Mean field games (MFG) have emerged as a viable tool in the analysis of large-scale self-organizing networked systems. In particular, MFGs provide a game-theoretic optimal control interpretation of the emergent behavior of noncooperative agents. The purpose of this paper is to study MFG models in which individual agents obey multidimensional nonlinear Langevin dynamics, and analyze the closed-loop stability of fixed points of the corresponding coupled forward-backward partial differential equation (PDE) systems. In such MFG models, the detailed balance property of the reversible diffusions underlies the perturbation dynamics of the forward–backward system. We use our approach to analyze closed-loop stability of two specific models. Explicit control design constraints, which guarantee stability, are obtained for a population distribution model and a mean consensus model. We also show that static state feedback using the steady-state controller can be employed to locally stabilize an MFG equilibrium.

中文翻译：

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具有Langevin动力学的代理商的平均场博弈

在大型自组织网络系统的分析中，平均现场游戏（MFG）已成为一种可行的工具。特别是，MFG为非合作代理的紧急行为提供了博弈论的最佳控制解释。本文的目的是研究其中的个体服从多维非线性Langevin动力学的MFG模型，并分析相应的前向后向偏微分方程（PDE）系统的固定点的闭环稳定性。在这样的MFG模型中，可逆扩散的详细平衡特性是前后系统扰动动力学的基础。我们使用我们的方法来分析两个特定模型的闭环稳定性。明确的控制设计约束，可以保证稳定性，获得了人口分布模型和均值共识模型。我们还表明，使用稳态控制器的静态反馈可用于局部稳定MFG平衡。