Study On Mean Field Social Control With Unmodeled Dynamics

The research of large population systems with a large number of agents is an important part of control theory,and has a wide range of applications in the fields of networks,economics and sociology.Due to the coupling between agents,the analysis and calculation complexity of the system increases significantly as the number of agents increases.Because of the limitation of communication distance and storage capacity,it is not practical to design centralized control strategies under centralized information structure for a single agent.To solve these problems,mean field models in the field of physics have been introduced to large population systems by many scholars,and the mean field games and control theory was developed and a lot of results was obtained.The mean field model models the population effect of the large population system as the mean field term.Agents interact through the mean field term.By mean field approxi-mations to achieve decoupling of agents to design decentralized control strategies under decentralized information structure,this greatly reduces the analysis and calculation complexity of the large population system.Mean field games consider that agents non-cooperatively optimize their own cost functionals,so this prob-lem is a Nash game and their strategies constitute a Nash equilibrium.The mean field social control problem refers to that the agents cooperatively optimize the social cost functional(the sum of agents' cost functionals),hence this is a team decision problem and therefore has more social value.At present,most of the literatures have studied the mean field games under the framework of accurate models.According to this,this paper mainly studies the mean field social control problem with model uncertainty.The mean field term is coupled not only in the dynamics but also in the cost functionals.The goal of agents is to coopera-tively optimize the social cost functional of the system-i.e.,the sum of the cost functionals of each agent.In order to consider the uncertainty of the system,we introduce a local adversarial disturbance for each agent.The disturbance can be seen as a control given by a virtual agent(disturber).Using the "soft constraint"method,the disturbance is penalized in the cost functionals.Therefore,the large population system includes a agent team and a disturber team.The agent team cooperates to minimize the social cost functional.The disturber team cooperates to maximize the social cost functional.The behavior between the two teams con-stitutes a zero-sum game.We consider the social optimal control strategies under the worst-case disturbance,which is the saddle point of the zero-sum game.The main results of this thesis are given as follows:1.We study the mean field social control problem with model uncertainty and additive noise by fix point approach.The auxiliary zero-sum game is obtained by mean filed approximations.The decentralized control strategies are designed by solving the auxiliary problem and proved that the the set of decentralized con-trol strategies has asymptotic social optimality.Applying the results to opinion dynamics,it proves that individual opinions converge to the average viewpoint of the system in the sense of probability.2.We study the mean field social control problem with model uncertainty and multiplicative noise by direct approach.The necessity and sufficient conditions with respect to the open-loop and closed-loop saddle points are obtained.The decentralized control strategies are designed by mean filed approximations.