- Research Article
- Open Access

# Strict Stability Criteria for Impulsive Functional Differential Systems

- Kaien Liu
^{1}Email author and - Guowei Yang
^{2}

**2008**:243863

https://doi.org/10.1155/2008/243863

© K. Liu and G. Yang. 2008

**Received:**4 September 2007**Accepted:**18 November 2007**Published:**15 December 2007

## Abstract

By using Lyapunov functions and Razumikhin techniques, the strict stability of impulsive functional differential systems is investigated. Some comparison theorems are given by virtue of differential inequalities. The corresponding theorems in the literature can be deduced from our results.

## Keywords

- Lyapunov Function
- Comparison System
- Comparison Theorem
- Differential Inequality
- Zero Solution

## 1. Introduction

Since time-delay systems are frequently encountered in engineering, biology, economy, and other disciplines, it is significant to study these systems [1]. On the other hand, because many evolution processes in nature are characterized by the fact that at certain moments of time they experience an abrupt change of state, the study of dynamic systems with impulse effects has been assuming greater importance [2–4]. It is natural to expect that the hybrid systems which are called impulsive functional differential systems can represent a truer framework for mathematical modeling of many real world phenomena. Recently, several papers dealing with stability problem for impulsive functional differential systems have been published [5–10].

The usual stability concepts do not give any information about the rate of decay of the solutions, and hence are not strict concepts. Consequently, strict-stability concepts have been defined and criteria for such notions to hold are discussed in [11]. Till now, to the best of our knowledge, only the following very little work has been done in this direction [12–15].

In this paper, we investigate strict stability for impulsive functional differential systems. The paper is organized as follows. In Section 2, we introduce some basic definitions and notations. In Section 3, we first give two comparison lemmas on differential inequalities. Then, by these lemmas, a comparison theorem is obtained and several direct results are deduced from it. An example is also given to illustrate the advantages of our results.

## 2. Preliminaries

where is the set of all positive integers, , is an open set in , here , and is continuous everywhere except for a finite number of points at which and exist and . for each , where denotes the norm of vector in , with as and denotes the right-hand derivative of . For each , is defined by , . For , , . We assume that and , so that is a solution of (2.1), which we call the zero solution.

if it is continuous and satisfies the differential equation in each , and at it satisfies .

Throughout this paper, we always assume the following conditions hold to ensure the global existence and uniqueness of solution of (2.1) through .

(H_{1})
is continuous on
for each
and for all
and
, the limits
exist.

(H_{2})
is Lipschitzian in
in each compact set in
.

(H_{3})
for all
and there exists
such that
implies that
for all
.

The function belongs to class if the following hold.

(A_{1})
is continuous on each of the sets
and for each
and
,
exists.

(A_{2})
is locally Lipschitzian in
and for
Let
,
along the solution
of (2.1) is defined as

Let us introduce the following notations for further use:

Definition 2.1.

Definition 2.2.

The zero solution of (2.1) is said to be strictly uniformly stable (SUS), if , and in (SS) are independent of .

Remark 2.3.

If in (SS) or (SUS), , we obtain nonstrict stabilities, that is, the usual stability or uniform stability, respectively. Moreover, strict stability immediately implies that the zero solution is not asymptotically stable.

where and are the minimal and maximal solutions of (2.5), (2.6), respectively.

The corresponding definitions of strict stability of the auxiliary systems (2.5), (2.6) are as follows.

Definition 2.4.

Definition 2.5.

The zero solutions of comparison systems (2.5),(2.6), as a system , are said to be strictly uniformly stable (SUS*), if , and in (SS*) are independent of .

## 3. Main Results

We first give two Razumikhin-type comparison lemmas on differential inequalities.

Lemma 3.1.

Assume that

Proof.

First, we prove that (3.2) holds. Otherwise, there exist such that

which contradicts (c). So (3.2) is correct. Equation (3.3) can be proved in the same way as above. Then Lemma 3.1 holds.

Lemma 3.2.

where and are the minimal and maximal solutions of (2.5), (2.6), respectively.

Proof.

By induction, (3.11) is correct. Similarly, (3.12) can be proved by using Lemma 3.1 and assumption (iii).

Using Lemma 3.2, we can easily get the following theorem about strict stability properties of (2.1).

Theorem 3.3.

- (iv)

Then the strict stability properties of comparison systems (2.5), (2.6) imply the corresponding strict stability properties of zero solution of (2.1).

Proof.

Next, choose such that and . We claim that with the choices of , and , the zero solution of (2.1) is strictly stable. That means that if is any solution of (2.1), implies that . If not, we have either of the following alternatives.

Case 1.

which is a contradiction.

Case 2.

We, therefore, obtain the strict stability of the zero solution of (2.1). If we assume that the zero solutions of comparison systems (2.5), (2.6) are (SUS*), since are independent of , we obtain, because of (iv), and in (3.16) are independent of , and hence, (SUS) of (2.1) holds.

