Systems, Control and Mean Field Games on Networks

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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.