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

Hanxiao Wang; Jingrui Sun; Jiongmin Yong

doi:10.3934/dcds.2019117

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2019, Volume 39, Issue 5: 2785-2805. Doi: 10.3934/dcds.2019117

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

1.
School of Mathematical Sciences, Fudan University, Shanghai 200433, China
2.
Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong 518055, China
3.
Department of Mathematics, University of Central Florida, Orlando FL 32816, USA

* Corresponding author: Jingrui Sun
* Corresponding author: Jingrui Sun

Received: May 31, 2018

Published: January 2019

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

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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.

Keywords:

Stochastic linear-quadratic optimal control,
Riccati equation,
open-loop solvability,
weak closed-loop solvability,
state feedback.

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

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