This paper discusses feedback Stackelberg strategies for the continuous-time mean-field type stochastic systems with multiple followers in infinite horizon. First, optimal control problems of the followers are studied in the sense of Nash equilibrium. With the help of a set of generalized algebraic Riccati equations (GAREs), sufficient conditions for the solvability are put forward. Then, the leader faces a constrained optimal control problem by transforming the cost functional into a trace criterion. Employing the Karush-Kuhn-Tucker (KKT) conditions, necessary conditions are presented in term of the solvability of the cross-coupled stochastic algebraic equations (CSAEs). Moreover, feedback Stackelberg strategies are obtained based on the solutions of the CSAEs. In addition, an iterative scheme is introduced to obtain efficiently the solutions of the CSAEs. Finally, an example is given to shed light on the effectiveness of the proposed results.

本文讨论了无限视界内多跟随器连续时间平均场型随机系统的反馈Stackelberg策略。首先，研究了纳什均衡意义上的追随者最优控制问题。借助一组广义代数 Riccati 方程（GARE），提出了可解性的充分条件。然后，领导者通过将成本函数转换为跟踪准则来面临约束最优控制问题。采用Karush-Kuhn-Tucker (KKT) 条件，根据交叉耦合随机代数方程(CSAE) 的可解性给出了必要条件。此外，反馈Stackelberg策略是基于CSAE的解决方案获得的。此外，引入迭代方案来有效地获得 CSAE 的解决方案。最后，给出一个例子来说明所提出的结果的有效性。