Journal of Applied Mathematics

Volume 2013 (2013), Article ID 313029, 12 pages

http://dx.doi.org/10.1155/2013/313029

Research Article

Stability in Terms of Two Measures for Nonlinear Impulsive Systems on Time Scales

Kexue Zhang¹ and Xinzhi Liu²

¹School of Control Science and Engineering, Shandong University, Jinan 250061, China

²Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada, N2L 3G1

Received 18 November 2012; Accepted 7 January 2013

Academic Editor: Qiru Wang

Copyright © 2013 Kexue Zhang and Xinzhi Liu. 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.

We investigate some stability problems in terms of two measures for nonlinear dynamic systems on time scales with fixed moments of impulsive effects. Sufficient conditions for (uniform) stability, (uniform) asymptotic stability, and instability in terms of two measures are derived by using the method of Lyapunov functions. Our results include the existing results as special cases when the time scale reduces to the set of real numbers. Particularly, our results provide stability criteria for impulsive discrete systems in terms of two measures, which have not been investigated extensively. Two examples are presented to illustrate the efficiency of the proposed results.

1. Introduction

It is well known that the theory of impulsive differential equations provides a general framework for mathematical modeling of many real world phenomena [1, 2]. In particular, it serves as an adequate mathematical tool for studying evolution processes that are subjected to abrupt changes in their states. At the present time, the qualitative theory of such equations has been extensively studied. Many results on the stability and boundedness of their solutions are obtained [1–4]. Due to the needs of applications, the concepts of Lyapunov stability have given rise to many new notions, for example, partial stability, conditional stability, eventual stability, practical stability, and so on. A notion which unifies and includes the above concepts of stability is the notion of stability in terms of two measures which was initialed by Movchan [5]. Since the publication of Salvadori's paper [6], this unified theory in terms of two measures became popular. For a systematic introduction to the theory of stability in terms of two measures, refer to [7].

On the other hand, a theory of time scales or calculus on measure chains was introduced by Hilger in his Ph.D. thesis [8] in 1988, with the purpose of incorporating both the existing theory of dynamic systems on continuous and discrete time scales, namely, time scale as arbitrary closed subset of real numbers, and extending the existing theory to dynamic systems on generalized hybrid (continuous/discrete) time scales. The theory of time scales recently has gained much attention and is undergoing rapid development. Recently, various work has been done on the stability problem of dynamic systems on time scales [9–14]. For more details about the theory of time scales, refer to [15–17].

Motivated by the above discussion, in this paper, we will consider the stability problems in terms of two measures for impulsive systems on time scales. Several new stability criteria and instability criteria are obtained by using the method of Lyapunov functions. As far as we know, there are very few studies on stability analysis of impulsive discrete systems in terms of two measures. Moreover, our results can be applied to impulsive systems on other time scales in addition to the set of integers and the set of real numbers.

The rest of this paper is organized as follows. In Section 2, we introduce some basic knowledge of dynamic systems on time scales. In Section 3, we formulate the problem and present several definitions of stability and instability in terms of two measures. In Section 4, several ()-stability and instability criteria are established by employing the Lyapunov function approach. For illustration of our results, two examples are shown in Section 5. Finally, some conclusions are drawn in Section 6.

2. Preliminaries

In this section, we briefly introduce some basic definitions and results concerning time scales for later use.

Let be the set of real numbers, be the set of nonnegative real numbers, be the set of integers, be the set of nonnegative integers, , and be an arbitrary nonempty closed subset of . We assume that is a topological space with relative topology induced from . Then, is called a time scale.

Definition 1. The mappings defined as

are called forward and backward jump operators, respectively.

A nonmaximal element is called right-scattered (rs) if and right-dense (rd) if . A nonminimal element is called left-scattered (ls) if and left-dense (ld) if . If has a ls maximum , then we define , otherwise, .

Definition 2. The graininess function is defined by

Definition 3. For and , one defines the delta derivative of , to be the number (when it exists) with the property that for any , there is a neighborhood of (i.e., for some ) such that

A function is rd-continuous if it is continuous at rd points in and its left-side limits exist at ld points in . The set of rd-continuous functions will be denoted by . If is continuous at each rd point and each ld point, is said to be continuous function on . If , then one defines the interval on by . Open intervals and half-open intervals can be defined similarly.

Definition 4. Let . A function is called the antiderivative of on if it is differentiable on and satisfies for all . In this case, one defines

One says that a function is regressive if for all . The set of all regressive and rd-continuous functions is denoted by , and the set of all positively regressive elements of is denoted by for all .

Definition 5. If , then one defines the exponential function on time scale by

where the cylinder transformation

where is the principal logarithm function.

It is known that is the unique solution of the initial value problem , .

