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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 853134, 10 pages
http://dx.doi.org/10.1155/2013/853134
Research Article

Stability of a Functional Differential System with a Finite Number of Delays

1CEITEC BUT, Brno University of Technology, Technicka 3058/10, 61600 Brno, Czech Republic
2Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technicka 8, 61600 Brno, Czech Republic

Received 22 December 2012; Accepted 23 May 2013

Academic Editor: Zhiming Guo

Copyright © 2013 Josef Rebenda and Zdeněk Šmarda. 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 paper is devoted to the study of asymptotic properties of a real two-dimensional differential system with unbounded nonconstant delays. The sufficient conditions for the stability and asymptotic stability of solutions are given. Used methods are based on the transformation of the considered real system to one equation with complex-valued coefficients. Asymptotic properties are studied by means of Lyapunov-Krasovskii functional. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one or more constant delays or one nonconstant delay were studied.

1. Introduction

There are a lot of papers dealing with the stability and asymptotic behaviour of -dimensional real vector equations with delay. Among others we should mention the recent results [113]. Since the plane has special topological properties different from those of -dimensional space, where or , it is interesting to study asymptotic behaviour of two-dimensional systems by using tools which are typical and effective for two-dimensional systems. The convenient tool is the combination of the method of complexification and the method of Lyapunov-Krasovskii functional. Using these techniques we obtain new and easy applicable results on stability, asymptotic stability, or boundedness of solutions of real two-dimensional differential system where are real functions, , are real square matrices, and , is a real vector function, , . It is supposed that the functions , are locally absolutely continuous on , are locally Lebesgue integrable on , and the function satisfies Carathéodory conditions on .

Delayed differential equations recently gain more importance in applications in science and real world. They can be found in applications in medicine (control of drug therapies and neurological, physiological, and epidemiological models), biology (predator-prey models and blowflies lifecycle), chemistry (chemical kinetics), physics (private communication and signal masking), and engineering (machining operation on a lathe). Equation (1) represents a generalization of many of these models. Particularly, (1) in this general form has an application in modeling of multiple regenerative effect in tool chatter. Obtained results on stability give the possibility to find the best spindle speeds and depth-of-cut for the machines for chatter-free high-productivity operation. For more details, see [14].

The main idea of the investigation, the combination of the method of complexification and the method of Lyapunov-Krasovskii functional, was introduced for ordinary differential equations in the paper by Ráb and Kalas [15]. The principle was transferred to differential equations with delay by Kalas and Baráková [16]. The results for several constant delays can be found in papers by Rebenda [17, 18]. Differential equations with one nonconstant delay are studied by Kalas [19] and Rebenda [20].

We extend such type of results to differential equations with a finite number of nonconstant delays. We introduce the transformation of the considered real system to one equation with complex-valued coefficients. We present sufficient conditions for stability and asymptotic stability of a solution and the conditions under which all solutions tend to zero. The applicability of the results is demonstrated with an example.

At the end of this introduction we append an overview of notations used in the paper and the transformation of the real system to one equation with complex-valued coefficients.

Consider the following:: set of all real numbers,: set of all positive real numbers, : set of all nonnegative real numbers, : set of all negative real numbers, : set of all nonpositive real numbers, : set of all complex numbers, : class of all continuous functions , : class of all locally absolutely continuous functions , : class of all locally Lebesgue integrable functions , : class of all functions satisfying Carathéodory conditions on ,   : real part of , : imaginary part of , : complex conjugate of .

Introducing complex variables , , we can rewrite system (1) into an equivalent equation with complex-valued coefficients: where for , , , , .

The relations between the functions are as follows:

Conversely, putting equation (2) can be written in real form (1) as well.

2. Preliminaries

We consider (2) in the case when and study the behavior of solutions of (2) under this assumption. This situation corresponds to the case when the equilibrium of the autonomous homogeneous system where is supposed to be regular constant matrix, is a centre or a focus.

This case is included in the case considered in [21], but in this special case, we are able to derive more useful results as we will see later in an example. The idea is based on the well-known result that the condition in an autonomous equation ensures that zero is a focus, a centre, or a node while under the condition zero can be just a focus or a centre. Details are found in [15].

A simple example shows that, in some cases, the results of this paper can be applied more suitably than those given in [21].

