Matrix measures, stability and contraction theory for dynamical systems on time scales

TL;DR

This paper develops a framework for analyzing the stability of dynamical systems on time scales using matrix measures, providing new bounds, stability conditions, and applications to epidemics and networks.

Contribution

It introduces the notion of matrix measures on time scales and applies this to derive stability criteria for linear and nonlinear systems, including epidemic and network models.

Findings

Established a time scale version of Coppel's upper bound.

Derived stability and input-to-state stability conditions for linear systems.

Provided convergence conditions for nonlinear systems and applications to epidemic and network dynamics.

Abstract

This paper is concerned with the study of the stability of dynamical systems evolving on time scales. We first {formalize the notion of matrix measures on time scales, prove some of their key properties and make use of this notion to study both linear and nonlinear dynamical systems on time scales.} Specifically, we start with considering linear time-varying systems and, for these, we prove a time scale analogous of an upper bound due to Coppel. We make use of this upper bound to give stability and input-to-state stability conditions for linear time-varying systems. {Then, we consider nonlinear time-varying dynamical systems on time scales and} establish a sufficient condition for the convergence of the solutions. Finally, after linking our results to the existence of a Lyapunov function, we make use of our approach to study certain epidemic dynamics and complex networks. For the former, we give a sufficient condition on the parameters of a SIQR model on time scales ensuring that its solutions converge to the disease-free solution. For the latter, we first give a sufficient condition for pinning controllability of complex time scale networks and then use this condition to study certain collective opinion dynamics. The theoretical results are complemented with simulations.