Global and localweighted homogeneity for time-delay systems

Abstract

The notion of weighted homogeneity is extended to time-delay systems. It is shown that the stability (resp., instability) of homogeneous functional systems on a sphere implies the global stability (resp., instability) of the system. The notion of local homogeneity is introduced, and a relation between stability and instability of the locally approximating dynamics and the original time-delay system is established using a Lyapunov-Razumikhin approach. An implication between homogeneity and input-to-state stability is investigated. Examples of applications of the proposed theory are given.