Optimal control and zero-sum differential game for stochastic uncertain systems with randomness and uncertainty

This paper presents a stochastic uncertain system characterized by a differential equation involving multiple Wiener processes and uncertain matrices. Our study delves into the realm of chance theory to address problems related to optimal control and zero-sum differential games within this stochastic uncertain framework. Based on the properties of both the Wiener processes and the uncertain matrices, we present an equation of optimality tailored for solving stochastic uncertain optimal control problems. Leveraging this equation, we derive analytical optimal solutions for both bang–bang and linear quadratic control problems. Furthermore, we extend our results to encompass the domain of zero-sum differential games, proposing equilibrium equations to identify the saddle point equilibrium in stochastic uncertain zero-sum games. The practical applicability of our theoretical framework is demonstrated through the analysis of a counterterrorism game scenario.