On Exponential Stability for Linear Difference Equations with Delays

Abstract

The article gives an overview of recent results on the stability of finite-difference equations with delay.

All results are compared with known signs of exponential stability of linear difference equations.

The results are obtained using the Bohl-Perron theorem and comparing the equation under study with an equation for which the Cauchy function is positive.

The Bohl-Perron theorem allows us to reduce the question of the exponential stability of a linear difference equation with delay to the solvability of an operator equation in one of the functional infinite-dimensional spaces.

That is, in fact, to an estimate of the norm or the spectral radius of a bounded linear operator in this space. For this estimation, different difference inequalities are used. One way to obtain such inequalities is to evaluate the fundamental solution in the event that this solution is positive.