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 ℓ1\ell _{1} and ℓ∞\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.

中文翻译：

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鲁棒非线性稳定性的非欧收缩理论


在本文中，我们研究了动力系统收缩和增量稳定性相对于非欧几里得规范的充分必要条件。首先，我们引入弱配对作为研究任意规范的收缩性的框架，并表征它们的属性。我们分别介绍并研究 ℓ1\ell _{1} 和 ℓ∞\ell _\infty 范数的符号和最大配对。使用弱配对，我们建立了五个等效的收缩表征，包括矢量场的单侧 Lipschitz 条件以及相应雅可比行列式的对数范数和 Demidovich 条件。第三，我们在两个方向上扩展了我们的收缩框架：我们证明了连续向量场收缩的等价性，并且我们形式化了平衡收缩的较弱概念，这确保了指数收敛到平衡。最后，作为一个应用，我们为契约系统提供增量输入状态稳定性和有限输入状态增益属性，以及关于契约系统 Lipschitz 互连的一般定理，其中增益矩阵的 Hurwitzness 意味着统。