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.

中文翻译：

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平均场随机线性二次最优控制问题：弱闭环可解性

本文涉及具有确定性系数的平均场随机线性二次（MF-SLQ）最优控制问题。介绍了弱闭环最优策略的概念。结果表明，开环可解性等同于弱闭环最优策略的存在。此外，当存在开环最优控制时，其中至少有一个承认状态反馈表示，这是弱闭环最优策略的结果。最后，给出一个例子来说明寻找弱闭环最优策略的过程。