# Absolute Stability of Discrete-Time Systems with Delay

## Abstract

We investigate the stability of nonlinear nonautonomous discrete-time systems with delaying arguments, whose linear part has slowly varying coefficients, and the nonlinear part has linear majorants. Based on the "freezing" technique to discrete-time systems, we derive explicit conditions for the absolute stability of the zero solution of such systems.

## Keywords

Lyapunov Function Homogeneous Equation Absolute Stability Linear Delay Cauchy Function## 1. Introduction

Over the past few decades, discrete-time systems with delay have drawn much attention from the researchers. This is due to their important role in many practical systems. The stability of time-delay systems is a fundamental problem because of its importance in the analysis of such systems.The basic method for stability analysis is the direct Lyapunov method, for example, see [1, 2, 3], and by this method, strong results have been obtained. But finding Lyapunov functions for nonautonomous delay difference systems is usually a difficult task. In contrast, many methods different from Lyapunov functions have been successfully applied to establish stability results for difference equations with delay, for example, see [3, 4, 5, 6, 7, 8, 9, 10, 11, 12].

This paper deals with the absolute stability of nonlinear nonautonomous discrete-time systems with delay, whose linear part has slowly varying coefficients, and the nonlinear part satisfies a Lipschitz condition.

The aim of this paper is to generalize the approach developed in [7] for linear nonautonomous delay difference systems to the nonlinear case with delaying arguments. Our approach is based on the "freezing" technique for discrete-time systems. This method has been used to investigate properties as well as to the construction of solutions for systems of linear differential equations. So, it is commonly used in analysing the stability of slowly varying initial-value problems as well as solving them, for example, see [13, 14]. However, its use to difference equations is rather new [7]. The stability conditions will be formulated assuming that we know the Cauchy solution (fundamental solution) of the unperturbed system.

The paper is organized as follows. After some preliminaries in Section 2, the sufficient conditions for the absolute stability are presented in Section 3. In Section 4, we reduce a delay difference system to a delay-free linear system of higher dimension, thus obtaining explicit stability conditions for the solutions.

## 2. Preliminaries

where Open image in new window is an integer Open image in new window and Open image in new window

where Open image in new window is a given vector-valued function, that is, Open image in new window

where Open image in new window ; Open image in new window ; Open image in new window , Open image in new window

Definition 2.1.

*absolutely stable*in the class of nonlinearities (2.6) if there is a positive constant Open image in new window , independent of Open image in new window (but dependent on Open image in new window ), such that

for any solution Open image in new window of (2.2) with the initial conditions (2.3).

It is clear that every solution Open image in new window of the initial-valued problem (2.2)-(2.3) exists, is unique and can be constructed recursively from (2.2).

In order to state and prove our main results, we need some suitable lemmas and theorems.

Lemma 2.2 (see [7]).

Lemma 2.3 (see [7]).

In [7], was established the following stability result in terms of the Cauchy solution Open image in new window of (2.9).

Theorem 2.4 (see [7]).

holds with constant Open image in new window , and Open image in new window independent of Open image in new window . If in addition, conditions (2.4), (2.5), and Open image in new window are fulfilled, then (2.16) is stable.

Our purpose is to generalize this result to the nonlinear problem (2.2)-(2.3).

Lemma 2.5 (see [9]).

## 3. Main Results

Theorem 3.1.

is valid, where Open image in new window , and Open image in new window

Proof.

This yields the required result.

Corollary 3.2.

where Open image in new window , and Open image in new window

Proof.

Now, Corollary 3.2 yields the following result.

Theorem 3.3.

hold. Then, the zero solution of (2.2)-(2.3) is absolutely stable in the class of nonlinearities in (2.6).

Proof.

where Open image in new window is a solution of (2.2) and Open image in new window

This fact proves the required result.

Remark 3.4.

Theorem 3.3 is exact in the sense that if (2.2) is a homogeneous linear stable equation with constant matrices Open image in new window , then Open image in new window , and condition (3.29) is always fulfilled.

It is somewhat inconvenient that to apply either condition (3.26) or (3.29), one has to assume explicit knowledge of the constants Open image in new window and Open image in new window . In the next theorem, we will derive sufficient conditions for the exponential growth of the Cauchy function associated to (2.9). Thus, our conditions may provide a useful tool for applications.

Theorem 3.5 (see [7]).

Now, we will consider the homogeneous equation (2.16), thus establishing the following consequence of Theorem 3.3.

Corollary 3.6.

hold. Then the zero solution of (2.16)–(2.3) is absolutely stable.

Example 3.7.

and Open image in new window . And Open image in new window , Open image in new window , are positive bounded sequences withthe following properties: Open image in new window and Open image in new window and Open image in new window ; Open image in new window , are nonnegative constants for Open image in new window . This yields that Open image in new window and Open image in new window , respectively, for Open image in new window . Thus Open image in new window .

Hence, Open image in new window

for a fixed Open image in new window tends to zero exponentially as Open image in new window that is, there exist constants Open image in new window and Open image in new window such that Open image in new window ; Open image in new window

If Open image in new window , then by Theorem 3.3, it follows that the zero solution of (3.39) is absolutely stable.

then it is not hard to check that the Cauchy solution of this system tends to zero exponentially as Open image in new window Hence, by Theorem 3.3, it follows that the zero solution of (3.39) is absolutely stable provided that the relation (3.29) is satisfied.

## 4. Linear Delay Systems

where Open image in new window Open image in new window and Open image in new window are variable Open image in new window -matrices.

In [4], were established very nice solution representation formulae to the system

assuming that Open image in new window and Open image in new window However, the stability problem was not investigated in this paper.

where Open image in new window are Open image in new window -matrices, Open image in new window Open image in new window By means of a characteristic equation, they established many results concerning the stability of the solutions of such equation. However, the case of variable coefficients is not studied in this article.

In the next corollary, we will apply Theorem 3.3 to this particular case of (2.2), thus obtaining the following corollary.

Corollary 4.1.

Under condition (3.25), one assumes that

(i)the matrices Open image in new window and Open image in new window satisfy Open image in new window and Open image in new window , respectively, for Open image in new window

Then, the zero solution of (4.1)-(2.3) is absolutely stable.

Remark 4.2.

I want to point out that this approach is just of interest for systems with "slowly changing" matrices.

where Open image in new window is the unit matrix in Open image in new window

holds for every nonnegative integer Open image in new window , where Open image in new window is the spectral radius of Open image in new window .

Theorem 4.3 (see [7]).

Assume that

where Open image in new window , with Open image in new window defined in (2.14).

where Open image in new window and Open image in new window .

Corollary 4.4.

where Open image in new window

Proof.

Hence, Open image in new window .

Remark 4.5.

This approach is usually not applicable to the time-varying delay case, because the transformed systems usually have time-varying matrix coefficients, which are difficult to analyze using available tools. Hence, our results will provide new tools to analyze these kind of systems.

## Notes

### Acknowledgments

The author thanks the referees of this paper for their careful reading and insightful critiques. This research was supported by Fondecyt Chile under Grant no. 1.070.980.

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