This paper is concerned with mean field linear quadratic social control with common noise, where the weight matrices of individual costs are indefinite We first obtain a set of forward–backward stochastic differential equations (FBSDEs) from variational analysis, and then construct a centralized feedback representation by decoupling the FBSDEs. By using solutions of two Riccati equations, we design a set of decentralized control laws, which is further shown to be asymptotically social optimal. The necessary and sufficient conditions are given for uniform stabilization of the systems by exploiting the relation between population state average and aggregate effect. An explicit expression of the optimal social cost is given in terms of two Riccati equations. Besides, the decentralized optimal solution is provided for mean field social control problem with finite agents. 

　　Publication: 

　　Automatica, Volume 146, December 2022, 110619 

　　Author: 

　　Bingchang Wang 

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

　　Huanshui Zhang 

　　College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China 

　　Ji-Feng Zhang 

　　Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China 

　　School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100149, China 

　　Email: jif@iss.ac.cn