  We consider the Linear-Quadratic-Regulator (LQR) problem in terms of
optimizing a real-valued matrix function over the set of feedback gains. Such a
setup facilitates examining the implications of a natural initial-state
independent formulation of LQR in designing first order algorithms. It is shown
that this cost function is smooth and coercive, and provide an alternate means
of noting its gradient dominated property. In the process, we provide a number
of analytic observations on the LQR cost when directly analyzed in terms of the
feedback gain. We then examine three types of well-posed flows for LQR:
gradient flow, natural gradient flow and the quasi-Newton flow. The coercive
property suggests that these flows admit unique solutions while gradient
dominated property indicates that the corresponding Lyapunov functionals decay
at an exponential rate; we also prove that these flows are exponentially stable
in the sense of Lyapunov. We then discuss the forward Euler discretization of
these flows, realized as gradient descent, natural gradient descent and the
quasi-Newton iteration. We present stepsize criteria for gradient descent and
natural gradient descent, guaranteeing that both algorithms converge linearly
to the global optima. An optimal stepsize for the quasi-Newton iteration is
also proposed, guaranteeing a $Q$-quadratic convergence rate--and in the
meantime--recovering the Hewer algorithm.
