Upper Bounds for the Distance between a Controllable Switched Linear System and the Set of Uncontrollable Ones

Abstract

The set of controllable switched linear systems is an open and dense set in the space of all switched linear systems. Therefore it makes sense to compute the distance from a controllable system to the nearest uncontrollable one. In the case of a standard system,x˙t=Axt+But, R. Eising, D. Boley, and W. S. Lu obtain some results for this distance, both in the complex and real cases. In this work we explore this distance, for switched linear systems in the real case, obtaining upper bounds for it. The main contribution of the paper is to prove that a natural generalization of the upper bound obtained by D. Boley and W. S. Lu is true in the case of switched linear systems.