Fuzzy Weighted Regularization for Fractional-Order Nonlinear Multiagent Systems Under Stackelberg–Nash Game

This study focuses on achieving hierarchical optimal synchronization in a fractional-order nonlinear multiagent system (FONMAS) composed of a single leader and multiple followers, analyzed within the framework of the Stackelberg–Nash game theory. The leader makes decisions by anticipating the optimal responses of all followers, whereas each follower concurrently reacts optimally to the leader’s strategy by engaging in a Nash game. To obtain the optimal control policy of the FONMAS, a regularized fuzzy reinforcement learning approach is proposed. First, a fractional-order (FO) Hamilton–Jacobi–Bellman (HJB) equation in coupled form is formulated, which serves as the basis for deriving the optimal control policies of both the leader and the followers. It is further proven that these strategies constitute a Stackelberg–Nash equilibrium. Next, due to the asymmetry among agents, solving the FO coupled HJB equations becomes challenging. To address this, a hierarchical learning framework grounded in FO value iteration is proposed, which depends solely on partial knowledge of the system dynamics. We demonstrate that, given mild coupling assumptions, this method converges asymptotically to equilibrium policies. Furthermore, a regularized actor–critic framework with fuzzy logic is employed to estimate the cost function and optimal control policy, and the FO weight update rules are developed by formulating a Lyapunov function for the optimal fuzzy weight deviation, ensuring convergence to optimal weight values. Ultimately, both theoretical analysis and simulation results support the efficiency of the proposed method.