The Scientific World Journal

Volume 2014 (2014), Article ID 241034, 7 pages

http://dx.doi.org/10.1155/2014/241034

Research Article

Practical Stability in terms of Two Measures for Set Differential Equations on Time Scales

Peiguang Wang¹ and Weiwei Sun²

¹College of Electronic and Information Engineering, Hebei University, Baoding 071002, China

²College of Mathematics and Computer Science, Hebei University, Baoding 071002, China

Received 8 November 2013; Accepted 24 December 2013; Published 20 January 2014

Academic Editors: N. Henderson and Y. Wu

Copyright © 2014 Peiguang Wang and Weiwei Sun. 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 present a new comparison principle by introducing a notion of upper quasi-monotone nondecreasing and obtain the practical stability criteria for set valued differential equations in terms of two measures on time scales by using the vector Lyapunov function together with the new comparison principle.

1. Introduction

Stability theory in the sense of Lyapunov is now well known. Its basic theory and applications can be found in the monographs of Lasalle and Lefschetz [1] and Rouche et al. [2]. To unify a variety of stability concepts and to offer a general framework for the investigation of the stability theory, the notion of stability in terms of two measures has been proved to be very powerful (see the monograph [3] and the papers [4–6]).

The practical stability is a very important problem in the field of application, which deals with the question of whether the system state evolves within certain subsets of the state-space. It is very useful in estimating the worst-case transient and steady-state responses and in verifying pointwise in time constraints imposed on the state trajectories. Thus practical stability is concerned with quantitative analysis as opposed to Lyapunov analysis which is qualitative in nature. There are some relative results for practical stability of various dynamic systems. We can refer to the monograph of Lakshmikantham et al. [7] and the papers of Zhang and Sun [8], Wang and Liu [9], Wang et al. [10], Sun et al. [11], and Hristova and Georgieva [12] and the references cited therein.

Recently, the study of set differential equations in a semilinear metric space has gained much attention due to its applicability to multivalued differential inclusions and fuzzy differential equations and its inclusion of ordinary differential systems as a special case [13], and some basic results of interest are obtained in [14–21]. However, we notice that there are very few results for set valued differential equation on time scales. For example, Girton [22] gave the results of the existence and uniqueness of the solution of an initial value problem that involve set valued differential equations on time scales. Ahmad and Sivasundaram [23] and Hong [24, 25] discussed some basic problems of set valued differential equation on time scales and obtained some stability criteria, respectively. In this paper, we present a new comparison principle by introducing a notion of upper quasi-monotone nondecreasing and obtain the practical stability criteria for set valued differential equations in terms of two measures on time scales by using the vector Lyapunov function together with the new comparison principle. Consequently, this paper is organized as follows. In Section 2, we introduce the concepts of the time scales and the set valued differential equations, the propositions of the set valued differential equations on time scales. In Section 3, the relatively new comparison principle and conditions of stability are given.

2. Preliminaries

Let be a time scale with as minimal element and no maximal element. Firstly, we give some relative definitions, which can be found in [1–3].

Definition 1. The mappings : defined as

are called jump operators.

Definition 2. A nonmaximal element is said to be right-scattered (rs) if and right-dense (rd) if . A nonminimal element is called left-scattered (ls) if and left-dense (ld) if .

Definition 3. Let

Definition 4. The mapping is called regulated, if in each left-dense the left sided and in each right-dense the right sided limit exist.

Definition 5. The mapping is called rd continuous if

(i)it is continuous at each right-dense ;(ii)at each left-dense point the left-sided limit exists.

Let denote the set of rd-continuous mappings from to .

Definition 6. The mapping is said to be right-dense (rd) continuous and is denoted by if

(i)it is continuous at each with right-dense or maximal ;(ii)the limits and exist at each with left-dense .

Let denote the collection of all nonempty, compact, and convex subsets of . Define the Hausdorff metric

where , are bounded sets in . We note that with this metric is a complete metric space.

It is known that if the space is equipped with the natural algebraic operations of addition and nonnegative scalar multiplication, then becomes a semilinear metric space which can be embedded as a complete cone into a corresponding Banach space.

The Hausdorff metric (3) satisfies the following properties:

for all and .

Definition 7. Let . The set satisfying is known as the Hukuhara difference of the sets and and is denoted by the symbol .

Definition 8. The forward derivative of : , denoted by , is defined by

Analogously, the backward derivative is defined as

Remark 9. The usual scalar subtraction occurs in the scalar coefficient at the beginning of the above expressions and the Hukuhara set difference occurs inside the brackets. Clearly, the existence of Hukuhara difference ensures the existence of the derivative in Definition 8; in the forthcoming analysis, we will be considering the forward derivative as the result for backward derivative followed immediately with suitable changes. The backward derivative has the following properties:

(i)let be differentiable at ; then ;(ii), .

Remark 10. If the sets in are singletons only, then there is only one selector possible, namely, itself. In this case, the integral reduces to the generalized integral from time scales into . The assumption generalizes into the integral for time scales. When is everywhere right-dense, then we have , which results in the conventional formulation for integration of set valued functions. Moreover, for , we have

(i);(ii) is closed and convex but need not be necessarily compact.

Definition 11. Let : . Then the Hausdorff distance between and , : , is -integrable and

Definition 12. Assume : is set valued function and . Let be an element of (provided it exists) with the property that given any , there exists a neighborhood of (i.e., for some ) such that

for all with , where . We call the -derivative of at . We say that is -differentiable at if its -derivative exists at . Moreover, we say is -differentiable on if its -derivative exists at each . The multivalued function : is then called the -derivative of on .

