Convergence Properties for Discrete-time Nonlinear Systems

TL;DR

This paper compares three convergence concepts in discrete-time nonlinear systems, analyzing their differences, relationships, and Lyapunov function characterizations to deepen understanding of system stability and convergence behaviors.

Contribution

It provides a comparative analysis of convergence notions, introduces Lyapunov characterizations, and offers conditions linking these properties for discrete-time nonlinear systems.

Findings

Differences between convergence notions are clarified through examples.

Lyapunov functions characterize each convergence property.

Sufficient conditions relate convergence properties via Lyapunov functions.

Abstract

Three similar convergence notions are considered. Two of them are the long established notions of convergent dynamics and incremental stability. The other is the more recent notion of contraction analysis. All three convergence notions require that all solutions of a system converge to each other. In this paper, we investigate the differences between these convergence properties for discrete-time, time-varying nonlinear systems by comparing the properties in pairs and using examples. We also demonstrate a time-varying smooth Lyapunov function characterization for each of these convergence notions, and, with appropriate assumptions, we provide several sufficient conditions to establish relationships between these properties in terms of Lyapunov functions.