On the stability of pairwise triangularisable and related switching systems

It was conjectured in Shorten and Cairbre (2000) that, a sufficient condition for the asymptotic stability of a given switching system is that the component systems are asymptotically stable, and a set of non-singular matrices exists such that the matrices are upper triangular (pairwise triangular). In this paper we show that while the general condition of pairwise triangularisability is not sufficient to guarantee asymptotic stability, the ideas developed in the above paper can be extended to prove asymptotic stability of the origin for a related class of switching system