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

平均场博弈理论诞生于本世纪头十年，现已迅速成为分析和控制大型多智能体系统聚集行为的最有效方法之一。这一想法可以通过鱼群的运动得到最好的解释，即随着鱼群中鱼的数量无限增加，一条鱼运动对鱼群其余部分的影响变得微不足道，每条鱼开始感知到它周围的“人群效应”。在这一点上，我们喜欢将通用鱼类视为与成本函数相关联的代理，该成本函数对其自身的状态相对于其他代理状态组的定位非常敏感，在此称为“平均场”。由于智能体周围的群体行为已经获得了惯性，给定这种行为，通用智能体必须解决一个最优控制问题来决定其最优移动策略。通过聚集代理的最佳响应，为了一致性，人们应该能够恢复最初假设的群体行为。这一结果在数学上被描述为定点计算。它是构造大规模博弈中基于分散控制律的近似纳什均衡的强大技术的基础。在处理所谓的线性二次(LQ)博弈情况时，结果特别有吸引力，因为集合的行为就像单个合适的动态代理。我们提出了一个研究计划，通过一种说明性的博弈方法，尽可能地扩展LQ博弈框架在分散控制群体动态方面的应用潜力，其中我们在设计成本函数时考虑了特定的期望总目标。我们建议从经典控制的目标，即参考信号跟踪和干扰抑制的角度来测试这种设计方法的极限。特别是，我们建议发展一种相当于“内部模型原理”的概念，即只有当系统本质上足够复杂，能够在内部初始条件的作用下产生该信号时，系统才能遵循给定的参考信号。此外，我们想调查逆纳什设计方法的可能性，根据该方法，当假设聚集遵循目标轨迹时，成本系数必须满足的微分方程式，并验证解的存在。虽然经典平均场博弈理论假设所有瞬时对所有代理人的影响，但我们希望将该理论扩展到网络，在网络中，通信并因此影响只传播，如在鱼群中，从一个对等到另一个对等。这导致了代理人之间相互影响的延迟。因此，我们希望在经典的平均场博弈动态模型中增加一个通信层，并研究这两个层之间的相互作用。这些应用被设想在智能电网、人群动力学的引导以及微型机器人群体的分散控制中。

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.