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

TL;DR

This paper investigates the conditions under which discrete-time stochastic LQ control problems are solvable in closed-loop form, establishing the equivalence of weak-closed loop strategies and open-loop solvability through Riccati equations and perturbation methods.

Contribution

It introduces the concept of weak-closed loop solvability and proves its equivalence to open-loop solvability for discrete-time stochastic LQ problems.

Findings

Closed-loop solvability is equivalent to the regular solution of a generalized Riccati equation.

Open-loop solvability is weaker than closed-loop solvability.

Weak-closed loop strategies are equivalent to open-loop solvability.

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.