Research Article | Open Access

# Global Exponential Stability of Impulsive Functional Differential Equations via Razumikhin Technique

**Academic Editor:**Yong Zhou

#### Abstract

This paper develops some new Razumikhin-type theorems on global exponential stability of impulsive functional differential equations. Some applications are given to impulsive delay differential equations. Compared with some existing works, a distinctive feature of this paper is to address exponential stability problems for any finite delay. It is shown that the functional differential equations can be globally exponentially stabilized by impulses even if it may be unstable itself. Two examples verify the effectiveness of the proposed results.

#### 1. Introduction

Functional differential equations (FDEs) which include delay differential equations (DDEs) play a very important role in formulation and analysis in mechanical, electrical, control engineering and physical sciences, economic, and social sciences [1, 2]. Therefore, the theory of FDEs has been developed very quickly. The investigation for FDEs has attracted the considerable attention of researchers and many qualitative theories of FDEs have been obtained. A large number of stability criteria of FDEs have been reported.

In addition to the delay effect, as is well known, impulsive effect is likely to exist in a wide variety of evolutionary processes in which states are changed abruptly at certain moments of time in the fields such as medicine and biology, economics, electronics, and telecommunications [3]. So far, a lot of interesting results on stability have been reported that have focused on the impulsive effect of FDEs (see, e.g., [4–16] and the references cited therein).

In particular, several papers devoted to the study of exponential stability of impulsive functional differential equations (IFDEs) have appeared during the past years. In [14, 15], the authors have investigated exponential stability of IFDEs by using the method of Lyapunov functions and Razumikhin techniques. In [16], the authors have also studied exponential stability by using the method of Lyapunov functional. However, some results in [15, 16] imposed a restrictive condition on time delays which were less than the length of all the impulsive intervals (see, e.g., [15, Theorems 3.1-3.2] and [16, Theorem 3.1]). The aim of this paper is to establish global exponential stability criteria for IFDEs by employing the Razumikhin technique which illustrate that impulses do contribute to the stability of some IFDEs and the restrictive condition that the time delays are less than the length of all the impulsive intervals can be removed in this paper.

#### 2. Preliminaries

Throughout this paper, unless otherwise specified, we use the following notations. Let , , , be the identity matrix, and be the maximal eigenvalue and the minimal eigenvalue of a matrix, respectively. If is a vector or matrix, its transpose is denoted by . For and , let be Euclidean vector norm, and denote the induced matrix norm by

Let and denote the family of all bounded continuous -valued functions defined on . is continuous for all but at most countable points and at these points , and exist and , where is an interval, and denote the right-hand and left-hand limits of the function at time , respectively. Especially, let with norm .

In this paper, we consider the following IFDEs:

where , , . The initial function . The impulsive function , and the impulsive moments satisfy , and .

In this paper, we assume that functions and , , satisfy all necessary conditions for the global existence and uniqueness of solutions for all . Denote by the solution of (2.2) such that . For the purpose of stability in this paper, we also assume that and , . So system (2.2) admits a zero solution or trivial solution . We further assume that all the solutions of (2.2) are continuous except at , at which is right continuous, that is, .

*Definition 2.1. *The trivial solution of system (2.2) is said to be globally exponentially stable, if there exist numbers and such that
whenever .

*Definition 2.2. * is said to belong to the class , if is continuous on each of the sets , exists, is locally Lipschitzian in all , and for all .

*Definition 2.3. * is said to belong to the class , if is continuous on each of the sets , for all , exists, are continuous, where ,

*Definition 2.4. *Given a function , the upper right-hand derivative of with respect to system (2.2) is defined by
for .

#### 3. Razumikhin-Type Theorems

In this section, we will present some Razumikhin-type theorems on global exponential stability for system (2.2) based on the Lyapunov-Razumikhin method.

Theorem 3.1. *Let and all be positive numbers, a real number, , and . Suppose that there exists a function such that*(i)* for all , and ;*(ii)* for all , , whenever for all ;*(iii)*, , .*

Then the trivial solution of system (2.2) is globally exponentially stable.

*Proof. *For any , we denote the solution of (2.2) by . Without loss of generality, we assume that .

Since and , there exist positive numbers and such that
Set , then .

