New Stability Conditions for Linear Difference Equations using Bohl-Perron Type Theorems

TL;DR

This paper develops new stability criteria for linear difference equations with multiple delays, extending Bohl-Perron theorems, and provides practical tests for exponential stability based on comparison equations and fundamental functions.

Contribution

It introduces novel exponential stability conditions for difference equations with variable delays, utilizing Bohl-Perron type theorems and solution representations.

Findings

New stability tests for equations with multiple delays

Conditions based on comparison equations and fundamental functions

Enhanced criteria for exponential stability in variable delay systems

Abstract

The Bohl-Perron result on exponential dichotomy for a linear difference equation $$ x(n+1)-x(n) + \sum_{l=1}^m a_l(n)x(h_l(n))=0, h_l(n)\leq n, $$ states (under some natural conditions) that if all solutions of the non-homogeneous equation with a bounded right hand side are bounded, then the relevant homogeneous equation is exponentially stable. According to its corollary, if a given equation is {\em close} to an exponentially stable comparison equation (the norm of some operator is less than one), then the considered equation is exponentially stable. For a difference equation with several variable delays and coefficients we obtain new exponential stability tests using the above results, representation of solutions and comparison equations with a positive fundamental function.