Research On Some Control Problems For Stochastic Mean Field Systems

Stochastic mean-field systems have been extensively studied because of their scientific significance and great importance in applications.In resent decades,meanfield models have attracted extensive attention.A lot of works have made great contribution to the study of mean-field systems.However,there are still some basic problems facing challenges.In this paper,we study some essence problems of linear quadratic(LQ)control.We propose a novel method for decoupling forwardbackward stochastic differential/difference equations(FBSDEs).The necessary and sufficient conditions for a series of optimal control problems are obtained,and the optimal controller’s analytical form are provided by solving the the FBSDEs.We considered some problems that have not been solved,including optimal control for mean-field systems with delay,indefinite mean-field game,social optimal control for mean-field systems with a major agent,output feedback control for mean-field systems.The specific contributions include the following aspects according to the order of chapters:For the continuous-time mean-field system with delay,the analytical optimal controller for the finite horizon problem is given,and the necessary and sufficient condition for stabilization in the mean square sense is obtained for the infinite horizon problem;For the discrete-time indefinite mean-field game with multiplicative noise,by using the property of convexity,the necessary and sufficient conditions for the existence of Nash equilibrium solution are obtained under the condition of indefinite weighting matrix.A distributed control strategy is designed,which is proved to be a ε-Nash equilibrium;For the social optimal control of discrete time mean-field system with a major agent,the necessary and sufficient condition for the centralized optimal controller is given,and the distributed controller is design for each agent under the standard assumption,which is further proved to be asymptotically optimal;For output feedback control problem for discrete-time mean-field system,the optimal estimator of each agent is obtained,and a set of output feedback type distributed controller is designed,which is proved to be asymptotically optimal.