Periodic open-loop stabilization of planar switched systems

In the context of the theory of switched systems, and especially of the open-loop stabilization problem, it is interesting to study the relationship between the placement of the eigenvalues of a matrix of the form H=t1A1+t2A2 and those of the matrix E= expt2A2expt1A1. It is wellknown that if all the eigenvalues of H have negative real part and t1+t2 is small enough, then the eigenvalues of E lie in the unit disc of the complex plane. In this paper we prove that in the two dimensional case a partial converse holds: if the eigenvalues of E lie in the unit disc of the complex plane for sufficiently small values of t1+t2, then there exist some s1, s2 (with, ingeneral, s1 not equal to t1, s2 not equal to t2) such that the eigenvalues of the matrix s1A1+s2A2 have negative real part