Remark 6. Let be a constant. If , then for all . If , then for all .

Definition 7. One says that a function is right-nondecreasing at a point provided

(i)if is rs, then ;(ii)if is rd, then there is a neighborhood of such that

Similarly, one says that is right-nonincreasing if above in (i) and in (ii) . If is right-nondecreasing (right-nonincreasing) at every , one says that is right-nondecreasing (right-nonincreasing) on .

Lemma 8. Let . Then is right-nondecreasing (right-nonincreasing) on if and only if for every , where

Proof. The condition is obviously necessary. Let us prove that it is sufficient. We only assume for as the second statement can be shown similarly.

If is rs, then

and hence .

Let now to be rd, and be a neighborhood of . We need to show that for with . This follows directly from Lemma 1.1.1 in [7].

Thus the proof of the lemma is complete.

3. Problem Formulation

Consider the following nonlinear impulsive system on time scale :

under the following assumptions.

(a) is a time scale with as minimal element and no maximal element.(b),   and .(c) and . If is rd point, denotes the right limit of at ; if is rs point, denotes the state of at with the impulse. If is ld point, denotes the left limit of at with if is ls point. Here, we assume that .(d) is continuous in for , , and for each , , ;(e) and .

Throughout this paper, we denote by the solution of system (10) satisfying initial condition . Obviously, system (10) admits the trivial solution. Moreover, is assumed to satisfy necessary assumptions so that the following initial value problems:

have unique solutions , , and , , , respectively (e.g., see [17] for existence and uniqueness results for dynamical systems on time scales.). Thus, if we define

then it is easy to see that is the unique solution of system (10).

Let us list the classes of functions and definitions for convenience.

, continuous on and exists for ; , strictly increasing and ;  for each and for each ;  for each for each and ; , continuous on , , and for all and , exists}.

For , , , we define the upper right-hand Dini delta derivative of relative to (10) as follows:

Definition 9. Let . Then one says that

(i)  is finer than if there exists a constant and a function such that implies ;(ii)  is weakly finer than if there exists a constant and a function such that implies .

Definition 10. Let and . Then is said to be

(i)-positive definite if there exist a and a function such that implies ;(ii)-decrescent if there exist a and a function such that implies ;(iii)-weakly decrescent if there exist a and a function such that implies .

Definition 11. The impulsive system (10) is said to be

(S₁)  -stable, if for each , , there exists a such that implies , for any solution of (10);(S₂)  -uniformly stable, if the in (S₁) is independent of ;(S₃)  -attractive, if for each , , there exist two positive constants and such that implies , ;(S₄)  -uniformly attractive, if (S₃) holds with and being independent of ;(S₅)  -asymptotically stable, if (S₁) and (S₃) hold simultaneously;(S₆)  -uniformly asymptotically stable, if (S₂) and (S₄) hold together;(S₇)  -unstable, if (S₁) fails to hold.

4. Main Results

Let us establish, in this section, sufficient conditions for -(uniform) stability, -(uniform) asymptotic stability, and -instability properties of impulsive systems (10) in the following subsections, respectively. Let

4.1. -(Uniform) Stability

Theorem 12. Assume that

(i), and is weakly finer than ; (ii), is -positive definite on , -weakly decrescent, locally Lipschtiz in for which is rd, and

 where , ; (iii) there exists , such that

 where , ; (iv) where ; (v) there exists a constant , , such that implies ; (vi) there exists a constant , , such that implies .

 Then system (10) is -stable.

Proof. Since is -weakly decrescent, there exist a constant and a function such that

There exists, in view of (ii), a function such that

By (i), there exist and such that

Let and be given. There exists such that

Choose . Let such that and be any solution of (10). Then, from (17) to (20), we get

which implies .

We now claim that, for every solution of (10), implies

If this is not true, then there exist a solution with and a such that , for some , satisfying

Since , it follows form condition (vi) that

where and . Next, we will show that there exists a , , such that

To do this, we consider the following two cases:

(1)there exists a , , such that ;(2) for all .

Case  1. Let . As , we know that . If is left-dense, from the selection of , we know that there exists a left-hand neighborhood for some , such that for all . Then, we can choose .

If is left-scattered, from the selection of and , we know that and . Here, we claim that . If this is not true, that is, , form condition (v), we know that

which is a contradiction. Thus, . Then, we can choose .

Case  2. If for all , then we can choose .

Hence, we can find a , , such that (25) holds.

For and by conditions (ii) and (iii), we obtain

By (27), we will show that

To do this, we apply the induction principle ([17], Theorem 1.7) on to the statement

(1) The statement is true since .

(2) Let be rs and be true. We have to prove that is true.

By the definition of upper right-hand derivative, we see that

then

which implies that is true.

