An optimal control problem is studied for a linear mean-field stochastic differential equation with a quadratic cost functional. The coefficients and the weighting matrices in the cost functional are all assumed to be deterministic. Closed-loop strategies are introduced, which require to be independent of initial states; and such a nature makes it very useful and convenient in applications. In this paper, the existence of an optimal closed-loop strategy for the system (also called the closed-loop solvability of the problem) is characterized by the existence of a regular solution to the coupled two (generalized) Riccati equations, together with some constraints on the adapted solution to a linear backward stochastic differential equation and a linear terminal value problem of an ordinary differential equation.

中文翻译：

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平均场随机线性二次最优控制问题：闭环可解性

研究了具有二次成本函数的线性平均场随机微分方程的最优控制问题。成本函数中的系数和加权矩阵均假定为确定性的。引入了闭环策略，要求独立于初始状态。这种性质使其在应用中非常有用和方便。在本文中，系统最优闭环策略的存在（也称为问题的闭环可解性）的特征在于存在耦合的两个（广义）Riccati方程的正则解，以及一些对线性倒向随机微分方程和普通微分方程的线性终值问题的自适应解的约束。