- Research Article
- Open Access

# Absolute Stability of Discrete-Time Systems with Delay

- Rigoberto Medina
^{1}Email author

**2008**:396504

https://doi.org/10.1155/2008/396504

© Rigoberto Medina. 2008

**Received:**18 October 2007**Accepted:**22 November 2007**Published:**30 December 2007

## 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 [1–3], 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 [3–12].

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 is an integer and

where is a given vector-valued function, that is,

where ; ; ,

Definition 2.1.

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

for any solution of (2.2) with the initial conditions (2.3).

It is clear that every solution 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 of (2.9).

Theorem 2.4 (see [7]).

holds with constant , and independent of . If in addition, conditions (2.4), (2.5), and 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 , and

Proof.

This yields the required result.

Corollary 3.2.

where , and

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 is a solution of (2.2) and

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 , then , 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 and . 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 . And , , are positive bounded sequences withthe following properties: and and ; , are nonnegative constants for . This yields that and , respectively, for . Thus .

Hence,

for a fixed tends to zero exponentially as that is, there exist constants and such that ;

If , 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 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 and are variable -matrices.

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

assuming that and However, the stability problem was not investigated in this paper.

where are -matrices, 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 and satisfy and , respectively, for

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 is the unit matrix in

holds for every nonnegative integer , where is the spectral radius of .

Theorem 4.3 (see [7]).

Assume that

- (i)
; and

- (ii)
= ,

where , with defined in (2.14).

where and .

Corollary 4.4.

where

Proof.

Hence, .

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.

## Declarations

### 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.

## Authors’ Affiliations

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