Abstract and Applied Analysis
Volume 2011, Article ID 134072, 9 pages
http://dx.doi.org/10.1155/2011/134072
Research Article

On Stability of Linear Delay Differential Equations under Perron's Condition

1Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, Brno 602 00, Czech Republic
2Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Brno 616 00, Czech Republic
3Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey

Received 18 January 2011; Accepted 22 February 2011

Academic Editor: Miroslava Růžičková

Copyright © 2011 J. Diblík and A. Zafer. 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 stability of the zero solution of a system of first-order linear functional differential equations with nonconstant delay is considered. Sufficient conditions for stability, uniform stability, asymptotic stability, and uniform asymptotic stability are established.

1. Introduction

We begin with a classical result for the linear system where is an matrix function defined and continuous on . By , we will denote the set of bounded functions defined and continuous on and by the Euclidean norm.

In 1930, Perron first formulated the following definition being named after him.

Definition 1.1 (see ). System (L1) is said to satisfy Perron's condition () if, for any given vector function , the solution of is bounded.

The following theorem by Bellman  is well known.

Theorem 1.2 (see ). If () holds and for some positive number , then the zero solution of (L1) is uniformly asymptotically stable.

The proof is accomplished by making use of the basic properties of a fundamental matrix, the Banach-Steinhaus theorem, and the adjoint system where denotes the transpose of .

It is shown by an example in  that Theorem 1.2 may not be valid if the function appearing in (N1) is replaced by a constant vector. However, such a theorem is later obtained in  under a Perron-like condition.

Theorem 1.2 is extended by Halanay  to linear delay systems of the form where , are matrix functions defined and continuous on and is a positive real number.

Definition 1.3. System (L2) is said to satisfy Perron's condition () if for any given vector function , the solution of satisfying , , is bounded.

Theorem 1.4 (see ). If () holds, , and for some positive numbers and , then the zero solution of (L2) is uniformly asymptotically stable.

The method used to prove Theorem 1.4 is similar to Bellman's except that the adjoint system is not constructed with respect to an inner product but the functional For some extensions to impulsive differential equations, we refer the reader in particular to [6, 7].

In this paper, we consider the more general linear delay system where and are matrix functions defined and continuous on and is a continuously differentiable increasing function defined on satisfying and . We set . Obviously, and increases on and .

Perron's condition takes the following form.

Definition 1.5. System (1.4) is said to satisfy Perron's condition () if, for any given vector function , the solution of satisfying , is bounded.

A natural question is whether the zero solution of (1.4) is uniformly asymptotically stable under Perron's condition (). It turns out that the answer depends on the delay function .

The paper is organized as follows. In Section 2, we only state our results; the proofs are included in Section 5. We define an adjoint system and give a variation of parameters formula in Section 3 to be needed in proving the main results. Section 4 contains also some lemmas concerning Perron's condition and a relation useful for changing the order of integration.

2. Stability Theorems

The conclusion obtained by Bellman and Halanay for systems (L1) and (L2), respectively, is quite strong. We are only able to prove the stability of the zero solution for more general equation (1.4) under Perron's condition. To get uniform stability or asymptotic stability or uniform asymptotic stability, we impose restrictions on the delay function.

For our purpose, we denote

Theorem 2.1. Let hold. If there are positive numbers and such that then the zero solution of (1.4) is stable.

Theorem 2.2. Let hold. If (2.2) is satisfied and if there exists a positive real number such that then the zero solution of (1.4) is uniformly stable.

Theorem 2.3. Let hold. If (2.2) and are satisfied, then the zero solution of (1.4) is asymptotically stable.

Theorem 2.4. Let hold. If (2.2), (2.3), and are satisfied, then the zero solution of (1.4) is uniformly asymptotically stable.

Remark 2.5. Note that if , then and hence the conditions (2.3), (2.4), and (2.5) are automatically satisfied. In this case, all theorems become equivalent, that is, the zero solution is uniformly asymptotically stable. Thus, the results obtained by Bellman and Halanay are recovered.

3. Variation of Parameters Formula

To establish a variation of parameters formula to represent the solutions of (1.5), one needs an adjoint system. The following lemma helps to define the adjoint of (1.4).

Lemma 3.1. Let be a solution of (1.4). If is a solution of then where

Proof. Verify directly.

Definition 3.2. The system (3.1) is said to be adjoint to system (1.4).

It is easy to see that the adjoint of system (3.1) is system (1.4); thus the systems are mutually adjoint to each other.

Lemma 3.3. Let be a matrix solution of (3.1) for satisfying and for . Then is a solution of (1.5) if and only if

Proof. Replacing by in (1.5) and then integrating the resulting equation multiplied by over , we have Comparing both sides and using which is true in view of for , we get and hence

It is not difficult to see from (3.4) that if is a matrix solution of (1.4) for satisfying and for , then Using this relation in Lemma 3.3 leads to the following variation of parameters formula.

Lemma 3.4. Let be a matrix solution of (1.4) for satisfying and for . Then is a solution of (1.5) if and only if

4. Auxiliary Results

Lemma 4.1. If holds, then there is a positive number such that

Proof. The proof follows as in . We provide only the steps for the reader's convenience.
Define for each rational number , .
In view of , the family of continuous linear operators from to is pointwise-bounded. For the space of bounded continuous functions , the usual sup norm is used.
By the Banach-Steinhaus theorem, the family is uniformly bounded. Thus, there is a positive number such that for every .
As the rational numbers are dense in the real numbers, for each there is such that as and so
The final step involves choosing a sequence of functions and using a limiting process.

Lemma 4.2. If (2.2) and (4.1) are true, then there is a positive number such that

Proof. From (3.1), we have Hence, by using (4.1), we see that for all ,

Lemma 4.3. Let be a continuous function satisfying for . Then

5. Proofs of Theorems

Let be given. For a given continuous vector function defined on , let denote the solution of (1.4) satisfying As usual,

Proof of Theorem 2.1. From Lemma 3.3, we may write In view of Lemma 4.2, it follows that Hence, the zero solution is stable.

Proof of Theorem 2.2. Using (2.3) in (5.4), we get from which the uniform stability follows.

Proof of Theorem 2.3. By Theorem 2.1, the zero solution is stable. We need to show the attractivity property.
From Lemma 3.3, for , we can write where Integrating with respect to from to , we have We change the order of integration by employing Lemma 4.3. After some rearrangements, we obtain It follows that In view of condition (2.4), we see from (5.10) that

Proof of Theorem 2.4. By Theorem 2.2, the zero solution is uniformly stable. From (5.10) and (2.3), we have Using condition (2.4) in the above inequality, we see that the zero solution is uniformly asymptotically stable as .

Acknowledgments

This research was supported by Grant P201/11/0768 of the Czech Grant Agency (Prague), by the Council of Czech Government MSM 0021630503 and MSM 00216 30519, and by Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology.

References

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5. A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York, NY, USA, 1966.
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