Stability control and recurrent switching rules

The recent paper [1] enlightens the interest of a class of switching rules with nice properties, called eventually periodic: 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 Γ1, . . . ,ΓH of R d . We shown that under the same assumptions as in [1], if a stabilizing eventually periodic switching rule for F is known, then Γ1, . . . ,ΓH can be constructed in such a way that F is stabilized by any recurrent switching rule subject to Γ1, . . . ,ΓH .