  This paper discusses the stabilizability, weak stabilizability, exact
observability and robust quadratic stabilizability of linear stochastic control
systems. By means of the spectrum technique of the generalized Lyapunov
operator, a necessary and sufficient condition is given for stabilizability and
weak stabilizability of stochastic systems, respectively. Some new concepts
called unremovable spectrums, strong solutions, and weakly feedback stabilizing
solutions are introduced. An unremovable spectrum theorem is given, which
generalizes the corresponding theorem of deterministic systems to stochastic
systems. A stochastic Popov-Belevith-Hautus (PBH) criterion for exact
observability is obtained. For applications, we give a comparison theorem for
generalized algebraic Riccati equations (GAREs), and two results on
Lyapunov-type equations are obtained, which improve the previous works.
Finally, we also discuss robust quadratic stabilization of uncertain stochastic
systems, and a necessary and sufficient condition is given for quadratic
stabilization via a linear matrix inequality (LMI).
