Hybrid stabilization of planar linear systems with one-dimensional outputs

We consider a linear control system x'=Ax+Bu with output y=Cx, where x is two-dimensional, u,y are one-dimensional, and give necessary and sufficient conditions in order that it can be stabilized by a hybrid, linear feedback, where the action of the "switch" just depends on the sign of y. We also show, on these conditions, that the use of two control functions is enough for getting the goal.