(3) Let be rd, be true and be a neighborhood of . We need to show that is true for , . By (27) and Remark 6, we get

which implies that is true.

(4) Let be ld and be true for all . We need to show that is true. By the continuous property of function and the exponential function, it follows that

which implies that is true.

Hence, we conclude that (29) is true.

Similarly, we can prove that

Then, by (28), (35), and (20), we obtain

that is, . Thus, by (18) and (25),

which is a contradiction. Therefore (22) is true and system (10) is -stable.

Theorem 13. Assume that all conditions of Theorem 12 hold with the following changes:

  is finer than ;   is -decrescent.

Then, system (10) is -uniformly stable.

Proof. From conditions (i)* and (ii)*, the number in the proof of Theorem 12 can be chosen independent of . Then following the same reasoning of Theorem 12, we can get the -uniform stability of system (10). The details are omitted.

If in condition (ii) of the previous theorems, then the Lyapunov function is monotone along the solutions of system (10) in each impulsive intervals. In this case, we have the following conservative result.

Corollary 14. Assume that

(i), , is -positive definite on , locally Lipschtiz in for each which is rd, and for and ; (ii) for , where and ,

and conditions (v), (vi) of Theorem 12 hold. Then

(A)if, in addition, is weakly finer than , and is -weakly decrescent, then system (10) is -stable;(B)if, in addition, is finer than , and is -decrescent, then system (10) is -uniformly stable.

Proof. Notice

where . Then, by Theorems 12 and 13, the result holds.

Remark 15. The continuous version of Corollary 14 with , , can be found in [7], while the discrete one for impulsive discrete systems is brand new, and the discrete version of Corollary 14 with , , reduces to Theorems and in [18] for discrete systems with no impulses.

4.2. -(Uniform) Asymptotic Stability

Theorem 16. Assume that conditions (v), (vi) of Theorem 12 and condition (i) of Corollary 14 hold and the following conditions are satisfied:

(i) is weakly finer than , and is -weakly decrescent; (ii) for , where , , , and if .

Then system (10) is -asymptotically stable.

Proof. By Theorem 12, system (10) is -stable. Thus, for , there exists such that implies , . To prove the theorem, it remains to show that .

Let . Then it follows from assumptions that is right-nonincreasing and bounded from below, and consequently exists. If for some solution of (10), we let . Then, by condition (ii), we have

Thus we obtain from (39) that

which implies, in view of the assumption , that . This is a contradiction. Thus we must have and consequently . Hence system (10) is -attractive and the proof is complete.

In the following theorems, two auxiliary functions of class are used to investigate the -asymptotic stability property of system (10).

Theorem 17. Let conditions , of Theorem 12 hold and assume that

(i) and is weakly finer than ; (ii) there exists a function such that is locally Lipschtiz in for each which is rd, -positive definite on , -weakly decrescent and

 where , ; (iii), , where and ; (iv) is -positive definite on , locally Lipschtiz in for every which is rd and

 where and .

Then system (10) is -asymptotically stable.

Proof. From Corollary 14, it follows that system (10) is -stable. Thus, for , there exists such that implies , . To prove the theorem, it remains to show that for every solution of (10) with , .

Suppose that this is not true. Then there exists a sequence diverging to as and such that for some positive number . From condition , we know that there exists a function such that , if . Then

For any given , there exists a such that . Then for , there exists a such that . If , from (43) and condition (iv), we have

where . If , we have

Following this procedure, we conclude that

Let

Then, by condition (ii),

which implies

for .

Hence, for , we obtain, form (46), (49), and condition (iii),

where . This is a contradiction, hence . Theorem 17 is proved.

In Theorem 17, the function may have a special form. In the case when and , , we deduce the following corollary.

Corollary 18. Let conditions (v), (vi) of Theorem 12 hold and assume that

(i) and is weakly finer than ; (ii)there exist , and function such that is locally Lipschtiz in for each which is rd, -positive definite on , -weakly decrescent and

(iii), , where and .

Then system (10) is -asymptotically stable.

Theorem 19. Assume that all conditions of Theorem 13 hold. Suppose further, that there exists a function such that is locally Lipschtiz in for each which is rd, and the following conditions hold:

(i), , , where , , , and ; (ii), , where and .

Then system (10) is -attractive.

Proof. By Theorem 13, system (10) is -uniformly stable. Thus, for , there exists a such that implies that , , where is any solution of (10).

Let be given, be the same as defined in the definition of -uniform stability, and . We claim that there exists a such that

If this is not true, then for all .

Let

for . By condition (i), we obtain

which implies that, for , ,

Then, it follows from (55) and condition (ii) that, for ,

where . Then, (56) implies that , for . This contradiction shows that (52) is true, and hence,

Thus we conclude that system (10) is -attractive.

