Minimum-time Set Stabilization for State-Based Games

This paper investigates the set stabilization of state-based games (SBGs) via the dynamic programming theory. A novel algebraic form-based optimal control method is proposed to drive all state-based action profiles of the SBG to the recurrent state equilibrium set in minimum time. First, by introducing an optimal time vector, the set stabilization of SBGs is transformed into an optimization problem. Then, a necessary and sufficient criterion is proposed to verify the set stabilizability of SBGs. Based on this, a control algorithm that utilizes dynamic programming is presented to compute the time-optimal feedback gain matrix. For any initial state-based action profiles, the control algorithm guarantees that the profiles of all players converge to the recurrent state equilibrium set in minimal time, while maintaining low computational complexity. Finally, a numerical example validates the effectiveness of the proposed approach, demonstrating its superiority in improving computational efficiency.