A Novel Mean-Field-Game-Type Optimal Control for Very Large-Scale Multiagent Systems

In this article, a decentralized adaptive optimal controller based on the emerging mean-field game (MFG) and self-organizing neural networks (NNs) has been developed for multiagent systems (MASs) with a large population and uncertain dynamics. This design can effectively break the “curse of dimensionality” and reduce the computational complexity by appropriately integrating emerging MFG theory with self-organizing NNs-based reinforcement learning techniques. First, the decentralized optimal control for massive MASs has been formulated into an MFG. To unfold the MFG, the coupled Hamilton–Jacobian–Bellman (HJB) equation and Fokker–Planck–Kolmogorov (FPK) equation needed to be solved simultaneously, which is challenging in real time. Therefore, a novel actor–critic–mass (ACM) structure has been developed along with self-organizing NNs subsequently. In the developed ACM structure, each agent has three NNs, including: 1) mass NN learning the mass MAS’s overall behavior via online estimating the solution of the FPK equation; 2) critic NN obtaining the optimal cost function through learning the HJB equation solution along with time; and 3) actor NN estimating the decentralized optimal control by using the critic and mass NNs along with the optimal control theory. To reduce the NNs’ computational complexity, a self-organizing NN has been adopted and integrated into a developed ACM structure that can adjust the NNs’ architecture based on the NNs’ learning performance and the computation cost. Finally, numerical simulation has been provided to demonstrate the effectiveness of the developed schemes.