Research Article | Open Access
Global Exponential Stability of Impulsive Functional Differential Equations via Razumikhin Technique
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.
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 . 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 , 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.
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.
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.
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.
This work was supported by the Natural Science Foundation of Guangdong Province (8351009001000002) and the National Natural Science Foundation of China (10971232).
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