  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.
