Advances in Difference Equations

, 2008:396504 | Cite as

Absolute Stability of Discrete-Time Systems with Delay

Open Access
Research Article

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

Let Open image in new window denote the set of nonnegative integers. Given a positive integer Open image in new window , denote by Open image in new window and Open image in new window the Open image in new window -dimensional space of complex column vectors and the set of Open image in new window matrices with complex entries, respectively. If Open image in new window is any norm on Open image in new window , the associated induced norm of a matrix Open image in new window is defined by
Consider the nonlinear discrete-time system with multiple delays of the form

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

We will consider (2.2) subject to the initial conditions

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

Throughout the paper, we will assume that the variable matrices Open image in new window have the properties

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

Definition 2.1.

The zero solution of (2.2) is 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).

The stability conditions for (2.2) will be formulated in terms of the Cauchy function Open image in new window (the fundamental solution) of
defined as follows. For a fixed Open image in new window let Open image in new window be the solution of (2.9) with initial conditions
Since the coefficients of (2.9) are constants for fixed Open image in new window , then the Cauchy function of (2.9) has the form
where Open image in new window is the solution of (2.9) with the initial conditions

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

Lemma 2.2 (see [7]).

where Open image in new window Open image in new window is a given function, subject to the initial conditions
has the form
where Open image in new window is the Cauchy function of (2.9) and Open image in new window is the solution of the homogeneous equation
with the same initial conditions:

Lemma 2.3 (see [7]).

The solution Open image in new window of (2.16) with initial conditions (2.14) has the form

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]).

Let the inequality

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]).

Let Open image in new window be a sequence of positive numbers such that
where Open image in new window is a constant. Then there exist constants Open image in new window and Open image in new window such that

3. Main Results

Now, we establish the main results of the paper, which will be valid for a family Open image in new window Open image in new window of slowly varying matrices. Let Open image in new window and Open image in new window With the notation
assume that
Consider the equation

Theorem 3.1.

Under conditions (2.4) and (2.5), let the inequality
holds. Then for any solution Open image in new window of problem (2.13)–(2.3), the estimate

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

Proof.

Fix Open image in new window and rewrite (3.3) in the form
A solution of the latter equation, subject to the initial conditions (2.3), can be represented as
where Open image in new window is the solution of the homogeneous equation (2.9) with initial conditions (2.3). Since Open image in new window is a solution of (2.9), we can write
This relation and (2.5) yield
since the Cauchy function is bounded by (3.2). Moreover,
From (3.10), it follows that
According to (2.4), we have
Take Open image in new window . Then, by the estimate
it follows that
we obtain
Condition (3.5) implies the inequality
Since Open image in new window is arbitrary, we obtain the estimate
Further,

This yields the required result.

Corollary 3.2.

Under conditions (2.4) and (2.5), let the inequality
hold, with constants Open image in new window and Open image in new window independent of Open image in new window . If, in addition,
Then, any solution Open image in new window of (2.13)–(2.3) satisfies the estimate

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

Proof.

Under condition (3.25), we obtain

Now, Corollary 3.2 yields the following result.

Theorem 3.3.

Let the conditions (2.4), (2.5), (2.6), (3.25), and, in addition,

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

Proof.

Condition (3.29) implies the inequality (3.26), and in addition
By (2.6), we obtain

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

then (2.2) takes the form (3.3). Thus, Corollary 3.2 implies
Thus, condition (3.29) implies

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]).

Assume that the Cauchy function Open image in new window of (2.9) satisfies
where Open image in new window is a constant. Then there exist constants Open image in new window and Open image in new window such that

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

Corollary 3.6.

Let conditions (2.4), (2.5), (3.25), and, in addition,

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

Example 3.7.

Consider the following delay difference system in the Euclidean space Open image in new window :

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 .

In addition, the function Open image in new window supplies the solvability and satisfies the condition

Hence, Open image in new window

Further, assume that the Cauchy solution Open image in new window of equation

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.

For instance, if the linear system with constant coefficients associated to the nonlinear system with variable coefficients (3.39) is

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

Now, we will consider an important particular case of (2.2), namely, the linear delay difference system

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.

Kipnis and Komissarova [6] investigated the stability of the system

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.

The purpose of this section is to apply a new method to investigate the stability of system (4.1), which combined with the "freezing technique," will allow us to derive explicit estimations to their solutions, namely, introducing new variables; one can reduce system (4.1) to a delay-free linear difference system of higher dimension. In fact, put
Then (4.1) takes the form

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

Let Open image in new window be the product of Open image in new window copies of Open image in new window Then we can consider (4.6) defined in the space Open image in new window . In Open image in new window define the norm
where Open image in new window is the Frobenius (Hilbert-Schmidt) norm of a matrix Open image in new window , Open image in new window Open image in new window , and Open image in new window are the eigenvalues of Open image in new window , including their multiplicities. Here Open image in new window is the adjoint matrix. If Open image in new window is normal, that is, Open image in new window , then Open image in new window If Open image in new window is a triangular matrix such that Open image in new window for Open image in new window , then
Due to [15, Theorem 2.1], for any Open image in new window -matrix Open image in new window , the inequality

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

Then, any solution Open image in new window of (4.1) is bounded and satisfies the inequality

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

Since the calculation of quantities Open image in new window and Open image in new window is not an easy task, by (4.11), some estimations to these formulae, namely, in terms of the eigenvalues of auxiliary matrices will be driven. In doing so, one assumes that
and denote

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

Corollary 4.4.

Under condition (i) of Theorem 4.3, let (4.13) and Open image in new window hold. Then, any solution Open image in new window of (4.1) is bounded. Moreover,

where Open image in new window

Proof.

By (4.11), we obtain
The relation
implies that
Simple calculations show that

Hence, Open image in new window .

On the other hand,
Thus, it follows that

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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Copyright information

© Rigoberto Medina. 2008

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Departamento de Ciencias ExactasUniversidad de Los LagosOsornoChile

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