Stability Analysis of Monotone Systems via Max-separable Lyapunov Functions

TL;DR

This paper demonstrates that for monotone nonlinear systems, various stability properties including D-stability and delay insensitivity are equivalent to the existence of max-separable Lyapunov functions, providing new insights and conditions for delay-independent stability.

Contribution

It establishes the equivalence between stability, Lyapunov functions, and delay insensitivity for monotone nonlinear systems, extending existing results.

Findings

Equivalence between asymptotic stability and max-separable Lyapunov functions.

Introduction of a new notion of D-stability for nonlinear systems.

Necessary and sufficient conditions for delay-independent stability.

Abstract

We analyze stability properties of monotone nonlinear systems via max-separable Lyapunov functions, motivated by the following observations: first, recent results have shown that asymptotic stability of a monotone nonlinear system implies the existence of a max-separable Lyapunov function on a compact set; second, for monotone linear systems, asymptotic stability implies the stronger properties of D-stability and insensitivity to time-delays. This paper establishes that for monotone nonlinear systems, equivalence holds between asymptotic stability, the existence of a max-separable Lyapunov function, D-stability, and insensitivity to bounded and unbounded time-varying delays. In particular, a new and general notion of D-stability for monotone nonlinear systems is discussed and a set of necessary and sufficient conditions for delay-independent stability are derived. Examples show how the results extend the state-of-the-art.