  In this paper, we analyze the optimization landscape of gradient descent
methods for static output feedback (SOF) control of discrete-time linear
time-invariant systems with quadratic cost. The SOF setting can be quite
common, for example, when there are unmodeled hidden states in the underlying
process. We first establish several important properties of the SOF cost
function, including coercivity, L-smoothness, and M-Lipschitz continuous
Hessian. We then utilize these properties to show that the gradient descent is
able to converge to a stationary point at a dimension-free rate. Furthermore,
we prove that under some mild conditions, gradient descent converges linearly
to a local minimum if the starting point is close to one. These results not
only characterize the performance of gradient descent in optimizing the SOF
problem, but also shed light on the efficiency of general policy gradient
methods in reinforcement learning.