Regarding (5) and since the delay functions satisfy , there are numbers , , and such that

Denote

Notice that, unlike the function introduced in [21], the previously defined function need not be positive.

Since and , the inequality is valid for . It can be easily verified that .

For the rest of this section, denote that

The stability and asymptotic stability are studied under the following assumptions.(i) The numbers , , and are such that (7) holds. (ii) There exist functions , , such that for , , , where , . (iii) is a function satisfying where is defined for by (iv) There exists a function which satisfies the inequalities , for almost all , where the function is defined by

If , , , are locally absolutely continuous on and on , the choice is admissible in (iii).

From the assumption (i), it follows that hence, the function is locally Lebesgue integrable on . Moreover, if and , then we can choose in (iv).

Finally, if in (ii), then (2) has the trivial solution . Notice that in this case the condition (ii) implies that the functions , are nonnegative on for , and due to this, on . The case is omitted since it can be replaced by .

3. Main Results

The aim is to generalize the results for ordinary differential equations published in [15] as well as the results contained in [16] (one constant delay), [18] (a finite number of constant delays), and [20] (one nonconstant delay). In the proof of the crucial theorem, we use the following auxiliary result.

Lemma 1. Let , , , and . Then for , .

The proof of Lemma 1 can be found, for example, in [15, page 131] or [17, page 101].

Theorem 2. Let the conditions (i), (ii), (iii), and (iv) hold and . (a) If then the trivial solution of (2) is stable on . (b) If then the trivial solution of (2) is asymptotically stable on .

Proof. Choose arbitrary . Let be any solution of (2) satisfying the condition for , where is a continuous complex-valued initial function defined on . Consider the Lyapunov functional where
To simplify the computations, denote that and write the functions of variable without brackets, for example, instead of .
From (20) we get for almost all for which is defined and exists.
Denote that and . It is clear that the derivative exists for almost all ; hence, we focus on the set .
In view of (9) we have for . For almost all , we compute where
Hence, has one-sided derivatives a.e. in . According to [22, Chapter IX., Theorem (1.1)] or [23], the set of all such that can be at most countable; thus, the derivative exists for almost all , and for these , .
In particular, the derivative exists for almost all for which is defined; thus, (22) holds for almost all for which is defined.
Now return the attention to the set . For almost all , it holds that . As is a solution of (2), we have for almost all .
Short computation gives , and from this we get for almost all .
Applying Lemma 1 to the last term, we obtain
Using this inequality together with (13), assumption (ii), and the relation , we obtain for almost all .
Consequently, for almost all .
Recalling that for almost all , we can see that inequality (29) is valid for almost all for which is defined.
From (29) we have
As fulfills condition (12), we obtain Hence, for almost all for which the solution exists.
Notice that, with respect to (9), for all for which is defined.
Suppose that condition (18) holds, and choose arbitrary . Put where , , and are the numbers from condition (i).
If the initial function of the solution satisfies , then the multiplication of (32) by and the integration over yield for all for which is defined. From (33) and (35) we get that is,
Thus, we have for all , and we conclude that the trivial solution of (2) is stable.
Now suppose that condition (19) is valid. Then, in view of the first part of Theorem 2, for , there is a such that implies that the solution of (2) exists for all and satisfies , where is arbitrary real constant. Hence, from this and (33), we have for all . This inequality, with condition (19), gives which completes the proof.

Remark 3. Theorem 2 represents a generalization of previous results.
If we take , , for , , , for , and , where , we get Theorem 4 from [16].
If we take , where , , we obtain Theorem 1 from [18].
If we take , , for , , , for , and , we get Theorem 2.2 from [20].

The next theorem involves the function in (ii); thus, it is more general than Theorem 2. A part of the proof of Theorem 2 is utilized in the proof of Theorem 4.

Theorem 4. Let the assumptions (i), (ii), (iii), and (iv) hold and where is any solution of (2) defined for . Then for , where .

Proof. Following the proof of Theorem 2, we have a.e. on . Using this inequality, we get a.e. on . Multiplying (43) by gives a.e. on . Integration over yields and multiplying (45) by , we obtain
The statement now follows from (33).

