Strict Stability Criteria for Impulsive Functional Differential Systems

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.

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

We consider the following impulsive functional differential system:

(2.1)

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.

Let for some and , a function   is said to be a solution of (2.1) with the initial condition

(2.2)

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 .

(H₁) is continuous on for each and for all and , the limits exist.

(H₂) is Lipschitzian in in each compact set in .

(H₃) for all and there exists such that implies that for all .

The function belongs to class if the following hold.

(A₁) is continuous on each of the sets and for each and , exists.

(A₂) is locally Lipschitzian in and for Let , along the solution of (2.1) is defined as

(2.3)

Let us introduce the following notations for further use:

(i)

(ii)

(iii)

(iv);

(v);

(vi).

Definition 2.1.

The zero solution of (2.1) is said to be strictly stable (SS), if for any and , there exists a such that implies for and for every , there exists an such that

(2.4)

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.

The preceding notions imply that the motion remains in the tube like domains. To obtain sufficient conditions for such stability concepts to hold, it is necessary to simultaneously obtain both lower and upper bounds of the derivative of Lyapunov function. Thus, we need to consider the following two auxiliary systems:

(2.5)

and

(2.6)

where , , for each .

From the theory of impulsive differential systems [2], we obtain that

(2.7)

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.

The zero solutions of comparison systems (2.5), (2.6), as a system, are said to be strictly stable (SS*), if for any and , there exist a and satisfying such that

(2.8)

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

(i) for each ;

(ii)there exists , where are continuous on and exist, , satisfying

(3.1)

Then

(3.2)

(3.3)

where and are the minimal and maximal solutions of systems (3.4) and (3.5), respectively,

(3.4)

(3.5)

Proof.

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

(a),

(b) and

(c).

By (a), (b), and (ii), applying the classical comparison theorem, we have

(3.6)

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.

Assume that (i) in Lemma 3.1 holds. Suppose further that

• (ii) there exists satisfying

  

  (3.7)

where and for any solution of (2.1), implies that

(3.8)

• (iii) there exists satisfying

  

  (3.9)

where , and for any solution of (2.1), implies that

(3.10)

Then

(3.11)

(3.12)

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

Proof.

Assume . First, we prove that (3.11) holds for , that is

(3.13)

Let . Equation (3.13) holds obviously by Lemma 3.1 for . By (ii), . The same proof as for leads to

(3.14)

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.

Assume that all the conditions of Lemma 3.2 hold. Suppose further that there exist functions , such that

• (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.

First, let us prove strict stability of the zero solution of (2.1). Suppose that and are given. Assume that (SS*) holds. Then, given , there exists , and satisfying such that

(3.15)

By (iv), there exist such that for ,

(3.16)

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.

There exists a such that

(3.17)

Then clearly . Thus, by Lemma 3.2, (i) and (ii) imply that

(3.18)

Using (3.15)–(3.18) and (iv), we get

(3.19)

which is a contradiction.

Case 2.

There exists a such that

(3.20)

(3.21)

By (H₃), (3.21) yields

(3.22)

Because of (3.20) and (3.22), there exists a such that

(3.23)

By Lemma 3.2, (i) and (iii) imply that

(3.24)

From (3.15), (3.23), (3.24), and (iv), we have the following contradiction:

(3.25)

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.

Under the given hypotheses, it is easy to obtain the solutions of (2.5) and (2.6):

(3.26)

Since are bounded, there exist two positive constants such that . Also, since , it follows that and , obviously . Given choose and for , choose . Then, if , we have

(3.27)

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.

Consider the system

(3.28)

where are continuous on . Assume that with , and .

Let , then

(3.29)

For any solution of (3.28) such that we have

(3.30)

and if , we have

(3.31)

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