Zero-sum games subject to time-delayed uncertain stochastic systems

This study focuses on time-delayed zero-sum games (ZSGs). Firstly, a zero-sum game (ZSG) for a time-delayed uncertain stochastic system is investigated. An approach is proposed to translate the time-delayed uncertain stochastic ZSG into an equivalent uncertain stochastic game without time delay. Then recursive equations are proposed accordingly to transform the uncertain stochastic game into a solution problem for deterministic difference equations. Based on recursive equations and chance theory, exact solutions for the saddle-point equilibrium solutions of ZSGs for time-delayed uncertain stochastic systems considering linear, quadratic and cubic control terms are provided. Finally, we illustrate the effectiveness of our approach to the time-delayed uncertain stochastic ZSGs through numerical examples.