Maximum Principle for Time-Delay Backward Doubly Stochastic Optimal Control Problems Under Partial Information

This paper investigates the optimal control problem of time-delay backward doubly stochastic systems under partial information. Partial information widely exists in practical control systems due to monitoring constraints, communication delays, and data acquisition costs. Combined with inherent system time delays, it greatly complicates state estimation and decision-making, which requires research. A new type of anticipated backward doubly stochastic differential equations is introduced to describe the system dynamics. Using stochastic analysis and the variational methods, the corresponding maximum principle for optimal control is derived. Furthermore, a verification theorem is established that provides rigorous sufficient optimality conditions: any admissible control satisfying the necessary conditions, along with reasonable convexity assumptions, indeed optimizes the cost functional, thereby bridging the gap between necessary and sufficient optimality criteria. As an application, we solve a time-delay linear-quadratic optimal control problem and obtain explicit analytical expressions; the results demonstrate the validity of the established theoretical framework.