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=θ1A1+θ2A2H=θ1A1+θ2A2 and those of the matrix E=eθ2A2eθ1A1E=eθ2A2eθ1A1. It is well known that if all the eigenvalues of H   have negative real part and θ1+θ2θ1+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+θ2θ1+θ2, then there exist some τ1,τ2τ1,τ2 (with, in general, τ1≠=θ1,τ2≠=θ2τ1≠=θ1,τ2≠=θ2) such that the eigenvalues of the matrix τ1A1+τ2A2τ1A1+τ2A2 have negative real part.