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Article Contents

# Weak closed-loop solvability of stochastic linear-quadratic optimal control problems

• * Corresponding author: Jingrui Sun

The first author is supported in part by the China Scholarship Council, while visiting University of Central Florida. The third author is supported in part by NSF DMS-1812921

• 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.

Mathematics Subject Classification: Primary: 93E20; Secondary: 49N10, 49N35.

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