Multiplayer Pareto optimal control with H_∞ constraint for nonlinear stochastic system via online synchronous reinforcement learning

This paper investigates a multiplayer Pareto game for affine nonlinear stochastic systems disturbed by both external and the internal multiplicative noises. The Pareto cooperative optimal strategies with the H_∞ constraint are resolved by integrating H₂/H_∞ theory with Pareto game theory. First, a nonlinear stochastic bounded real lemma (SBRL) is derived, explicitly accounting for non-zero initial conditions. Through the analysis of four cross-coupled Hamilton–Jacobi equations (HJEs), we establish necessary and sufficient conditions for the existence of Pareto optimal strategies with the H_∞ constraint. Secondly, to address the complexity of solving these nonlinear partial differential HJEs, we propose a neural network (NN) framework with synchronous tuning rules for the actor, critic, and disturbance components, based on a reinforcement learning (RL) approach. The designed tuning rules ensure convergence of the actor–critic-disturbance components to the desired values, enabling the realization of robust Pareto control strategies. The convergence of the proposed algorithm is rigorously analyzed using a constructed Lyapunov function for the NN weight errors. Finally, a numerical simulation example is provided to demonstrate the effectiveness of the proposed methods and main results.