Abstract and Applied Analysis

Volume 2013 (2013), Article ID 103894, 5 pages

http://dx.doi.org/10.1155/2013/103894

Research Article

-Exponential Stability of Nonlinear Impulsive Dynamic Equations on Time Scales

Veysel Fuat Hatipoğlu,¹ Deniz Uçar,² and Zeynep Fidan Koçak¹

¹Department of Mathematics, Faculty of Science, Muğla University, Kötekli Campus, 48000 Muğla, Turkey

²Department of Mathematics, Faculty of Sciences and Arts, Usak University, 1 Eylul Campus, 64200 Usak, Turkey

Received 26 November 2012; Accepted 15 March 2013

Academic Editor: Stefan Siegmund

Copyright © 2013 Veysel Fuat Hatipoğlu 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.

The purpose of this paper is to present the sufficient -exponential, uniform exponential, and global exponential stability conditions for nonlinear impulsive dynamic systems on time scales.

1. Introduction

In recent years, a significant progress has been made in the stability theory of impulsive systems [1, 2], and in [3] authors studied the -exponential stability for nonlinear impulsive differential equations. There are various types of stability of dynamic systems on time scales such as asymptotic stability [4, 5], exponential and uniform exponential stability [6–8], and -stability [9]. In the past decade, many authors studied impulsive dynamic systems on time scales [10–14]. There are some papers on the theory of the stability of impulsive dynamic systems on time scales. In [15], stability criteria for impulsive systems are given and in [16], authors studied -uniform stability of linear impulsive dynamic systems.

In this paper, we consider the -exponential stability of the zero solution of the first-order nonlinear impulsive dynamic system

where is a time scale which has at least finitely many right-dense points of impulsive , is a nonlinear function and rd continuous in , , and are fixed moments of impulsive effect. Let , , be rd continuous functions and let . Throughout the paper, we assume that , for all in the time scale interval , and call the zero function the trivial solution of (1) and we consider . Existence and uniqueness of solutions of (1) have been studied in [10].

In the following part we present some basic concepts about time scale calculus and we refer the reader to resource [17] for more detailed information on dynamic equations on time scales.

2. Preliminaries

A time scale is an arbitrary nonempty closed subset of the real numbers . For we define the forward jump operator by

while the backward jump operator is defined by

If , we say that is right scattered, while if , we say that is left scattered. Also, if , then is called right dense, and if , then is called left dense. The graininess function is defined by

We introduce the set which is derived from the time scale as follows. If has a left-scattered maximum , then ; otherwise .

A function on is said to be delta differentiable at some point if there is a number such that for every there is a neighborhood of such that

The function is said to be regressive provided for all . The set of all regressive rd-continuous functions is denoted by .

Let and for all . The exponential function on , defined by

is the solution to the initial value problem , . Properties of the exponential function on are given in [6].

In [6] authors defined the Lyapunov function on time scales, type I Lyapunov function as,

and derivative of type I Lyapunov function as follows:

We start introducing notations that will be used in the following sections. In the Euclidean -space, norm of a vector is given by . The induced norm of an matrix is defined to be .

Now, we give definition of -exponential, -uniform exponential, -global exponential stability, and stability conditions for the solution of nonlinear impulsive dynamic system (1).

3. -Exponential Stability

Definition 1. The trivial solution to (1) is exponentially stable on if any solution of the system (1) satisfies for all , ,

where is a positive constant and is a nonnegative increasing function, . If the function is independent of , then the trivial solution to system (1) is said to be uniformly exponentially stable on .

Definition 2. The trivial solution to (1) is globally exponentially stable on if there exist some constants and such that any solution of (1), for all , , we have

Now, we shall present sufficient conditions for the -exponential stability, uniformly exponential stability, and globally exponentially stability of(1).

Theorem 3. Assume that contains the origin and there exists a type I Lyapunov function such that, for all and , ,

where , , and are positive functions, where is nondecreasing; , and are positive constants; is a nonnegative constant, and . Then the trivial solution to (1) is exponentially stable on .

Proof. Let be a solution to (1) that stays in for all . As ,   is well defined and positive. Thus . Consider

Integrating both sides of above inequality from to with , we obtain, for ,

From condition

Letting

we get,

By condition (11), we have

And by the fact that , we obtain

From (18) and (20) we obtain the result for all, , ,

By Definition 1 system (1) is exponentially stable.

If we consider as scaler function independent of , then we get a sufficient condition for uniformly exponential stability as stated below.

Theorem 4. In Theorem 3 if is a constant function independent of and , , are positive constants, then the trivial solution to system (1) is uniformly exponentially stable on .

Proof. The proof is similar to proof of Theorem 3 by taking and , hence omitted.

Theorem 5. Assume that contains the origin and there exists a type I Lyapunov function such that, for all and , ,

where is a constant function independent of . ,   are constants and . Then the trivial solution to (1) is uniformly exponentially stable on .

Proof. Let be a solution to (1) that stays in for all . Since , is well defined and positive. Now consider

Integrating both sides of the above inequality from to , we obtain, for ,

This implies that

From (26) and by invoking condition (22) we obtain, for all , ,

By Definition 1 system (1) is uniformly exponentially stable.

Theorem 6. Assume that contains the origin and there exists a type I Lyapunov function such that, for all and , ,

where , and are positive constants, , is a nonnegative constant, and . Then the trivial solution to (1) is globally exponentially stable on .

Proof. Let be a solution to (1) that stays in for all . Since , is well defined and positive. For all , , consider

Integrating both sides of the above inequality from to , with , we obtain,

This implies that

From (32), and by invoking condition (28), we obtain, for all , ,

If we set , then (33) can be written as

Since , by Definition 2 system (1) is globally exponentially stable.

4. Examples

Example 7. We consider Example (35) in [7] and extend the example by using impulse condition,

where is a constant . If there is a constant such that

for some constant and all , (35) is uniformly exponentially stable.

Under above assumptions, we will show that the conditions of Theorem 4 are satisfied. Let , choose and , , then (11) holds with , . If we calculate , for all ,

we have the following comparison:

Dividing and multiplying the right-hand side by , we see that (12) holds under the above assumptions with and . Also, since , we have

for all . Therefore (13) is satisfied. Hence, all hypotheses of Theorem 4 are satisfied and we conclude that the trivial solution to (35) is uniformly exponentially stable. We consider following two special cases of (35).

Case 1. If , then . It is easy to see that (37) holds for . Also for , condition (38) is satisfied. Hence, we conclude that if , then the trivial solution to (35) is uniformly exponentially stable.

Case 2. If , then . In this case rewriting (37) we have

then (37) holds for . Also for , condition (38) is satisfied. Therefore for , then the trivial solution to (35) is uniformly exponentially stable.

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