This paper studies open-loop and feedback solutions to leader-follower mean field linear-quadratic-Gaussian games with multiplicative noise by the direct approach. The leader-follower game involves a leader and many followers, where the state and control weight matrices in their costs are not limited to be positive definite. From variational analysis with mean field approximations, we obtain a set of open-loop controls in terms of solutions to mean field forward-backward stochastic differential equations. By applying the matrix maximum principle, a set of decentralized feedback strategies is constructed. Different from traditional works, a cross term has appeared in derivation due to the presence of mean field terms. For open-loop and feedback solutions, the corresponding optimal costs of all players are explicitly given in terms of the solutions to two Riccati equations, respectively.

Publication:

SIAM JOURNAL ON CONTROL AND OPTIMIZATION

http://dx.doi.org/10.1137/24M170017X

Author:

Wang, Bing-Chang

School of Control Science and Engineering, Shandong University, Jinan 250061, China

Email：bcwang@sdu.edu.cn

Zhang, Huanshui

College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266510, China

Email：hszhang@sdu.edu.cn

Zhang, Ji-Feng

Corresponding author

School of Automation and Electrical Engineering, Zhongyuan Universityof Technology, Zhengzhou 450007, Henan Province, China

State Key Laboratory of MathematicalSciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing100190, China

Email：jif@iss.ac.cn