Internally positive representations and stability analysis of linear differential systems with multiple time-varying delays

Abstract

This work introduces the internally positive representation of linear time-varying delay differential systems, in the general case of multiple time-varying delays. The technique, previously established for the delay-free case and recently extended to various classes of linear delay systems, aims at building a positive representation of systems whose dynamics is, in general, not definite in sign, in order to export results that only hold for positive systems to arbitrary ones. In the special case of constant matrices, this leads to a simple and easy to check condition for the delay-independent stability of differential systems with multiple time-varying delays. The condition is shown to be less conservative than some well-known conditions available in the literature. Numerical examples are proposed to validate the theoretical results.