  We study the distributed Linear Quadratic Gaussian (LQG) control problem in
discrete-time and finite-horizon, where the controller depends linearly on the
history of the outputs and it is required to lie in a given subspace, e.g. to
possess a certain sparsity pattern. It is well-known that this problem can be
solved with convex programming within the Youla domain if and only if a
condition known as Quadratic Invariance (QI) holds. In this paper, we first
show that given QI sparsity constraints, one can directly descend the gradient
of the cost function within the domain of output-feedback controllers and
converge to a global optimum. Note that convergence is guaranteed despite
non-convexity of the cost function. Second, we characterize a class of Uniquely
Stationary (US) problems, for which first-order methods are guaranteed to
converge to a global optimum. We show that the class of US problems is strictly
larger than that of strongly QI problems and that it is not included in that of
QI problems. We refer to Figure 1 for details. Finally, we propose a tractable
test for the US property.
