Asymptotically Optimal Decentralized Control for Interacted ARX Multi-Agent Systems

We consider the decentralized control for a class of stochastic multi-agent systems described by coupled first order auto-regression models with exogenous inputs (ARX models). A stochastic time-averaged group-tracking-like performance index is adopted for each agent, with which the individual and population average states are coupled nonlinearly. A decentralized control law is designed based on the estimate of the population average state and the Nash certainty equivalence principle. By probability limit theory, it is shown that: 1) the estimate of the population average state is strongly consistent. 2) the closed-loop system is almost surely uniformly stable, and bounded independently of the number of agents. 3) when the nonlinear coupling function in the indexes is globally Lipschitz continuous, the decentralized control law is asymptotically optimal almost surely; when locally Lipschitz continuous, the control law is asymptotically optimal in probability.