Non-Euclidean Contraction Theory for Robust Nonlinear Stability

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