Non-Euclidean Contraction Theory for Monotone and Positive Systems

TL;DR

This paper introduces a non-Euclidean norm framework using conic matrix measures to analyze stability, contractivity, and exponential convergence in monotone and positive systems, including neural networks.

Contribution

It develops a novel conic matrix measure approach for stability analysis of monotone and positive systems with non-Euclidean norms, extending existing theories.

Findings

Conic matrix measures characterize contractivity of monotone systems.

Provided conditions for exponential convergence of positive systems.

Applied framework to neural networks and interconnected systems.

Abstract

In this note we study contractivity of monotone systems and exponential convergence of positive systems using non-Euclidean norms. We first introduce the notion of conic matrix measure as a framework to study stability of monotone and positive systems. We study properties of the conic matrix measures and investigate their connection with weak pairings and standard matrix measures. Using conic matrix measures and weak pairings, we characterize contractivity and incremental stability of monotone systems with respect to non-Euclidean norms. Moreover, we use conic matrix measures to provide sufficient conditions for exponential convergence of positive systems to their equilibria. We show that our framework leads to novel results on (i) the contractivity of excitatory Hopfield neural networks, and (ii) the stability of interconnected systems using non-monotone positive comparison systems.