Weak and Semi-contraction theory with application to network systems

作     者：Jafarpour, Saber Velarde, Cisneros Bullo, Francesco 

作者机构：Center of Control Dynamical Systems and Computation UC Santa Barbara CA93106-5070 United States 

出 版 物：《arXiv》 (arXiv)

年 卷 期：2020年

核心收录：

主　　题：Trajectories 

摘      要：We develop two generalizations of contraction theory: semi-contraction and weak-contraction theory. First, using the notion of semi-norm, we propose a geometric framework for semi-contraction theory. We introduce matrix semi-measures, characterize their properties, and provide techniques to compute them. We also propose two optimization problems for the spectral abscissa of a given matrix using its semi-measures. For dynamical systems, we use the semi-measure of their Jacobian to study convergence of their trajectories to invariant subspaces. Second, we provide a comprehensive treatment of weakly-contracting systems;we prove a dichotomy for asymptotic behavior of their trajectories and show that, for their equilibrium points, local asymptotic stability implies global asymptotic stability. Third, we introduce the class of doubly-contracting systems and show that every trajectory of a doubly-contracting system converges to an equilibrium point. Finally, we apply our results to various important network systems including affine averaging and affine flow systems, continuous-time distributed primal-dual algorithms, and networks of diffusively-coupled oscillators. Copyright © 2020, The Authors. All rights reserved.