Robust Mean Field Social Optimal Control with Applications to Opinion Dynamics

This paper investigates the social optimal problem in linear quadratic mean field control systems with unmodeled dynamics. The objective of the agents in the social system is to optimize the social cost, which is the sum of the costs of all the agents. By the variational method, the social optimal problem is analyzed, and the equivalent robust optimal control problems are obtained for each agent. The decentralized strategies are obtained by solving the auxiliary problem with the mean field approximation, and the asymptotic social optimality is proved under mild conditions. The above results are applied into opinion dynamics, and all opinions are shown to converge to the mean opinion in probability.