Abstract

In this paper we study the impulsive stabilization of dynamic equations on time scales via the Lyapunov’s direct method. Our results show that dynamic equations on time scales may be -exponentially stabilized by impulses. Furthermore, we give some examples to illustrate our results.

1. Introduction

Differential equations with impulse effect provide an adequate mathematical description of various real-world phenomena in physics, engineering, biology, economics, neutral network, social sciences, and so forth. Since the 1960s, the theory of impulsive differential or difference equations has been studied by many authors [13].

Aulbach and Hilger [4, 5] introduced the theory of time scales (measure chains) in order to create a theory that can unify continuous and discrete analysis. The theory of dynamic systems on time scales has been developed as a generalization of both continuous and discrete dynamic systems simultaneously and applied to many different fields of mathematics [6, 7].

It is widely known that the various types of stability of nonlinear impulsive differential equations or impulsive difference equations can be characterized by using Lyapunov’s second method and inequalities [811]. In recent years, some authors studied the stability of impulsive dynamic systems on time scales [1216]. Furthermore, Hatipoğlu et al. [12] studied the -exponential stability of nonlinear impulsive dynamic equations on time scales. Liu [17] investigated the impulsive stabilization of nonlinear systems by employing Lyapunov’s direct method and obtained sufficient conditions for both stabilization and destabilization.

In this paper we study the impulsive stabilization of dynamic equations on time scales via the Lyapunov’s direct method. Our results show that dynamic equations on time scales may be -exponentially stabilized by impulses. We give some examples to illustrate our results.

2. Preliminaries

We refer the reader to [6] for all the basic definitions and results from time scales calculus that we will use in the sequel (e.g., delta differentiability, rd-continuity, and exponential function and its properties).

It is assumed throughout that a time scale will be unbounded above and is bounded. Let be the -dimensional real Euclidean space. denotes the set of all rd-continuous functions from to and . Also, for any , let . We denote by (resp., ) the set of all regressive (resp., positively regressive) functions from to . The set of all rd-continuous and regressive functions from to is denoted by . Also, let

We consider the impulsive dynamic system with impulses at constant times with the following conditions:(i), with , for .(ii) The function is rd-continuous in and for .(iii) The function is continuous and for ;(iv) represents the right limit of at .The solution of the impulsive dynamic equation with impulse effect (2) depends not only on the initial condition but also on the moments of impulses for each . Let be the unique solution of (2) satisfying the initial condition . For the existence and continuation of solutions of impulsive dynamic equations, see [3, 18].

We need the following well-known impulsive inequality of Gronwall’s type to prove our main results.

Lemma 1 (see [14]). Let , let , let , and let for each . Then, implies

Lemma 2 (see [19]). For every positive constant with , the following inequalities hold: where .

Lemma 3 (see [6]). If , then we have, for all ,(i) and ;(ii);(iii) and ;(iv);(v) and .

Akinyele [20] introduced the notion of -stability of degree with respect to a function , increasing and differentiable on and such that for and , .

Now, we give notions of -exponential, -uniformly exponential, and -globally exponential stability for solutions of nonlinear impulsive dynamic equations on time scales.

Definition 4 (see [12]). Let . System (2) is called -exponentially stable if any solution of (2) satisfies where the function is increasing in , ,  , and is a positive constant.
Moreover, system (2) is said to be -uniformly exponentially stable if is independent of .
System (2) is said to be -globally exponentially stable if system (2) is -exponentially stable for each and the function is independent on each and in the definition of -exponential stability; that is, there exist constants with and such that for any initial value , where is any solution of system (2).

Remark 5. System (2) is exponentially stable if we set in the definition of -exponential stability.
Moreover, system (2) is uniformly exponentially stable if we set in the definition of -uniformly exponential stability.

For the Lyapunov-like function , we recall the following definition.

Definition 6 (see [7, Definition 3.1.1]). We define the generalized derivative of relative to system (2) as follows: given , there exists a neighborhood of such that where is any solution of system (2) and the upper right Dini derivative of is given by where .

Then it is well-known that if is Lipschitzian in for each [21].

In case is right-dense, we have

In case is right-scattered and is continuous at , we have

In fact, if is a solution of system (2), then we have by the chain rule of a differentiable function [22, Theorem 1].

