Stability control and recurrent switching rules

Recently, it has been enlightens the interest of a class of switching rules with nice properties, called eventually periodic: more precisely, it is proven that a finite family F of linear vector fields of R^d can be stabilized by means of eventually periodic switching rules, provided that it is asymptotically controllable and satisfies an additional finite time controllability condition. Unfortunately, simple examples point out that in general, eventually periodic switching rules are not robust with respect to state measurement errors. In this paper, we introduce a new type of switching rules with improved robustness properties, called recurrent switching rules. They are subject to the construction of a finite sequence of complete cones Gamma_1,...,Gamma_H of R^d. We shown that under natural assumptions, if a stabilizing eventually periodic switching rule for F is known, then Gamma_1,...,Gamma_H can be constructed in such a way that F is stabilized by any recurrent switching rule subject to Gamma_1,...,Gamma_H.