Dynamical Aspects of Initial/Boundary Value Problems for Ordinary Differential EquationsView this Special Issue
A Generalized Nonuniform Contraction and Lyapunov Function
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.
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  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 . 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 , Perron , Coppel  and in the case of Banach spaces by Daletskiĭ and Kreĭn . 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.
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
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 andfor 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.
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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