In this paper we consider the discrete‐time mean‐field stochastic linear‐quadratic (MF‐LQ) optimal control problem with indefinite weighting matrices. First, we establish the maximum principle, and by the solvability of mean‐field forward‐backward stochastic difference equations derived from the maximum principle, we characterize the existence of the open‐loop optimal control for the MF‐LQ problem. Then, by virtue of introducing the linear matrix inequalities condition, we obtain the solvability of the generalized difference Riccati equations (GDREs). Moreover, we show that the indefinite MF‐LQ problem is well‐posed if and only if the GDREs are solvable. Finally, a numerical example is used to show the effectiveness of the obtained results.

中文翻译：

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有限水平离散时间平均场随机线性二次最优控制问题

在本文中，我们考虑了具有不确定权重矩阵的离散时间平均场随机线性二次（MF-LQ）最优控制问题。首先，我们建立了最大原理，并且通过从最大原理导出的均值场向前-向后随机差分方程的可解性，我们刻画了MF-LQ问题的开环最优控制的存在性。然后，通过引入线性矩阵不等式条件，我们获得了广义差分Riccati方程（GDRE）的可解性。此外，我们表明，当且仅当GDRE可解决时，不确定的MF-LQ问题才是适当的。最后，通过算例说明了所得结果的有效性。