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May  2019, 39(5): 2785-2805. doi: 10.3934/dcds.2019117

## 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

Received  June 2018 Published  January 2019

Fund Project: 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.

Citation: Hanxiao Wang, Jingrui Sun, Jiongmin Yong. Weak closed-loop solvability of stochastic linear-quadratic optimal control problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2785-2805. doi: 10.3934/dcds.2019117
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