Weak and Semi-Contraction for Network Systems and Diffusively-Coupled Oscillators

TL;DR

This paper introduces semi-contraction and weak-contraction theories to analyze network systems, providing new conditions for convergence and synchronization, with applications to diffusively-coupled oscillators and distributed algorithms.

Contribution

It develops a geometric semi-contraction framework using semi-norms, characterizes spectral properties via semi-measures, and applies these to improve synchronization criteria in network systems.

Findings

Weakly contracting systems exhibit a dichotomy in asymptotic behavior.

Doubly-contracting systems' trajectories always converge to an equilibrium.

Semi-contraction theory yields sharper synchronization conditions for diffusively-coupled systems.

Abstract

We develop two generalizations of contraction theory, namely, 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 and characterize their properties. We show that the spectral abscissa of a matrix is the infimum over weighted semi-measures. For dynamical systems, we use the semi-measure of their Jacobian to characterize the contractivity properties of their trajectories. Second, for weakly contracting systems, we prove a dichotomy for the asymptotic behavior of their trajectories and novel sufficient conditions for convergence to an equilibrium. Third, we show that every trajectory of a doubly-contracting system, i.e., a system that is both weakly and semi-contracting, 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 dynamical systems. For diffusively-coupled systems, the semi-contraction theory leads to a sufficient condition for synchronization that is sharper, in general, than previously-known tests.