Systems, Control and Mean Field Games on Networks

大智能体群体系统的平均场博弈理论对非合作博弈理论和大规模系统控制都产生了革命性的影响。MFG纳什均衡理论的核心是一对偏微分方程，由(i)通用代理的最优控制的Hamilton-Jacobi-Bellman （HJB）方程和（ii）这种代理状态的概率分布的Fokker-Planck-Kolmogorov （FPK）方程（或等效的随机SDE）组成，两者由该分布联系起来。本文提出了复杂大系统分析与控制的三个新方向：分别是新的石墨网络控制理论（GNC）和新引入的石墨平均场博弈理论（GMFG）及其各种新的应用领域；它们的主要共同特征是将基于MFG的方法扩展到网络上的系统。首先，利用无限节点图极限（graphon）理论和无限维系统控制理论，对我们新引入的（基于MFG的）复杂网络系统控制GNC理论进行了发展规划。这种方法允许通过对简单的无限极限系统的推导来设计大型复杂有限网络系统的控制。GNC理论的研究课题包括石墨系统可控性、可观察性和系统辨识的进展。其次，我们提出了作者最近提出的MFG理论的新研究方向，它极大地推广了标准理论，使其适用于分布在无界网络上的种群，其中均衡由所谓的Graphon MFG （GMFG）方程给出。目前GMFG理论的关键研究领域包括存在唯一性理论、epsilon - Nash近似理论、通过石墨结构对GMFG系统进行分类，以及主-小智能体系统理论、具有不同行为模式的混合智能体系统和主要智能体不完全可观察的系统。第三，将研究新的应用领域，包括：(i) GNC：来自现实世界网络（如电网、神经元和社交网络）的基于数据的GNC方法。（ii） MFG：状态估计（即过滤）理论在混合动态主要主体和大人口次要主体融资中的应用。（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、进化博弈以及联盟的形成和衰落等系统实例。

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.