  In this paper we demonstrate how certain structured feedback gains
necessarily emerge as the optimal controller gains in two linear optimal
control formulations for multi-agent systems. We consider the cases of linear
optimal synchronization and linear optimal centroid stabilization. In the
former problem, the considered cost functional integrates squared
synchronization error and input, and in the latter, the considered cost
functional integrates squared sum of the states and input. Our approach is to
view the structures in the feedback gains in terms of a static output feedback
with suitable output matrices and to relate this fact with the optimal control
formulations. We show that the two considered problems are special cases of a
more general case in which the optimal feedback to a linear quadratic regulator
problem with cost functionals integrating squared outputs and inputs is a
static output feedback. A treatment in this light leads to a very simple and
general solution which significantly generalizes a recent result for the linear
optimal synchronization problem. We illustrate the general problem in a
geometric light.
