The controllability and stabilization of unstable LTI systems with input saturation

We study the controllable region of a general unstable continuous-time LTI system with input saturation and we also study the stabilization of such a system by a saturated linear state feedback. We give simple exact descriptions of the controllable regions for certain classes of unstable systems. The study on stabilization is quite preliminary. We only deal with anti-stable planar systems. We conjecture that for such a system its asymptotic stability region (domain of attraction) under a saturated linear state feedback can be easily obtained from a stable limit cycle of its time-reversed system. We conjecture that for such a system a saturated linear state feedback can be designed so that the asymptotic stability region is arbitrarily close to its controllable region