On the global attractivity of a class of switching systems

We investigate the stability properties of a class of switching systems of the form x˙=A_ix, A_i∈IR^n×n, A_i∈𝒜Δ=A{A₁, ..., A_m}. We consider sets of matrices 𝒜, where no single matrix T exists that simultaneously transforms each A_i∈ 𝒜 to upper triangular form, but where a set of nonsingular matrices T_ij exist such that the matrices T_ijA_iT_ij ^-1, T_ijA_jT_ij^-1, are upper triangular. We show that, for a special class of such systems, the origin of the switching system is globally attractive