Remark 5. Theorem 4 generalizes theorems contained in previous papers.
If we take , , for , , , for , and , where , we get Theorem 2 from [16].
If we take , where , , we obtain Theorem 2 from [18].
If we take , , for , , , for , and , we get Theorem 2.7 from [20].

The last of the main propositions gives the conditions under which all solutions of (2) tend to zero.

Theorem 6. Let the assumptions (i), (ii), (iii), and (iv) be satisfied. Let a.e. on , where . If then any solution of (2) existing for satisfies

Proof. Choose arbitrary . According to (47), there is such that for and for . From (47) we have as ; hence, there is such that for . Considering this fact and (41), we get for . This completes the proof.

Remark 7. Theorem 6 is a generalization of results published in the papers [16, 18, 20].
If we take , , for , , , for , and , where , we get Theorem 3 from [16].
If we take , where , , we obtain Theorem 3 from [18].
If we take , for , , for , and , we get Theorem 2.14 from [20].

4. Corollaries and Examples

From Theorem 2 we easily obtain several corollaries. We give an example which shows that it is worth to consider the case (5).

Corollary 8. Let , , . Suppose that , , for , where . Let be such that for and .
Let satisfy a.e. on for . If then the trivial solution of (2) is stable. If then the trivial solution of (2) is asymptotically stable.

Proof. Choose such that for . Denote that and again. Since are constants, then also and are constants, and we have . Using condition (51) we get and it follows that condition (ii) holds with and .
Condition (53) implies that . Since in view of (14) we obtain and the assertion follows from (16) and Theorem 2.

Now we show an example that, under certain circumstances, Corollary 8 is more useful than Corollary 1 from [21].

Example 9. Consider (2), where , , , , for , and

Assume that and , . Put . Then, , . We have for where ; hence, we cannot apply Corollary 1 from [21].

On the other hand, if we use where , we have thus, Corollary 8 guarantees the stability and also asymptotic stability of the trivial solution of the considered equation.

The following corollary gives sufficient conditions for stability of the trivial solution of (2).

Corollary 10. Assume that the conditions (i), (ii), and (iii) are valid with . If is monotone and bounded on and if where , then the trivial solution of (2) is stable.

Proof. Suppose firstly that is nonincreasing on . Then, a.e. on .
If for some , then on . Consequently, has to satisfy only the inequality a.e. on , so we may choose on . It follows that .
On the other hand, if on , we may put . Then,
In both cases, satisfies condition (iv) and the inequality on ; hence,
Now assume that is nondecreasing on . Then, a.e. on .
If on , we may treat it as previously mentioned.
Otherwise, there is some such that on , and we may choose on . Clearly satisfies condition (iv) on . Since a.e. on , it follows that a.e. on . Hence, and then since is bounded on .
The statement follows from Theorem 2.

We can derive several consequences from Theorem 4.

Corollary 11. Let the conditions (i), (ii), (iii), and (iv) be fulfilled and for some .
If is any solution of (2) existing for , then

Proof. From the assumptions and (45) we can see that there are and such that for we have
Then,

Corollary 12. Let the assumptions (i), (ii), (iii), and (iv) hold, and let where . If is any solution of (2) existing for , then .

Proof. In view of (73), there are , , and such that for and for . From (41) we get

Remark 13. If , we can take in the proof of Corollary 12, and taking inequalities (74) into account we obtain the following statement: there is an such that holds for the solution of (2).

5. Conclusion

We studied asymptotic behavior of real two-dimensional differential system with a finite number of nonconstant delays. We considered the case corresponding to the situation when the equilibrium point of autonomous system (6) is a stable focus or a stable centre. We utilized the method of complexification and the method of Lyapunov-Krasovskii functional. Criteria for stability and asymptotic stability of the solutions as well as conditions ensuring that all solutions of (2) tend to zero are derived. At the end we supplied several corollaries and an example which shows that in some cases the criteria obtained in this paper are more applicable than the criteria presented in [21].

Acknowledgments

The first author was supported by the Project CZ.1.07/2.3.00/30.0039 of Brno University of Technology. The work of the second author was realised in CEITEC—Central European Institute of Technology—with research infrastructure supported by the Project CZ.1.05/1.1.00/02.0068 financed from European Regional Development Fund and by the Project FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology. This support is gratefully acknowledged.

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