# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2020026

## Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability

 1 Department of Mathematics, Southern University of Science and Technology, Shenzhen Guangdong 518055, China 2 School of Mathematical Sciences, Fudan University, Shanghai 200433, China

* Corresponding author: Hanxiao Wang

Received  June 2019 Revised  January 2020 Published  June 2020

Fund Project: The first author is supported by NSFC Grant 11901280

This paper is concerned with mean-field stochastic linear-quadratic (MF-SLQ, for short) optimal control problems with deterministic coefficients. The notion of weak closed-loop optimal strategy is introduced. It is shown that the open-loop solvability is equivalent to the existence of a weak closed-loop optimal strategy. Moreover, when open-loop optimal controls exist, there is at least one of them admitting a state feedback representation, which is the outcome of a weak closed-loop optimal strategy. Finally, an example is presented to illustrate the procedure for finding weak closed-loop optimal strategies.

Citation: Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2020026
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