In the present work, we study deterministic mean field games (MFGs) with finite time horizon in which the dynamics of a generic agent is controlled by the acceleration. They are described by a system of PDEs coupling a continuity equation for the density of the distribution of states (forward in time) and a Hamilton–Jacobi equation for the optimal value of a representative agent (backward in time). The state variable is the pair \((x,v)\in {\mathbb {R}}^N\times {\mathbb {R}}^N\) where x stands for the position and v stands for the velocity. The dynamics is often referred to as the double integrator. In this case, the Hamiltonian of the system is neither strictly convex nor coercive, hence the available results on MFGs cannot be applied. Moreover, we will assume that the Hamiltonian is unbounded w.r.t. the velocity variable v. We prove the existence of a weak solution of the MFG system via a vanishing viscosity method and we characterize the distribution of states as the image of the initial distribution by the flow associated with the optimal control.

在当前的工作中，我们研究有限时间范围内的确定性平均场博弈（MFG），其中通用代理的动力学由加速度控制。它们由PDE系统描述，该系统耦合状态分布密度的连续性方程（时间向前）和代表代理的最优值的Hamilton-Jacobi方程（时间向前）。状态变量是\（（x，v）\ in {\ mathbb {R}} ^ N \ times {\ mathbb {R}} ^ N \}中的对，其中x代表位置，v代表速度。动态常被称为双积分器。在这种情况下，系统的哈密顿量既不是严格凸的也不是强制的，因此无法应用MFG的可用结果。此外，我们将假设哈密顿量不受速度变量v的限制。我们通过消失粘度方法证明了MFG系统的弱解的存在，并且通过与最优控制相关的流将状态分布表征为初始分布的图像。