Study On Linear Quadratic Mean Field Social Control With Unmodeled Dynamics And Multiplicative Noise

Stochastic optimal control and multi-agent model are important research topics in modern control theory,which have been extensively studied by many scholars at home and abroad.In recent years,the mean field theory has been introduced to large-population games,in which the mean field game theory was developed.So far,mean field game and control has been widely applied in the transmitted power regulation of CDMA networks,charging control of plug-in electric vehicles,production output adjustment and many other fields.Under the linear quadratic(LQ)framework,the model uncertainty(or called "unmodeled dynamics")and the social optima are two meaningful research hotspots in the mean field game.The unmodeled dynamics is helpful to study the robust mean field game and the social optima characterizes the situation that all players cooperatively optimize the social cost(generally,the social cost is the sum of all the individual costs),both of which are widespread in practice.This paper studies the LQ mean field social control with unmodeled dynamics and multiplicative noise,in which the decentralized control strategies are designed and asymptotic social optimality is proved.The organization of this paper is as follows:In the first two chapters,we introduce the backgrounds,basic methods and theorems related to optimal control and mean field game as the preliminaries for the later chapters;In the third chapters,the problem formulation and the main conclusions are given.At the same time,the numerical simulation are given to show the efficiency of the decentralized strategies.The last chapter summarizes the paper and looks forward to future works.The main work and contributions of this paper are as follows:Firstly,the social optima in LQ mean field control with unmodeled dynamics is considered.The decentralized strategies are designed by the analysis of person-by-person optimality and mean field approximation;Secondly,for the case of uniform agents,we give the conditions which ensure that the consistency equations admit a solution.Finally,it is proved that the designed decentralized strategies have asymptotically robust social optimality.