Integral reinforcement learning solutions for a synchronisation system with constrained policies

A class of the differential games is considered where the agents employ constrained control strategies, and the mutual interactions between the agents are restricted by an undirected graph topology. The dynamical behaviour of the agents and the applied control policies are evaluated using local non-linear performance indices. The solution of the differential game is obtained via a game-theoretic mathematical framework based on adaptive integral reinforcement learning (IRL) schemes. The constrained optimality conditions for the graphical game are found using Bellman's optimality principles. It is demonstrated that, solving the game's coupled IRL-Bellman optimality equations with constrained control policies yields a Nash equilibrium solution. Online adaptive learning solutions are developed using value iteration processes and means of the adaptive critics. Neural network structures are adopted to approximate the constrained optimal control strategies and the respective optimal value functions for each agent in a distributed fashion. The robustness of the proposed solutions is tested using uncertain dynamical learning environment and graph with large time-varying deviations in the connectivity weights.