Recurrent switching rules for pairs of linear planar vector fields

Given a switched system formed by a pair of (not asymptotically stable) linear vector fields of $\R^2$, we define a special class of hybrid feedback laws (called recurrent switching rules) whose structure depends on some conic regions with nonempty interior. We prove that if the system is asymptotically controllable and radially controllable, then it is robustly stabilized by any recurrent switching rule, provided that the conic regions are properly designed.