Control of collective dynamics via mean field and and inverse mean field game theory

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Mean field game theory has emerged in the first decade of the century and has quickly become one of the most effective ways of analyzing and controlling the aggregate behavior of large multi-agent systems. The idea can best be explained through the motion of fish schools whereby as the number of fish in a fish school increases indefinitely, the impact of one fish motion on the rest of the school becomes negligible, and each fish starts perceiving a "crowd effect" around it. At that point, we like to think of a generic fish as an agent associated with a cost function sensitive to how its own state is positioned with respect to the group of other agent states, hereby called the "mean field". Since the group behavior around the agent has gained inertia, given that behavior, the generic agent has to solve an optimal control problem to decide for its optimal moving strategy. By aggregating the optimal responses of agents, for consistency, one should be able to recover the group behavior assumed in the first place. This  is mathematically characterized as a fixed-point calculation. It is the basis of a powerful technique for constructing approximate Nash equilibria in large-scale games based on decentralized control laws.   The result is particularly attractive when dealing with the so-called linear quadratic (LQ) games situation because the aggregate behaves like a single appropriate dynamic agent. We propose a research program to extend as far as possible the application potential of the LQ games framework for the decentralized control of group dynamics through a prescriptive game approach where we design the cost functions with a specific desired aggregate goal in mind. We propose to test the limits of this design approach in terms of the objectives of classical control namely reference signal following  and disturbance rejection. In particular, we propose to develop the equivalent of a notion of "internal model principle", whereby  a system can follow a given reference signal only if it is intrinsically complex enough to generate that signal under the actions of internal initial conditions. In addition, we would like to investigate the possibilities of inverse-Nash design approaches, whereby one develops the differential equations that the cost coefficients must  satisfy when the aggregate is assumed to follow a target trajectory, and verify existence of solutions.    While classical mean field game theory has assumed instantaneous all to all agent influences, we would like to extend the theory over networks where communication and thus influences  propagate only, as in fish schools, from peer to peer. This results in delayed mutual influences of agents. Thus we would like to enhance the classical mean field game dynamic model with a communication layer, and study the mutual interactions of these two layers.  Applications are envisioned in smart grids, the channeling of crowd dynamics, and the decentralized control of micro-robotic swarms.