  A decentralized linear quadratic system with a major agent and a collection
of minor agents is considered. The major agent affects the minor agents, but
not vice versa. The state of the major agent is observed by all agents. In
addition, the minor agents have a noisy observation of their local state. The
noise processes is \emph{not} assumed to be Gaussian. The structures of the
optimal strategy and the best linear strategy are characterized. It is shown
that major agent's optimal control action is a linear function of the major
agent's MMSE (minimum mean squared error) estimate of the system state while
the minor agent's optimal control action is a linear function of the major
agent's MMSE estimate of the system state and a "correction term" which depends
on the difference of the minor agent's MMSE estimate of its local state and the
major agent's MMSE estimate of the minor agent's local state. Since the noise
is non-Gaussian, the minor agent's MMSE estimate is a non-linear function of
its observation. It is shown that replacing the minor agent's MMSE estimate by
its LLMS (linear least mean square) estimate gives the best linear control
strategy. The results are proved using a direct method based on conditional
independence, common-information-based splitting of state and control actions,
and simplifying the per-step cost based on conditional independence,
orthogonality principle, and completion of squares.
