Systems, Control and Mean Field Games on Networks

大规模智能体系统的平均场博弈理论是非合作博弈理论和大规模系统控制的一个革命。MFG纳什均衡理论的核心是一对偏微分方程，包括（i）一般代理的最优控制的哈密尔顿-雅可比-贝尔曼（HJB）方程，和（ii）这样的代理状态的概率分布的福克-普朗克-柯尔莫哥洛夫（FPK）方程（或等效的随机微分方程），两者都由该分布联系起来。在这个计划中，提出了三个新的方向，在分析和控制的复杂大系统：分别是新的Graphon网络控制（GNC）理论，新引入的Graphon平均场博弈（GMFG）理论，和他们的各种新的应用领域;其关键的共同特点是MFG为基础的方法扩展到网络上的系统。首先，我们新引入的（基于MFG）GNC理论的复杂网络系统的控制，使用无限节点图极限（graphon）理论和无限维系统控制理论的进步计划。这种方法允许设计大型复杂的有限网络系统的控制，通过他们的推导更简单的无限极限系统。GNC理论的研究主题包括图子系统的可控性、可观性和系统辨识的进展。其次，我们提出了研究的新方向MFG理论最近推出的提议者，大大推广了标准理论，使其适用于人口分布在无界网络的平衡，由所谓的Graphon MFG（GMFG）方程。GMFG理论的主要研究领域现在包括存在性和唯一性理论，Nash- Nash近似理论，以及通过graphon结构的GMFG系统分类，以及主要-次要代理系统理论，具有不同行为模式的混合代理系统和主要代理不完全可观察的系统。第三，将研究新的应用领域，包括：（i）GNC：基于数据的GNC从真实的世界网络，如电网，神经和社交网络的方法。(ii)MFG：状态估计（即过滤）理论在混合动态主要代理和大人口次要代理金融中的应用。(iii)GMFG：建模，估计，动力学和控制将应用于金融，经济，交通，流行病学，神经系统，群集，进化游戏和联盟的形成和衰退的系统实例。

Mean Field Game (MFG) theory for large agent population systems has revolutionized both non-cooperative game theory and large scale systems control. The core of the MFG Nash equilibrium theory is the pair of PDEs consisting of (i) the Hamilton-Jacobi-Bellman (HJB) equation of optimal control for a generic agent, and (ii) the Fokker-Planck-Kolmogorov (FPK) equation (or the equivalent stochastic SDE) for the probability distribution of the state of such an agent, both linked by that distribution. In this program three new directions are proposed in the analysis and control of complex large scale systems: respectively, the new Graphon Network Control (GNC) theory, the newly introduced Graphon Mean Field Game (GMFG) theory, and their various New Application domains; their key common feature is the extension of MFG based methods to systems on networks. First, the advancement is planned for our newly introduced (MFG based) GNC theory of the control of complex network systems using infinite node graph limit (graphon) theory and infinite dimensional systems control theory. This methodology permits the design of controls for large complex finite network systems via their derivation for the simpler infinite limit systems. Research topics in GNC theory include advances in graphon system controllability, observability, and system identification. Second, we propose research in the new direction in MFG theory recently introduced by the proposer which greatly generalizes standard theory so that it applies to populations distributed on unbounded networks where the equilibria are given by the so-called Graphon MFG (GMFG) equations. Key areas of research in GMFG theory now include existence and uniqueness theory, epsilon - Nash approximation theory, and GMFG systems classification via graphon structures, together with major-minor agent system theory, systems with hybrid agents with distinct modes of behaviour and systems where the major agents are not completely observable. Third, new applications domains will be investigated, including: (i) GNC: Methodologies for data based GNC from real world networks, e.g. electricity grids, and neuronal and social networks. (ii) MFG: applications of state estimation (i.e. filtering) theory to finance with hybrid dynamical major agents and large population minor agents. (iii) GMFG: Modelling, estimation, dynamics and control will be employed in applications to instances of systems in finance, economics, transportation, epidemiology, neuronal systems, flocking, evolutionary games and to the formation and decay of coalitions.