Dual Seminorms, Ergodic Coefficients and Semicontraction Theory

TL;DR

This paper develops a unified theory of semicontraction in dynamical systems using seminorms, linking ergodic coefficients to stability and contraction rates in both linear and nonlinear systems.

Contribution

It introduces dual notions of seminorms, relates ergodic coefficients to matrix seminorms, and provides comprehensive criteria for strong semicontractivity in various dynamical systems.

Findings

Ergodic coefficients are induced matrix seminorms.

Markov-Dobrushin coefficient explains contraction rates.

Theorems for semicontractivity in linear and nonlinear systems.

Abstract

Dynamical systems that are contracting on a subspace are said to be semicontracting. Semicontraction theory is a useful tool in the study of consensus algorithms and dynamical flow systems such as Markov chains. To develop a comprehensive theory of semicontracting systems, we investigate seminorms on vector spaces and define two canonical notions: projection and distance semi-norms. We show that the well-known lp ergodic coefficients are induced matrix seminorms and play a central role in stability problems. In particular, we formulate a duality theorem that explains why the Markov-Dobrushin coefficient is the rate of contraction for both averaging and conservation flows in discrete time. Moreover, we obtain parallel results for induced matrix log seminorms. Finally, we propose comprehensive theorems for strong semicontractivity of linear and non-linear time-varying dynamical systems with invariance and conservation properties both in discrete and continuous time.