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In this article, we study necessary and sufficient conditions for contraction and incremental stability of dynamical systems with respect to non-Euclidean norms. First, we introduce weak pairings as a framework to study contractivity with respect to arbitrary norms, and characterize their properties. We introduce and study the sign and max pairings for the   $\ell _{1}$   and   $\ell _\infty$   norms, respectively. Using weak pairings, we establish five equivalent characterizations for contraction, including the one-sided Lipschitz condition for the vector field as well as logarithmic norm and Demidovich conditions for the corresponding Jacobian. Third, we extend our contraction framework in two directions: we prove equivalences for contraction of continuous vector fields, and we formalize the weaker notion of equilibrium contraction, which ensures exponential convergence to an equilibrium. Finally, as an application, we provide incremental input-to-state stability and finite input-state gain properties for contracting systems, and a general theorem about the Lipschitz interconnection of contracting systems, whereby the Hurwitzness of a gain matrix implies the contractivity of the interconnected system.

Non-Euclidean Contraction Theory for Robust Nonlinear Stability