Equivalence of several characteristics of dual switched linear systems

This paper is concerned with the equivalence of several dynamic characteristics between mutually dual switched linear systems, both discrete- and continuous-time cases are considered. Two systems are mutually dual means that their system matrices are transpose of each other. The dynamic properties considered in this paper include four types: exponential stability, global uniform asymptotic stability, attractivity, and weak attractivity. It is shown that every one of the four dynamic properties of a switched linear system implies each of the four dynamic properties of its dual system, and vice versa. The main results enable us to investigate these dynamic properties of switched system by virtue of study related to the corresponding properties of its dual system, thus providing an alternative way to explore the dynamics of switched systems. A numerical example is provided to illustrate the obtained theoretical conclusions.