Weak Closed-loop Solvability for Discrete-time Stochastic Linear-Quadratic Optimal Control
Yue Sun, Xianping Wu, Xun Li
TL;DR
The work addresses the solvability of finite-horizon discrete-time stochastic LQ control, showing that closed-loop solvability corresponds to a regular solution of a generalized Riccati equation while open-loop solvability is weaker. It introduces a perturbation method that connects open-loop solvability to a weak closed-loop feedback structure with $u^*=K^*x^*+v^*$, and proves the equivalence of open-loop and weak closed-loop solvability. A numerical example demonstrates a case that is open-loop solvable but not closed-loop solvable and illustrates the computation of the weak closed-loop gain. The results clarify when Riccati-based feedback can generate the optimal control and when a weak feedback form suffices, with potential implications for robust design under uncertainty.
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.