Definition 7 (see [23]). is said to belong to the class if (i) is rd-continuous in and for each , ,   exists.(ii) is locally Lipschizian in and for .

3. Main Results

In this section we investigate -exponential stability for impulsive dynamic equations on time scales via Lyapunov’s direct method.

The following result shows that dynamic equations on time scales may be -exponentially stabilized by impulses. It is adapted from Theorem 3.1 in [11].

Theorem 8. Assume that there exists a function and constants and , with such that the following conditions hold:(i) for ;(ii) for all ;(iii), where each is a positive constant;(iv) and for each .Then the zero solution of system (2) is -exponentially stable.

Proof. Let be any solution of system (2) with initial value , and .
We will show that We can choose such that We first show that In view of conditions (i) and (15), we have
Next, we show that From conditions (i)–(iv), (17) and Lemma 2, we have Now we assume that (14) holds for ; that is, From conditions (iii) and (20), we have Thus (14) holds for each . Then it follows from mathematical induction that (14) holds for each .
In view of conditions (i) and (14), we get where , , and . Hence the trivial solution of system (2) is -exponentially stable. This completes the proof.

Remark 9. We obtain the following results from Theorem 8.(i)If we set for each in Theorem 8, then the zero solution of system (2) is exponentially stable.(ii)If the conditions of Theorem 8 hold and , then the zero solution of system (2) is globally -exponentially stable.

Also, we can obtain the following result as a discrete version of Theorem 8 for .

Corollary 10. Assume that there exists a function and constants and , such that the following conditions hold:(i) is locally Lipschizian in the second variable and for each ;(ii) for ;(iii) for all ;(iv) , where each is a positive constant;(v) and for each .Then the zero solution of system (2) is -exponentially stable.

We can obtain the following result which can be proved as in the similar manner of Theorem 8.

Corollary 11. Assume that all conditions of Theorem 8 are satisfied with the condition (ii) replaced by (ii)′: for all , .Then the zero solution of system (2) is also -exponentially stable.

Remark 12. If we set in the condition (i) of Corollary 11, then the zero solution of system (2) is also exponentially stable.

Next, we obtain the following result that the stability properties of dynamic systems can be preserved under certain impulsive perturbations. It is adapted from Theorem 1 in [13].

Theorem 13. Assume that there exist a function and constants and , with such that the following conditions hold: (i) for ;(ii) for all ;(iii) , where each () is a positive constant and .Then the zero solution of system (2) is -exponentially stable.

Proof. Let be any solution of system (2) with , and .
It follows from condition (ii) that By integrating both sides of (23) from to and condition (iii), we obtain From condition (ii), we have In view of conditions (iii) and (25), we have It follows from mathematical induction that Thus we obtain where .
In view of conditions (i) and (28), we have where , , and . The proof is complete.

Remark 14. We obtain the following results in [13] from Theorem 13.(i)If we set for each in Theorem 13, then the zero solution of system (2) is exponentially stable.(ii)If we set and in condition (i) of Theorem 13, then the zero solution of system (2) is exponentially stable.

4. Examples

In this section we give two examples which illustrate our results from the previous section. Let .

Example 15 (see [24, Example 2]). We consider the impulsive dynamic equation on time scales where for each and .
Let and , then it follows that
We consider two cases: and .

Case 1 (). Letting , , ,  , , , we note that all conditions of Theorem 8 are satisfied. Hence the zero solution of system (30) is -exponentially stable by Theorem 8.

Case 2 ( with ). Then system (30) rewrites where . Then we have Letting , ,  , , , , it follows that all conditions of Theorem 8 are satisfied. Hence the zero solution of system (32) is also -exponentially stable by Theorem 8.

Remark 16. It follows from Example 15 that the zero solution of system (30) without impulses is unstable; however, after impulsive effect, the zero solution becomes -exponentially stable. This implies that impulses may be used to exponentially stabilize dynamic equations on time scales.

We give the following example to illustrate Theorem 13.

Example 17 (see [13, Example]). Let and . We consider the impulsive dynamic system on time scales where for each . In case with , then the system (34) rewrites where . Letting and and employing similar manner in [13, Example], it follows that the zero solution of (34) is exponentially stable.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The first author was supported by Research Fund for the Doctoral Program of Harbin University of Commerce (no. 14RW06). The third author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2007585). The authors are thankful to the anonymous referees for their valuable comments and corrections to improve this paper.