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Journal of Applied Mathematics
Volume 2012, Article ID 723893, 9 pages
http://dx.doi.org/10.1155/2012/723893
Research Article

Asymptotic Stability of a Class of Impulsive Delay Differential Equations

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received 8 June 2012; Revised 22 August 2012; Accepted 9 September 2012

Academic Editor: Jaime Munoz Rivera

Copyright © 2012 G. L. Zhang et al. 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.

Abstract

This paper is concerned with a class of linear impulsive delay differential equations. Asymptotic stability of analytic solutions of this kind of equations is studied by the property of delay differential equations without impulsive perturbations. New numerical methods for this kind of equations are constructed. The convergence and asymptotic stability of the methods for this kind of equations are studied.

1. Introduction

Impulsive differential equations arise widely in the study of medicine, biology, economics, and engineering, and so forth In recent years, theory of impulsive differential delay equations (IDDEs) has been an object of active research (see [118] and the reference therein). The results about the existence and uniqueness of IDDEs have been studied in [2, 3, 10]. And the stability of IDDEs have attracted increasing interest in both theoretical research and practical applications (see [1, 3, 4, 618] and the reference therein). In this paper, we use the property of delay differential equations without impulsive perturbations to study asymptotic stability of analytic solutions of a class of linear impulsive delay differential equations.

There are a few papers on numerical methods of impulsive differential equations. In [5], Covachev et al. obtained a convergent difference approximation for a nonlinear impulsive ordinary system in a Banach space. In [9, 19], the authors studied the stability of Runge-Kutta methods for linear impulsive ordinary differential equations. In [20], Ding et al. studied the convergence property of an Euler method for IDDEs. In this paper, we construct new numerical methods for a class of linear impulsive delay differential equations. The convergence and asymptotic stability of the methods for this kind of equations are studied.

2. Stability of Analytic Solutions of a Class of Linear IDDEs

In this paper, we consider the following equation: where , , and are real constants, is a continuous function on , and denotes the right-hand derivative of .

Definition 2.1. The zero solution of (2.1) is said to be asymptotically stable, if where is the solution of (2.1) for any initial function .

We define and the delay differential equations without impulsive perturbations as follows: In the rest of the paper, is defined as where denotes the greatest integer function.

Theorem 2.2. Assume that is the solution of (2.1) and , then is the solution of (2.3). On the other hand, assume that is the solution of (2.3) and , then is the solution of (2.1).

Proof. (i) We prove that is continuous if is the solution of (2.1). Obviously, is continuous on . We know that is continuous. Therefore, Hence, . So is continuous at . It follows from that is continuous at . Hence is continuous on .
(ii) We prove that is the solution of (2.3) if is the solution of (2.1). For , we have Because denotes the right-hand derivative of , , hence is the solution of (2.3).
(iii) We prove that is the solution of (2.1) if is the solution of (2.3). For , Obviously, So . Consequently, is the solution of (2.1).

By Theorem 2.2, we obtain the following corollary.

Corollary 2.3. Assume that , where is the solution of (2.1). Then , where , and also is the solution of (2.3). On the other hand, assume that , where is the solution of (2.3). Then , where , and also is the solution of (2.1).

Lemma 2.4 (see [21]). Suppose and is the solution of (2.3). Then, for any , there is a constant such that

Theorem 2.5. Assume that Then the zero solution of (2.1) is asymptotically stable.

Proof. First we prove that , where Suppose that . Then there exist a such that and . So And implies that Hence (2.12) and (2.13) imply that , which is a contradiction.
Consequently, . By Lemma 2.4, we know that , where the solution of (2.3). By Corollary 2.3, we know that , where is the solution of (2.1).

Corollary 2.6. Assume that there is a constant , such that and . Then the zero solution of (2.1) is asymptotically stable.

Proof. Obviously, . And because , so By Theorem 2.5, we have that the zero solution of (2.1) is asymptotically stable.

Similarly, we can get the following corollary.

Corollary 2.7. Assume that there is a constant , such that and . Then the zero solution of (2.1) is asymptotically stable.

3. Numerical Solutions of IDDEs

We consider the Runge-Kutta methods for (2.3) as follows: where , is a given stepsize with integer . is referred to as the number of stages. The weights , the abscissas , and the matrix will be denoted by . , , and is an approximation of , .

Define Then is the numerical solution of (2.1).

Theorem 3.1. Assume that the Runge-Kutta method is of order . Then the order of (3.1) and (3.2) for (2.1) is also , when .

Proof. In [22], it has been proved that the order of (3.1) is , if the Runge-Kutta method (A, b, c) is of order . And for any fixed , and any , we have , where . Hence the order of (3.1) and (3.2) is .

By (3.2), we know that and , if and only if . The definition of P-stability region and P-stable has been given in many papers, for example, [23]. The P-stability region of (3.1) is the set of pairs of numbers , such that the numerical solution satisfies for all constant delays and all initial function .

The Runge-Kutta method (3.1) is P-stable, if Therefore we obtain the following theorem.

Theorem 3.2. If then the solutions of (2.3) and (2.1) are asymptotically stable. In addition, if the Runge-Kutta method is P-stable, then the process (3.1) preserves asymptotically stable of (2.3); consequently, the process (3.1)-(3.2) preserves asymptotically stable of (2.1).

4. Numerical Experiments

Consider the following IDDE: The analytic solution of (4.1) (see Figure 1) as is Note that , , , and are the errors of the analytic solution and the numerical solution (3.1) (3.2) for (4.1) at , respectively.

723893.fig.001
Figure 1: The analytic solution of (4.1) and the real part analytic solution of (4.3).

In Tables 1 and 2, we have listed the errors at and 2 of the explicit Euler method and the Trapezoid method.

tab1
Table 1
tab2
Table 2

Obviously, , so the zero solution of (4.1) is asymptotically stable by Theorem 2.5. Assume that is the solution of (4.1) and , for . Then is the solution of the following equation: Because , we know that as (see Figure 1 ).

The numerical process (3.1) and (3.2) for (4.1) is asymptotically stable as (3.2) is the Trapezoid method (see Figure 2) and the implicit Euler method (see Figure 3).

723893.fig.002
Figure 2: The numerical solution of (4.1), when (3.1) is the Trapezoid method as h = .
723893.fig.003
Figure 3: The numerical solution of (4.1), when (3.1) is the implicit Euler method as h = .

Acknowledgment

The authors wish to thank referees for valuable comments. The research was supported by the NSF of China no. 11071050.

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