Weak Closed-Loop Solvability of Stochastic Linear-Quadratic Optimal Control Problems

TL;DR

This paper introduces the concept of weak closed-loop optimal strategies for stochastic LQ problems, establishing their equivalence to open-loop solvability and providing methods to find such strategies.

Contribution

It defines weak closed-loop strategies for stochastic LQ problems, linking open-loop solvability to a new feedback-based approach where traditional Riccati methods may fail.

Findings

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

At least one open-loop optimal control admits a feedback representation.

An example demonstrates how to find weak closed-loop strategies.

Abstract

Recently it has been found that for a stochastic linear-quadratic optimal control problem (LQ problem, for short) in a finite horizon, open-loop solvability is strictly weaker than closed-loop solvability which is equivalent to the regular solvability of the corresponding Riccati equation. Therefore, when an LQ problem is merely open-loop solvable not closed-loop solvable, which is possible, the usual Riccati equation approach will fail to produce a state feedback representation of open-loop optimal controls. The objective of this paper is to introduce and investigate the notion of weak closed-loop optimal strategy for LQ problems so that its existence is equivalent to the open-loop solvability of the LQ problem. Moreover, there is at least one open-loop optimal control admitting a state feedback representation. Finally, we present an example to illustrate the procedure for finding weak closed-loop optimal strategies.