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.

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

Let denote the set of nonnegative integers. Given a positive integer , denote by and the -dimensional space of complex column vectors and the set of matrices with complex entries, respectively. If is any norm on , the associated induced norm of a matrix is defined by

(2.1)

Consider the nonlinear discrete-time system with multiple delays of the form

(2.2)

where is an integer and

We will consider (2.2) subject to the initial conditions

(2.3)

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

Throughout the paper, we will assume that the variable matrices have the properties

(2.4)

(2.5)

In addition, is a given function satisfying the growth condition

(2.6)

where ; ; ,

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 , independent of (but dependent on ), such that

(2.7)

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).

Put

(2.8)

The stability conditions for (2.2) will be formulated in terms of the Cauchy function (the fundamental solution) of

(2.9)

defined as follows. For a fixed let be the solution of (2.9) with initial conditions

(2.10)

Since the coefficients of (2.9) are constants for fixed , then the Cauchy function of (2.9) has the form

(2.11)

where is the solution of (2.9) with the initial conditions

(2.12)

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

Lemma 2.2 (see [7]).

The solution of

(2.13)

where is a given function, subject to the initial conditions

(2.14)

has the form

(2.15)

where is the Cauchy function of (2.9) and is the solution of the homogeneous equation

(2.16)

with the same initial conditions:

(2.17)

Lemma 2.3 (see [7]).

The solution of (2.16) with initial conditions (2.14) has the form

(2.18)

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

Theorem 2.4 (see [7]).

Let the inequality

(2.19)

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]).

Let be a sequence of positive numbers such that

(2.20)

where is a constant. Then there exist constants and such that

(2.21)

3. Main Results

Now, we establish the main results of the paper, which will be valid for a family of slowly varying matrices. Let and With the notation

(3.1)

assume that

(3.2)

Consider the equation

(3.3)

where is a bounded function such that

(3.4)

Theorem 3.1.

Under conditions (2.4) and (2.5), let the inequality

(3.5)

holds. Then for any solution of problem (2.13)–(2.3), the estimate

(3.6)

is valid, where , and

Proof.

Fix and rewrite (3.3) in the form

(3.7)

Making

(3.8)

we get

(3.9)

A solution of the latter equation, subject to the initial conditions (2.3), can be represented as

(3.10)

where is the solution of the homogeneous equation (2.9) with initial conditions (2.3). Since is a solution of (2.9), we can write

(3.11)

This relation and (2.5) yield

(3.12)

since the Cauchy function is bounded by (3.2). Moreover,

(3.13)

From (3.10), it follows that

(3.14)

According to (2.4), we have

(3.15)

Take . Then, by the estimate

(3.16)

it follows that

(3.17)

Hence,

(3.18)

where

(3.19)

Making

(3.20)

we obtain

(3.21)

Condition (3.5) implies the inequality

(3.22)

Since is arbitrary, we obtain the estimate

(3.23)

Further,

(3.24)

This yields the required result.

Corollary 3.2.

Under conditions (2.4) and (2.5), let the inequality

(3.25)

hold, with constants and independent of . If, in addition,

(3.26)

Then, any solution of (2.13)–(2.3) satisfies the estimate

(3.27)

where , and

Proof.

Under condition (3.25), we obtain

(3.28)

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,

(3.29)

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

(3.30)

By (2.6), we obtain

(3.31)

where is a solution of (2.2) and

Let

(3.32)

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

(3.33)

Thus, condition (3.29) implies

(3.34)

where

(3.35)

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]).

Assume that the Cauchy function of (2.9) satisfies

(3.36)

where is a constant. Then there exist constants and such that

(3.37)

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,

(3.38)

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 :

(3.39)

where

(3.40)

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

In addition, the function supplies the solvability and satisfies the condition

(3.41)

Hence,

Further, assume that the Cauchy solution of equation

(3.42)

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.

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

(3.43)

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

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

(4.1)

where and are variable -matrices.

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

(4.2)

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

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

(4.3)

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

(ii)

• (iii)
  

  (4.4)

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

(4.5)

Then (4.1) takes the form

(4.6)

where

(4.7)

where is the unit matrix in

Let be the product of copies of Then we can consider (4.6) defined in the space . In define the norm

(4.8)

For an -matrix , denote

(4.9)

where is the Frobenius (Hilbert-Schmidt) norm of a matrix , , and are the eigenvalues of , including their multiplicities. Here is the adjoint matrix. If is normal, that is, , then If is a triangular matrix such that for , then

(4.10)

Due to [15, Theorem 2.1], for any -matrix , the inequality

(4.11)

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

Theorem 4.3 (see [7]).

Assume that

• (i) ; and

• (ii) =,

Then, any solution of (4.1) is bounded and satisfies the inequality

(4.12)

where , with defined in (2.14).

Since the calculation of quantities and 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

(4.13)

and denote

(4.14)

where and .

Corollary 4.4.

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

(4.15)

where

Proof.

By (4.11), we obtain

(4.16)

The relation

(4.17)

implies that

(4.18)

Simple calculations show that

(4.19)

Thus,

(4.20)

Hence, .

On the other hand,

(4.21)

But

(4.22)

Thus, it follows that

(4.23)

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.