Proposition 13. Let  : ,   (provided it exists). Then some easy and useful relationships concerning the -derivative hold.

(i)If the  -derivative of    at    exists, then it is unique. Hence, the  -derivative is well defined.(ii)Assume  :     is a multivalued function and let  . Then one has the following.

(1)If    is  -differentiable at  , then    is continuous at  .(2)If    is continuous at    and    is right-scattered, then    is  -differentiable at    with

(3)If     is right-dense,   then    is  -differentiable at    if the limits

 exist and satisfy the equations

(4)If    is differentiable at  ,   then

Remark 14. Let : , (provided it exists). We consider the two cases and , where stands for the set consisting of all integers.

(1)If , then : is -differentiable at if and only if

exists, that is, if and only if is differentiable in the Hukuhara sense at . In this case, we have

(2)If , then : is -differentiable at with

where is the usual forward multivalued difference operator.

3. Main Results

Consider the space

together with either of the following metrics on the space :

where , .

Consider the initial value problem

where , , such that for each , , .

First of all, we define the following classes of function:

In order to discuss the stability of the solution of set valued differential systems (18), we state some notions and definitions.

On the vector Lyapunov function on time scales, we defined the Dini derivative of the function along with the solutions of (18) by

Definition 15. A function : is said to be upper quasi-monotone nondecreasing in , if and imply , where .

In the following, we will prove the comparison result in terms of vector Lyapunov functions relative to the set differential system on time scales.

Theorem 16. Assume that

, is locally Lipschitzian in ; that is, for , one has , where is an matrix of nonnegative elements, . Here, by one means the vector , where are the components of , ; satisfies

where is upper quasi-monotone nondecreasing in for each ;

, is the solution of

Then for any solution of (18), one has

provided .

Proof. Let be the solution of (22) for , . Set as an application of the properties of Hausdorff metric; we obtain the estimation

Letting , where is the Hukuhara difference of and for small and is assumed to exist, hence,

Consequently, we find that

where is the right-derivative of . Hence, we have

It is said that

Because , we can obtain that

The proof is complete.

Now, we list some definitions about stability which will be used in the following discussion.

Definition 17. Let , . Then one says that

(i) is finer than if there exists a and a function such that

(ii) is uniformly finer than if in (i) is independent of .

Definition 18. Let be positive constants . The system (18) is said to be

(i)practically stable if for any , the condition implies , for , where is any solution of (18);(ii)practically quasi-stable if for any and some with , the condition implies , , ;(iii)strongly practically stable if (i) and (ii) hold simultaneously;(iv)practically asymptotically stable if (i) holds and for any there exists such that and implies , .

Definition 19. Let be positive constants , . The system (18) is said to be

(PS1)-practically stable if for any , the condition implies , , for some , where is any solution of (18);(PS2)-practically quasi-stable if for any and some with , the condition implies , , ;(PS3)-strongly practically stable if (PS1) and (PS2) hold simultaneously;(PS4)-practically asymptotically stable if (PS1) holds and for any there exists such that and implies , .

One can similarly define corresponding notion for the system (22).

Definition 20. Let be positive constants , . Then we say that the system (22) is -practically stable if for any , the condition implies , , , where is any solution of (22).

Other practical stability notions can be defined similarly.

Theorem 21. Assume that

are positive constants and ;, , is nondecreasing in , and is uniformly finer than , , , where ;there exists , such that is locally Lipschitzian in for each right-dense and for . It holds that

for ,

where is upper quasi-monotone nondecreasing in for each and is the solution of (22);

 .

Then practical stability properties of system (22) imply the corresponding -practical stability properties of (18).

Proof. Assume that (22) is practically stable; then for given , we have

Then by and , it follows that . We claim that .

Indeed, if this were not true, there would exist a solution of (18) with and , , such that , , . As , Theorem 16 together with , implies that

This contradiction proves that implies , .

Next we prove that system (18) is -strongly practically stable. For given positive numbers , , , and , suppose that (22) is strongly practically stable for positive numbers , , , and ; this means we only need to prove -practical quasi-stability of system (18). Practical quasi-stability of (22) means that implies , with .

From the foregoing argument, since , if , we have for all if ; thus we have , provided . Hence system (18) is -strongly practically stable.

Finally, we show that system (18) is -practically asymptotically stable. Now, let us suppose that (22) is practically asymptotically stable. This implies we only need to prove that for any given , there exists with such that and implies , for system (18). Practical asymptotic stability of (22) means that

From the argument above, since whenever , we obtain

for all , if . Thus we have , , provided . Hence system (18) is -strongly practically stable.

The proof is complete.

Theorem 22. Suppose that the conditions of Theorem 21 are satisfied except that condition (A₃) is replaced by

and for

Then -practical stability properties of system (22) imply the corresponding -practical stability properties of the system (18).

Proof. Assume that (22) is -practically stable. Then we have , , such that

Suppose that the -practical stability of (18) does not hold; then, arguing as in Theorem 16, by (A ₂), (A ₆), we can show that

Then using (A ₆), we have

which is a contradiction. The proof is complete.

Conflict of Interests

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

Authors’ Contribution

All authors completed the paper together. All authors read and approved the final paper.

Acknowledgments

The authors would like to thank the reviewers for their valuable suggestions and comments. This paper is supported by the National Natural Science Foundation of China (11271106) and the Natural Science Foundation of Hebei Province of China (A2013201232).

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