Feedback Stackelberg Solution for Mean-Field Type Stochastic Systems with Multiple Followers

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.