Proving conditions for controllability Let's say I have the following LTI system: 
$$\dot{x}(t) = \mathbf{A} x(t) + \mathbf{B} u(t)$$
I need to somehow show the following is true or false (proof): 

This system is controllable if and only if for any $\mu$ such that $(−\mu\mathbf{I − A})$ is Hurwitz, there exists a unique positive deﬁnite solution $\mathbf W$ to the Lyapunov equation:

$$\mathbf A \mathbf W + \mathbf W \mathbf A^T − \mathbf B\mathbf B^T = −2\mu\mathbf W$$

Can anyone see the proof? I tried replacing $\mathbf A$ in the Lyapunov equation with $(−\mu\mathbf I − \mathbf A)$, and then adding the $−2\mu\mathbf W$, to the left side, but the $\mu$'s end up cancelling out, so I'm not sure. 
I'd really appreciate any help.
Not really a direct help, but this paper might provide some insight, particularly the paragraph below.