Set , we have
Let be a fixed number. In the following, we will prove that
We first prove that
It is noted that , . So, it only needs to prove that for . On the contrary, there exist some such that . Set , then we have and . Set . Then and . For , we have
Hence
By condition (ii) and (3.2), it follows that for
So, we obtain . This is a contradiction, so (3.4) holds.

Now, we assume that for some ,
We will prove that
Suppose not, there exist some such that . Set . From condition (iii) and (3.8), we have . Hence and . Set . Then we have . Furthermore, we have for . Thus, (3.7) holds. Then , which yields a contradiction. Therefore, (3.9) holds.

By mathematical induction, we have
Hence
where , . The proof is therefore complete.

Theorem 3.2. *Let and all be positive numbers, , and . Suppose that there exists a function such that*(i)* for all , and ,*(ii)* for all , whenever for all ;*(iii)*, , .*

Then the trivial solution of system (2.2) is globally exponentially stable.

*Proof. *Since and , there exist positive numbers and such that
Set and , then . Set , where is defined as in the proof of Theorem 3.1.

Now, let , we will prove that (3.3) holds.

We first prove that (3.4) holds. In fact, it is noticed that , for . So it only needs to prove for . On the contrary, there exists such that . Set , then we have . Set . Then and . For , we have
Hence
It follows that for
which leads to . This is a contradiction to the fact , so (3.4) holds.

Now, we assume that for some , (3.8) holds. We will prove that (3.9) holds. In order to do this, we first claim that
Suppose not, then we have . There are two cases to be considered.*Case 1. * for all .

By (3.8), we have for and . Thus, we get for which leads to . This is a contradiction.*Case 2. *There is some such that .

Set . Then and . Since for , , . By (3.15), we have for , which gives . This is also a contradiction.

Hence, (3.16) holds. It follows from (iii) that . Now, we assume that (3.9) is not true, set . Hence and . If for , set , otherwise, set . Thus, we have for . Hence, by (3.15), for . Then , which yields a contradiction. Therefore, (3.9) holds. The rest is the same as in the proof of Theorem 3.1.

*Remark 3.3. *By Theorems 3.1 and 3.2, we can design impulsive control for the following FDEs
such that the system can be impulsively stabilized to its trivial solution. In Theorem 3.1, the constant may be chosen as a positive number. In the stability theory of FDEs, the condition allows the derivative of the Lyapunov function to be positive which may not even guarantee the stability of functional differential system (see, e.g., [4, 15]). However, as we can see from Theorem 3.1, impulses play an important role in making a functional differential system globally exponentially stable even if it may be unstable itself.

*Remark 3.4. *It is important to emphasize that, in contrast with some existing exponential stability results for IFDEs in the literature [15, 16], Theorems 3.1 and 3.2 are also valid for any finite delay. Therefore, our new results are more practically applicable than those in the literature, since the restrictive condition that the supper bound of time delay is less than the length of all the impulsive intervals is actually removed here.

#### 4. Applications and Examples

Now, we will apply the general Razumikhin-type theorems established in Section 3 to deal with the global exponential stability of impulsive delay differential equations (IDDEs).

Consider a delay system of the form

where , , are all continuous, and

is continuous. We also assume that (4.1) has a global solution which is again denoted by , and .

Theorem 4.1. *Let , be a real number, and be all positive numbers, . Suppose that there exists a function such that*(i)* for all , and ;*(ii)*
for all ;*(iii)*, , .**If , then the trivial solution of (4.1) is globally exponentially stable.*

*Proof. *For , let
Then system (4.1) becomes system (2.2), and becomes
If , then there exists a constant , such that
So, if and for all , then
where . By Theorem 3.1, the trivial solution of (4.1) is globally exponentially stable.

Corollary 4.2. *Let . Assume that there exist scalar numbers , and such that
**
for all , , .**If
**
then the trivial solution of (4.1) is globally exponentially stable.*

*Proof. *Let , then we can easily see that condition (i) of Theorem 4.1 holds.

For and , from (4.8) and (4.9), we can calculate that
Let , by the inequality , we have
Substituting (4.13) into (4.12), we obtain
where , , .

From (4.11), we have
The conclusion follows from Theorem 4.1 immediately and the proof is completed.

*Remark 4.3. *Let . If , by Theorem 3.2, we can also give some other results on global exponential stability for (4.1). For the details, we omit them here.

*Example 4.4. *Consider a scalar nonlinear impulsive delay differential equation
on , where is a continuous function, , and
with , .

