$\mu$-Stability of Nonlinear Positive Systems With Unbounded Time-Varying Delays

TL;DR

This paper investigates the stability of nonlinear positive systems with unbounded time-varying delays, introducing a novel transformation to analyze their dynamics and establish criteria for global -stability.

Contribution

A new transformation method is proposed to simplify the stability analysis of nonlinear positive systems with delays, leading to novel -stability criteria.

Findings

Derived criteria for global -stability of the systems.

The transformation reduces complex nonlinear functions to simpler forms.

Numerical example validates the theoretical results.

Abstract

Stability of the zero solution plays an important role in the investigation of positive systems. In this note, we revisit the $\mu$-stability of positive nonlinear systems with unbounded time-varying delays. The system is modelled by continuous-time differential equations. Under some assumptions on the nonlinear functions like homogeneous, cooperative, nondecreasing, we propose a novel transform, by which the nonlinear system reduces to a new system. Thus, we analyze its dynamics, which can simplify the nonlinear homogenous functions with respect to (w.r.t.) arbitrary dilation map to those w.r.t. the standard dilation map. We finally get some criteria for the global $\mu$-stability. A numerical example is given to demonstrate the validity of obtained results.