Safe optimal control for multiplayer Stackelberg–Nash games of nonlinear time-delay systems via adaptive dynamic programming

This paper investigates a safe optimal control method for time-delay nonlinear systems based on multiplayer Stackelberg-Nash games (SNGs). Initially, considering different roles of players in the multiplayer SNGs, a hierarchical decision-making process is described as the designed value functions for the leader and all followers. Next, this paper incorporates a control barrier function into the value function to ensure that the system states remain within a safe range, thereby guaranteeing safety while achieving the optimization. Meanwhile, new cost functions are constructed which include the Lyapunov-Krasovskii (L-K) function to eliminate time-delay effects, and a single critic neural network is used to approximate the optimal controller for the leader and the N-followers. Based on the Lyapunov stability theory, it is proven that all signals in the closed-loop system are uniformly ultimately bounded (UUB). Finally, a simulation example is provided to demonstrate the effectiveness of the proposed optimal control method.