Using Theorem 3.3, we can get two direct results on strictly uniform stability of zero solution of (2.1) and the first one is Theorem 3.3 in [15].

Corollary 3.4.

In Theorem 3.3, suppose that , where , and .

Then the zero solution of (2.1) is strictly uniformly stable.

Corollary 3.5.

In Theorem 3.3, suppose that , where and are bounded, and are just the same as in Corollary 3.4.

Then the zero solution of (2.1) is strictly uniformly stable.

Proof.

That is, the zero solutions of (2.5), (2.6) are strictly uniformly stable. Hence, by Theorem 3.3, the zero solution of (2.1) is strictly uniformly stable.

Example 3.6.

where are continuous on . Assume that with , and .

By Corollary 3.5, the zero solution of (2.1) is strictly uniformly stable.

## Declarations

### Acknowledgments

This project is supported by the National Natural Science Foundation of China (60673101) and the Natural Science Foundation of Shandong Province (Y2007G30). The authors are grateful to the referees for their helful comments.

## Authors’ Affiliations

## References

- Hale J:
*Theory of Functional Differential Equations, Applied Mathematical Sciences*.*Volume 3*. 2nd edition. Springer, New York, NY, USA; 1977:x+365.View ArticleGoogle Scholar - Lakshmikantham V, Baĭnov DD, Simeonov PS:
*Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics*.*Volume 6*. World Scientific, Teaneck, NJ, USA; 1989:xii+273.View ArticleGoogle Scholar - Fu XL, Yan BQ, Liu YS:
*Introduction to Impulsive Differential Systems*. Science Press, Beijing, China; 2005.Google Scholar - Fu XL, Wang KN, Lao HX:
**Boundedness of perturbed systems with impulsive effects.***Acta Mathematica Scientia. Series A*2004,**24**(2):135–143.MathSciNetGoogle Scholar - Shen J, Yan J:
**Razumikhin type stability theorems for impulsive functional-differential equations.***Nonlinear Analysis: Theory, Methods & Applications*1998,**33**(5):519–531. 10.1016/S0362-546X(97)00565-8MATHMathSciNetView ArticleGoogle Scholar - Shen JH:
**Razumikhin techniques in impulsive functional-differential equations.***Nonlinear Analysis: Theory, Methods & Applications*1999,**36**(1):119–130. 10.1016/S0362-546X(98)00018-2MATHMathSciNetView ArticleGoogle Scholar - Yan J, Shen J:
**Impulsive stabilization of functional-differential equations by Lyapunov-Razumikhin functions.***Nonlinear Analysis: Theory, Methods & Applications*1999,**37**(2):245–255. 10.1016/S0362-546X(98)00045-5MATHMathSciNetView ArticleGoogle Scholar - Luo Z, Shen J:
**New Razumikhin type theorems for impulsive functional differential equations.***Applied Mathematics and Computation*2002,**125**(2–3):375–386.MATHMathSciNetView ArticleGoogle Scholar - Xing Y, Han M:
**A new approach to stability of impulsive functional differential equations.***Applied Mathematics and Computation*2004,**151**(3):835–847. 10.1016/S0096-3003(03)00540-XMATHMathSciNetView ArticleGoogle Scholar - Liu K, Fu X:
**Stability of functional differential equations with impulses.***Journal of Mathematical Analysis and Applications*2007,**328**(2):830–841. 10.1016/j.jmaa.2006.05.071MATHMathSciNetView ArticleGoogle Scholar - Lakshmikantham V, Leela S:
*Differential and Integral Inequalities: Theory and Applications. Vol. I: Ordinary Differential Equations, Mathematics in Science and Engineering*.*Volume 55*. Academic Press, New York; 1969:ix+390.MATHGoogle Scholar - Lakshmikantham V, Vasundhara Devi J:
**Strict stability criteria for impulsive differential systems.***Nonlinear Analysis: Theory, Methods & Applications*1993,**21**(10):785–794. 10.1016/0362-546X(93)90123-AMATHMathSciNetView ArticleGoogle Scholar - Lakshmikantham V, Zhang Y:
**Strict practical stability of delay differential equation.***Applied Mathematics and Computation*2001,**122**(3):341–351. 10.1016/S0096-3003(00)00049-7MATHMathSciNetView ArticleGoogle Scholar - Lakshmikantham V, Mohapatra RN:
**Strict stability of differential equations.***Nonlinear Analysis: Theory, Methods & Applications*2001,**46**(7):915–921. 10.1016/S0362-546X(00)00122-XMATHMathSciNetView ArticleGoogle Scholar - Zhang Y, Sun J:
**Strict stability of impulsive functional differential equations.***Journal of Mathematical Analysis and Applications*2005,**301**(1):237–248. 10.1016/j.jmaa.2004.07.018MATHMathSciNetView ArticleGoogle Scholar

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