  This paper studies social optimal control of mean field LQG
(linear-quadratic-Gaussian) models with uncertainty. Specially, the uncertainty
is represented by a uncertain drift which is common for all agents. A robust
optimization approach is applied by assuming all agents treat the uncertain
drift as an adversarial player. In our model, both dynamics and costs of agents
are coupled by mean field terms, and both finite- and infinite-time horizon
cases are considered. By examining social functional variation and exploiting
person-by-person optimality principle, we construct an auxiliary control
problem for the generic agent via a class of forward-backward stochastic
differential equation system. By solving the auxiliary problem and constructing
consistent mean field approximation, a set of decentralized control strategies
is designed and shown to be asymptotically optimal.
