Weak Closed-loop Solvability for Discrete-time Stochastic Linear-Quadratic Optimal Control

Yue Sun; Xianping Wu; Xun Li

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arXiv · 2502.16229

Weak Closed-loop Solvability for Discrete-time Stochastic Linear-Quadratic Optimal Control

Yue Sun·Xianping Wu·Xun Li

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Abstract

In this paper, the solvability of discrete-time stochastic linear-quadratic (LQ) optimal control problem in finite horizon is considered. Firstly, it shows that the closed-loop solvability for the LQ control problem is optimal if and only if the generalized Riccati equation admits a regular solution by solving the forward and backward difference equations iteratively. To this ends, it finds that the open-loop solvability is strictly weaker than closed-loop solvability, that is, the LQ control problem is always open-loop optimal solvable if it is closed-loop optimal solvable but not vice versa. Secondly, by the perturbation method, it proves that the weak-closed loop strategy which is a feedback form of a state feedback representation is equivalent to the open-loop solvability of the LQ control problem. Finally, an example sheds light on the theoretical results established.

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BibTeX RIS

Yue Sun, Xianping Wu, Xun Li. 2025-02-22. Weak Closed-loop Solvability for Discrete-time Stochastic Linear-Quadratic Optimal Control. https://arxiv.org/abs/2502.16229

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