Linear quadratic zero-sum game for time-delayed uncertain stochastic systems

This study focuses on the analysis of a linear quadratic zero-sum game (LQZSG) for time-delayed uncertain stochastic systems. To begin, we introduce a general time-delayed zero-sum game. Employing an algebraic transformation method, we transform the time-delayed zero-sum game into an equivalent uncertain random zero-sum game without time delay. Subsequently, we present equilibrium equations that streamline the transformation of the uncertain random zero-sum game into problems solvable as deterministic difference equations. We then investigate the LQZSG involving a linear time-delayed uncertain stochastic system with a quadratic objective function. Within this framework, we provide a unified framework for solving this type of game and obtaining the analytic expression for the saddle-point equilibrium of the LQZSG. Additionally, we present a numerical example and a counter-terrorism economic game to illustrate the applicability of our findings.