A complete characterization of exponential stability for discrete dynamics

TL;DR

This paper provides a comprehensive characterization of exponential stability for discrete linear dynamics using operator invertibility and admissible exponents, highlighting differences between uniform and nonuniform stability behaviors.

Contribution

It introduces a complete characterization of exponential stability for discrete dynamics via operator invertibility and admissible exponents, extending previous spectral approaches.

Findings

Exponential stability is characterized by invertibility of a specific operator.

Spectral properties determine stability for bounded orbits.

Differences between uniform and nonuniform behavior are demonstrated through examples.

Abstract

For a discrete dynamics defined by a sequence of bounded and not necessarily invertible linear operators, we give a complete characterization of exponential stability in terms of invertibility of a certain operator acting on suitable Banach sequence spaces. We connect the invertibility of this operator to the existence of a particular type of admissible exponents. For the bounded orbits, exponential stability results from a spectral property. Some adequate examples are presented to emphasize some significant qualitative differences between uniform and nonuniform behavior.