  This paper revisits the classical Linear Quadratic Gaussian (LQG) control
from a modern optimization perspective. We analyze two aspects of the
optimization landscape of the LQG problem: 1) connectivity of the set of
stabilizing controllers $\mathcal{C}_n$; and 2) structure of stationary points.
It is known that similarity transformations do not change the input-output
behavior of a dynamical controller or LQG cost. This inherent symmetry by
similarity transformations makes the landscape of LQG very rich. We show that
1) the set of stabilizing controllers $\mathcal{C}_n$ has at most two
path-connected components and they are diffeomorphic under a mapping defined by
a similarity transformation; 2) there might exist many \emph{strictly
suboptimal stationary points} of the LQG cost function over $\mathcal{C}_n$ and
these stationary points are always \emph{non-minimal}; 3) all \emph{minimal}
stationary points are globally optimal and they are identical up to a
similarity transformation. These results shed some light on the performance
analysis of direct policy gradient methods for solving the LQG problem.
