Bang‐Bang and Linear Quadratic Zero‐Sum Game for Noncausal and Causal Systems

ABSTRACT This paper investigates zero‐sum games under the constraints of noncausal and causal systems, providing a comprehensive theoretical framework for such models. Initially, a general framework of zero‐sum game models constrained by noncausal and causal systems is first established. For single‐stage scenarios, the saddle‐point equilibrium is derived with relative ease. In the case of multistage scenarios, we derive equilibrium equations governing the saddle‐point equilibrium solutions, and these equations are used to obtain the equilibrium of bang‐bang and linear quadratic zero‐sum games. The study is further extended to multistage time‐delayed zero‐sum games, where an equivalent transformation approach is proposed to determine equilibrium solutions under noncausal and causal system constraints. Finally, the effectiveness of the results is demonstrated through numerical examples.