Uncertain stochastic linear quadratic control subject to forward and backward multi-stage systems

Abstract

This paper investigates linear quadratic (LQ) optimal control problems of forward uncertain and backward stochastic multi-stage systems. First, leveraging dynamic programming, we derive stochastic recursive equations tailored to address an LQ optimal control problem for backward stochastic multi-stage systems within a probabilistic framework. Subsequently, we extend our analysis to an equivalent LQ optimal control problem formulated under chance theory, incorporating both forward uncertain and backward stochastic dynamics. From this, we derive uncertain stochastic recursive equations to solve the equivalent problem and provide explicit analytical expressions for the optimal control strategies and corresponding optimal values. Additionally, we explore the effects of variable separability within uncertain random variables, demonstrating that under chance theory, the optimal solutions of LQ optimal control problems remain consistent when uncertain and random variables are separable. Finally, a numerical example is provided to validate our results.