Safe tracking in games: Achieving optimal control with unknown dynamics and constraints

This paper introduces mix-zero-sum differential (MZSD) game theory to address multi-player tracking systems, offering a better understanding of the coexistence of cooperation and competition among players. Within this framework, we present an optimal safety tracking control (OSTC) method, which incorporates a control barrier function (CBF) into the value function to ensure that the tracking error remains within a specified range, thus guaranteeing safety while achieving optimization. Simultaneously, to eliminate the need for system dynamics, we propose a novel approach leveraging off-policy integral reinforcement learning (IRL) technology to obtain the Nash equilibrium solution of the MZSD games. We establish a unique critics–actors neural network (NN) structure that updates concurrently. Furthermore, we analyze stability and convergence using the Lyapunov method. We conduct two simulations to demonstrate the effectiveness of the proposed algorithm.