  This paper investigates the social optimality of linear quadratic mean field
control systems with unmodeled dynamics. The objective of agents is to optimize
the social cost, which is the sum of costs of all agents. By variational
analysis and direct decoupling methods, the social optimal control problem is
analyzed, and two equivalent auxiliary robust optimal control problems are
obtained for a representative agent. By solving the auxiliary problem with
consistent mean field approximations, a set of decentralized strategies is
designed, and its asymptotic social optimality is further proved. Next, the
results are applied into the study of opinion dynamics in social networks. The
evolution of opinions is analyzed over finite and infinite horizons,
respectively. All opinions are shown to reach agreement with the average
opinion in a probabilistic sense. Finally, local interactions among multiple
populations are examined via graphon theory.