Remark 20. When , and , , Theorem 19 contains Theorem  3.4 in [18] for discrete systems without impulse effects.

Next, we will give two results on uniform asymptotic stability in terms of two measures.

Theorem 21. Let all the conditions of Theorem 19 and the following additional conditions hold:

(i) is -decrescent, and   ( is a positive constant), for ; (ii)there exists a constant such that

Then system (10) is uniformly asymptotically stable.

Proof. Since is -decrescent, there exist and a function such that

By Theorem 19, system (10) is -uniformly stable. Thus, there exists a such that implies , , for any solution of (10).

Let be given and be the same as defined in the definition of -uniformly stability. Let be the smallest integer such that

where and .

Choose and let be any solution of (10) with . We claim that there exists a such that . If this is not true, then for all . By (56), (60), and condition (ii), we have

which is a contradiction. Thus, our claim is true and by the uniform stability we have

Hence, system (10) is -uniformly attractive. This completes the proof.

Theorem 22. Let conditions (v), (vi) of Theorem 12 hold and assume that

(i) and is finer than ; (ii)there exist a , and function such that is locally Lipschtiz in for each which is rd, -positive definite on , -decrescent and

(iii), , where and .

Then system (10) is -uniformly asymptotically stable.

Proof. Since is -decrescent, there exist a constant and a function such that

The fact that is -positive definite on implies that there exists a function such that

It follows from Corollary 14 that system (10) is -uniformly stable. Thus for , there exist a such that implies

By the choice of , we get

To prove the theorem, it is enough to show that system (10) is -uniformly attractive.

Given , let be the same as defined in the definition of -uniformly stability. Then for any solution of system (10) with , we claim that there exists a such that, for some ,

where . Suppose that this is false. Then for any there exists a solution of (10) satisfying , such that

By setting

and condition (ii), we have

which implies , for . Then, for , we get

where . Hence, for , we obtain

which is a contradiction. Hence there exists a number such that . Then for , thus for as well, we have

that is, holds for which means that system (10) is uniformly attractive. Theorem 22 is proved.

4.3. -Instability

Theorem 23. Assume that

(i), , is locally Lipschtiz in for each which is rd, -positive definite on , and

 and for any and , there is such that for implies for ; (ii) for , ,

 where with , is nondecreasing and , for .

Then system (10) is -unstable.

Proof. Let us assume on the contrary that system (10) is -stable. Then for , there exists a such that implies that for .

By setting , we know from condition (i) and (ii) that is right-nondecreasing for . Then, , if . Thus, it follows from condition (i) that there exists a such that for . Since is -positive definite on , there exists such that , if . Then, we have

From condition (ii) and (77), we have

and so

which implies that, for a given number , there exists a such that for . Thus by condition (i), there is a such that for which is a contradiction to the -stability. Therefore, system (10) is -unstable.

Theorem 24. Assume that

(i), , is locally Lipschtiz in for each which is rd, -decrescent on , and

 where and ; (ii) there exist , , such that , for , and

 (iii) for any ,

 where and .

Then system (10) is -unstable.

Proof. For the sake of contradiction, we assume that system (10) is -stable. Then for , there exists a such that implies for . Since is -decrescent on , there exists a function such that

Let and , for , . Then it follows from condition that

which implies

Since , it follows from (85), that

From condition (ii), we have

which, together with (86) and condition (iii), yields

Thus,

which contradicts (83). Therefore, system (10) is -unstable and the proof is complete.

5. Examples

In this section, as applications of the above-derived theoretical criteria, two representative examples are given as follows.

Example 25. Consider the system

on time scale .

Let . Then we have

where , and

that is,

where and .

If , then by (91) we have

Then for and , we have

From Corollary 14, we conclude that system (90) is -uniformly stable. It should be noted that -stability in this case implies, by the choice of and , that the trivial solution of (90) is partially stable with respect to .

If, for a given positive constant , , then we have

Then for and , we have

By Corollary 18, we conclude that system (90) is -asymptotically stable. Moreover, since , we conclude, by Theorem 22, that system (90) is -uniformly asymptotically stable.

Example 26. Consider the following system [14]

on time scale such that , for all .

Let , then we get

where , . Substituting (98) into (99) yields

When , we have

where and .

For and , we have

Then, for any and , implies that . Hence, all the conditions of Theorem 23 are satisfied, and system (98) is -unstable.

6. Conclusions

We have generalized the concepts of stability in terms of two measures relative to ordinary impulsive systems to impulsive systems on time scales. By employing the approach of Lyapunov function, we have established some criteria ensuring stability and instability in terms of two measures for nonlinear impulsive systems on time scales. Two examples have been worked out to demonstrate the main results.

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