lagrangian over a suitable class of closed probability measure. This characterization leads us to define the ergodic MFG problem as a fixed-point problem on the set of closed probability measures. Then we also show that this MFG ergodic problem has at least one solution, that the associated ergodic constant is unique under the standard monotonicity assumption and that the time-average of the value function of the time-dependent MFG problem with control of acceleration converges to this ergodic constant.

本文的目的是研究具有加速度控制的一阶均值现场博弈（MFG）系统解的长时间行为。这样做的主要问题是问题的时间可控性不足，从而无法以标准方式定义相关的遍历均场游戏问题。为了克服这个问题，我们首先研究了最优控制问题的长期平均值，并对其进行了加速度控制：我们证明了值函数的时间平均值收敛于遍历常数，并将该遍历常数表示为拉格朗日常数的最小值。一类合适的封闭概率测度。这种表征使我们将遍历遍历的MFG问题定义为一组封闭概率测度上的定点问题。