This paper is concerned with the exact controllability of linear mean-field stochastic systems with time-variant random coefficients. We prove that the exact controllability, the validity of the observability inequality for the dual equation, the unique solvability of a family of optimal control problems, the unique solvability of a family of mean-field forward-backward stochastic differential equations (MF-FBSDEs), and the unique solvability of a family of norm optimal control problems are all equivalent. Therefore, some approaches are provided to investigate the issue of exact controllability.

中文翻译：

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线性平均场随机系统的精确可控性和平均场后向随机微分方程的可观性不等式

本文关注具有时变随机系数的线性平均场随机系统的精确可控性。我们证明了精确可控性、对偶方程的可观测性不等式的有效性、最优控制问题族的唯一可解性、平均场正反向随机微分方程 (MF-FBSDE) 族的唯一可解性，并且一族范数最优控制问题的唯一可解性都是等价的。因此，提供了一些方法来研究精确可控性的问题。