Periodic open-loop stabilization of planar switched systems

In the context of the theory of switched systems, and especially of the openloop stabilization problem, it is interesting to study the relationship between the placement of the eigenvalues of a matrix of the form H = θ1A1 + θ2A2 and those of the matrix E = e22e11 . It is well known that if all the eigenvalues of H have negative real part and θ1 + θ2 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 θ1 + θ2, then there exist some τ1, τ2 (with, in general, τ1 6= θ1, τ2 6= θ2) such that the eigenvalues of the matrix τ1A1 + τ2A2 have negative real part.