Abstract and Applied Analysis

Volume 2012 (2012), Article ID 613038, 14 pages

doi:10.1155/2012/613038

Research Article

A Generalized Nonuniform Contraction and Lyapunov Function

Fang-fang Liao,¹ Yongxin Jiang,² and Zhiting Xie²

¹Nanjing College of Information Technology, Nanjing 210046, China

²Department of Mathematics, College of Science, Hohai University, Nanjing 210098, China

Received 19 November 2012; Accepted 1 December 2012

Academic Editor: Juntao Sun

Copyright © 2012 Fang-fang Liao et al. 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.

For nonautonomous linear equations , we give a complete characterization of general nonuniform contractions in terms of Lyapunov functions. We consider the general case of nonuniform contractions, which corresponds to the existence of what we call nonuniform -contractions. As an application, we establish the robustness of the nonuniform contraction under sufficiently small linear perturbations. Moreover, we show that the stability of a nonuniform contraction persists under sufficiently small nonlinear perturbations.

1. Introduction

We consider nonautonomous linear equations

where is a continuous function with values in the space of bounded linear operators in a Banach space . Our main aim is to characterize the existence of a general nonuniform contraction for (1.1) in terms of Lyapunov functions.

We assume that each solution of (1.1) is global, and we denote the corresponding evolution operator by , which is the linear operator such that

for any solution of (1.1). Clearly, and

We shall say that an increasing function is a growth rate if

Given two growth rates , we say that (1.1) admits a nonuniform -contraction if there exist constants and such that

We emphasize that the notion of nonuniform -contraction often occurs under reasonably weak assumptions. We refer the reader to [1] for details.

In this work, we mainly consider more general nonuniform contractions (see (2.1) below) and we give a complete characterization of such contractions in terms of Lyapunov functions, especially in terms of quadratic Lyapunov functions, which are Lyapunov functions defined in terms of quadratic forms. The importance of Lyapunov functions is well established, particularly in the study of the stability of trajectories both under linear and nonlinear perturbations. This study goes back to the seminal work of Lyapunov in his 1892 thesis [2]. For more results, we refer the reader to [3–6] for the classical exponential contractions and dichotomies, [7–9] for the nonuniform exponential contractions and nonuniform exponential dichotomies.

The proof of this paper follows from the ideas in [9, 10]. As an application, we provide a very direct proof of the robustness of the nonuniform contraction, that is, of the persistence of the nonuniform contraction in the equation

for any sufficiently small linear perturbation . We remark that the so-called robustness problem also has a long history. In particular, the problem was discussed by Massera and Schäffer [11], Perron [12], Coppel [3] and in the case of Banach spaces by Daletskiĭ and Kreĭn [13]. For more recent work we refer to [14–16] and the references therein.

Furthermore, for a large class of nonlinear perturbations with for every , we show that if (1.1) admits a nonuniform contraction, then the zero solution of the equation

is stable. The proof uses the corresponding characterization between the nonuniform contractions and quadratic Lyapunov functions.

2. Lyapunov Functions and Nonuniform Contractions

Given a growth rate and a function , we say that (1.1) admits a nonuniform -contraction if there exists a constant such that

The nonuniform -contraction is a special case of nonuniform -contraction with .

Now we introduce the notion of Lyapunov functions. We say that a continuous function is a strict Lyapunov function to (1.1) if

(1)for every and ,

(2)for every and ,

(3)there exists a constant such that for every and ,

The following result gives an optimal characterization of nonuniform -contractions in terms of strict Lyapunov functions.

Theorem 2.1. (1.1) admits a nonuniform -contraction if and only if there exists a strict Lyapunov function for (1.1).

Proof. We assume that there exists a strict Lyapunov function for (1.1). By and , for every and , we have

Therefore, (1.1) admits a nonuniform -contraction with .

Next we assume that (1.1) admits a nonuniform -contraction. For and , we set

By (2.1), we have . Moreover, setting , we obtain . This establishes . Furthermore, for , we have

Therefore, is a strict Lyapunov function for (1.1).

Next we consider another class of Lyapunov functions, namely, those defined in terms of quadratic forms.

Let be a symmetric positive-definite operator for each . A quadratic Lyapunov function is given as

Given linear operators , we write if for .

Theorem 2.2. Assume that there exist constants and such that

Then (1.1) admits a nonuniform -contraction (up to a multiplicative constant) if and only if there exist symmetric positive definite operators and constants such that is of class in and

Proof. We first assume that (1.1) admits a nonuniform -contraction. Consider the linear operators

for some constant . Clearly, is symmetric for each . Moreover, by (2.8), we have

Since is symmetric, we obtain

and therefore (2.10) holds. Since

we find that is of class in with derivative

which implies that

Therefore,

which establishes (2.11) with .

Now we assume that conditions (2.9) and (2.10)-(2.11) hold. Set . By (2.10), we have

Lemma  2.3. There exists a constant such that

Proof of Lemma  2.3. Note that

Hence, by condition (2.11), and the fact that we obtain

Now given , take such that with as in (2.9). Then

It follows from (2.9) that

Since , we have

which yields (2.20) with .

Lemma  2.4. For , one has

Proof of Lemma  2.4. By conditions (2.11) and (2.21), we have

Therefore,

It follows from Gronwall’s lemma that

which yields the desired result.

By Lemmas 2.3 and 2.4 together with (2.19), we obtain

and therefore,

which implies that (1.1) admits a nonuniform -contraction.

As an application of Theorem 2.2, we establish the robustness of nonuniform -contractions. Roughly speaking, a nonuniform contraction for (1.1) is said to be robust if (1.6) still admits a nonuniform contraction for any sufficiently small perturbation .

Theorem  2.5. Let be continuous functions such that (1.1) admits a nonuniform -contraction with condition (2.9). Suppose further that for every and

for some sufficiently small. Then (1.6) admits a nonuniform -contraction.

Proof. Let be the evolution operator associated to (1.6). It is easy to verify that

For every with , we have

Using Gronwall’s inequality, we obtain

for every with . Therefore condition (2.9) also holds for the perturbed equation (1.6).

Now we consider the matrices in (2.12). Condition (2.10) can be obtained as in the proof of Theorem 2.2. For condition (2.11), it is sufficient to show that

for some constant . Using (2.10) and (2.32), we have

and taking sufficiently small, we find that (2.36) holds with some .

3. Stability of Nonlinear Perturbations

Before stating the result, we fist prove an equivalent characterization of property . Given matrices for each , we consider the functions

whenever the derivatives are well defined and are given as (2.8).

Lemma 3.1. Let be functions. Then property is equivalent to

Proof. Now we assume that property holds. If and , then

Similarly, if is such that , then

This establishes (3.2).

Next we assume that (3.2) holds. We rewrite (3.2) in the form

which implies that

and hence property holds.

Theorem 3.2. Assume that (1.1) admits a nonuniform -contraction satisfying (2.9). Suppose further that there exists a constant such that and

Then for each , there exists such that

for every solution of (1.7).

Proof. For as in (2.12) and as in (2.8), we have, for every ,

Since , we have

Applying Lemma 3.1, we obtain

In particular, for ,

From the identity that for every and , we have

On the other hand,

Therefore,

and hence

Therefore, if is a solution of (1.7), then

If is small enough such that , then

and hence

It follows from Gronwall’s inequality that

Now given , take such that with as in (2.9). Then

Taking

then

It follows from (2.13) and (3.20) that

Now the proof is finished.

Acknowledgments

The authors would like to deliver great thanks to Professor Jifeng Chu for his valuable suggestions and comments. Y. Jiang was supported by the Fundamental Research Funds for the Central Universities.

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