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In this note we study contractivity of monotone systems and exponentialconvergence of positive systems using non-Euclidean norms. We first introducethe notion of conic matrix measure as a framework to study stability ofmonotone and positive systems. We study properties of the conic matrix measuresand investigate their connection with weak pairings and standard matrixmeasures. Using conic matrix measures and weak pairings, we characterizecontractivity and incremental stability of monotone systems with respect tonon-Euclidean norms. Moreover, we use conic matrix measures to providesufficient conditions for exponential convergence of positive systems to theirequilibria. We show that our framework leads to novel results on (i) thecontractivity of excitatory Hopfield neural networks, and (ii) the stability ofinterconnected systems using non-monotone positive comparison systems.

Non-Euclidean Contraction Theory for Monotone and Positive Systems