  We consider team optimal control of decentralized systems with linear
dynamics, quadratic costs, and arbitrary disturbance that consist of multiple
sub-populations with exchangeable agents (i.e., exchanging two agents within
the same sub-population does not affect the dynamics or the cost). Such a
system is equivalent to one where the dynamics and costs are coupled across
agents through the mean-field (or empirical mean) of the states and actions
(even when the primitive random variables are non-exchangeable). Two
information structures are investigated. In the first, all agents observe their
local state and the mean-field of all sub-populations, in the second, all
agents observe their local state but the mean-field of only a subset of the
sub-populations. Both information structures are non-classical and not
partially nested. Nonetheless, it is shown that linear control strategies are
optimal for the first and approximately optimal for the second, the
approximation error is inversely proportional to the size of the
sub-populations whose mean-fields are not observed. The corresponding gains are
determined by the solution of K+1 decoupled standard Riccati equations, where K
is the number of sub-populations. The dimensions of the Riccati equations do
not depend on the size of the sub-populations, thus the solution complexity is
independent of the number of agents. Generalizations to major-minor agents,
tracking cost, weighted mean-field, and infinite horizon are provided. The
results are illustrated using an example of demand response in smart grids.
