Non-Euclidean Contraction Theory for Monotone and Positive Systems

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 the contractivity of excitatory Hopfield neural networks and the stability of interconnected systems using nonmonotone positive comparison systems.

Automated summary

In this paper, we investigate the contractivity of monotone systems and the exponential convergence of positive systems using non-Euclidean norms. We introduce the concept of conic matrix measure as a framework to study stability of monotone and positive systems, and study its properties and its connection with weak pairings and standard matrix measures. By using conic matrix measures and weak pairings, we characterize the contractivity and incremental stability of monotone systems with respect to non-Euclidean norms. Moreover, we provide sufficient conditions for the exponential convergence of positive systems to their equilibria using conic matrix measures. We present novel results on the contractivity of excitatory Hopfield neural networks and the stability of interconnected systems using nonmonotone positive comparison systems.