论文标题
控制和平均野外游戏问题的崇高行为取决于加速度
Ergodic behavior of control and mean field games problems depending on acceleration
论文作者
论文摘要
本文的目的是研究以控制加速度的一阶平均野外游戏（MFG）系统的长期行为。这样做的主要问题是问题缺乏时间的可控性，该问题可防止以标准方式定义相关的厄运平均野外游戏问题。为了克服这个问题，我们首先研究了最佳控制问题的长期平均水平与加速度的控制：我们证明，值函数的时间平均值会收敛到ergodic常数，并表示该千古常量是在适当的封闭概率措施的情况下，将其作为Lagrangian的最小值。这种表征使我们在封闭概率度量集中定义了千古的MFG问题为定点问题。然后，我们还表明，这个MFG ergodic问题至少具有一个解决方案，相关的沿阵贡常数在标准的单重点假设下是唯一的，并且与时间相关的MFG问题的价值功能的时间平均值随着加速度的控制收敛到此ergodic常数。
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 mono-tonicity 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.