From (4.8)-(4.9), we can see , . By Corollary 4.2, if there exists a scalar number , such that
then the trivial solution of (4.16) is global exponentially stable.

*Example 4.5. *Consider the following linear impulsive delay system:
where
is a continuous function. Since and , we can not use the results in [5, 14] to determine the exponential stability. From Corollary 4.2, we can choose , , , by (4.11), we obtain that if , then trivial solution of (4.19) is globally exponentially stable.

*Remark 4.6. *For (4.19), the authors in [15] chose , , such that . However, if , the results in [15, 16] fail to determine the global exponential stability.

#### 5. Conclusion

In this paper, some new Razumikhin-type theorems on global exponential stability for IFDEs are obtained by employing Lyapunov-Razumikhin technique. Some applications to IDDEs are also given. It should be mentioned that our results may allow us to develop an effective impulsive control strategy to stabilize an underlying delay dynamical system even if it may be unstable in practice, which is particularly meaningful for applications in engineering and technology. Two examples are also given to demonstrate the effectiveness of the theoretical results.

#### Acknowledgments

This work was supported by the Natural Science Foundation of Guangdong Province (8351009001000002) and the National Natural Science Foundation of China (10971232).

#### References

- J. K. Hale and S. M. V. Lunel,
*Introduction to Functional Differential Equations*, vol. 99 of*Applied Mathematical Sciences*, Springer, New York, NY, USA, 1993. View at: Zentralblatt MATH | MathSciNet - V. B. Kolmanovskiĭ and V. R. Nosov,
*Stability of Functional Differential Equations*, vol. 180 of*Mathematics in Science and Engineering*, Academic Press, London, UK, 1986. View at: Zentralblatt MATH | MathSciNet - A. M. Samoĭlenko and N. A. Perestyuk,
*Impulsive Differential Equations*, vol. 14 of*World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises*, World Scientific, Singapore, 1995. View at: Zentralblatt MATH | MathSciNet - X. Liu and G. Ballinger, “Uniform asymptotic stability of impulsive delay differential equations,”
*Computers & Mathematics with Applications*, vol. 41, no. 7-8, pp. 903–915, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - X. Liu, “Stability of impulsive control systems with time delay,”
*Mathematical and Computer Modelling*, vol. 39, no. 4-5, pp. 511–519, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Zhang and J. Sun, “Strict stability of impulsive functional differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 301, no. 1, pp. 237–248, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. Liu and X. Fu, “Stability of functional differential equations with impulses,”
*Journal of Mathematical Analysis and Applications*, vol. 328, no. 2, pp. 830–841, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. Chen and X. Wen, “Asymptotic stability for impulsive functional differential equation,”
*Journal of Mathematical Analysis and Applications*, vol. 336, no. 2, pp. 1149–1160, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. Liang and J. Shen, “Uniform stability theorems for delay differential equations with impulses,”
*Journal of Mathematical Analysis and Applications*, vol. 326, no. 1, pp. 62–74, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. Chen and X. Fu, “New Razumikhin-type theorems on the stability for impulsive functional differential systems,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 66, no. 9, pp. 2040–2052, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Zhang and J. Sun, “Stability of impulsive functional differential equations,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 68, no. 12, pp. 3665–3678, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. Luo and J. Shen, “Stability of impulsive functional differential equations via the Liapunov functional,”
*Applied Mathematics Letters*, vol. 22, no. 2, pp. 163–169, 2009. View at: Publisher Site | Google Scholar | MathSciNet - K. Liu and G. Yang, “The improvement of Razumikhin type theorems for impulsive functional differential equations,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 70, no. 9, pp. 3104–3109, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Q. Wang and X. Liu, “Exponential stability for impulsive delay differential equations by Razumikhin method,”
*Journal of Mathematical Analysis and Applications*, vol. 309, no. 2, pp. 462–473, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Q. Wang and X. Liu, “Impulsive stabilization of delay differential systems via the Lyapunov-Razumikhin method,”
*Applied Mathematics Letters*, vol. 20, no. 8, pp. 839–845, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - X. Liu and Q. Wang, “The method of Lyapunov functionals and exponential stability of impulsive systems with time delay,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 66, no. 7, pp. 1465–1484, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2010 Shiguo Peng and Liping Yang. 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.