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

• * Corresponding author: Hanxiao Wang

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.

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

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