A Converse Lyapunov Theorem and Robustness with Respect to Unbounded Perturbations for Exponential Dissipativity
 Xudong Li^{1} and
 Yuan Guo^{1}Email author
DOI: 10.1155/2010/726347
© X. Li and Y. Guo. 2010
Received: 18 December 2009
Accepted: 17 April 2010
Published: 23 May 2010
Abstract
This paper is concerned with the nonlinear system . We give a converse Lyapunov theorem and prove robustness of uniform exponential dissipativity with respect to unbounded external perturbations, without assuming being globally Lipschitz in .
1. Introduction
where is always assumed to be a continuous vector field which is locally Lipschitz in space variable . Our main aim is twofold: one is to give a converse Lyapunov theorem for uniform exponential dissipativity, and the other is to study robustness of uniform exponential dissipativity to unbounded perturbations.
In [1] Lyapunov introduced his famous sufficient conditions for asymptotic stability of (1.1), where we can also find the first contribution to the converse question, known as converse Lyapunov theorems. The answers have proved instrumental, over the years, in establishing robustness of various stability notions and have served as the starting point for many nonlinear control systems design concepts.

(F_{1}) There exists an such that(1.2)
where denotes the inner product in .
Then we study robustness property of uniform exponential dissipativity to perturbations. A basic problem in the dynamical theory concerns the robustness of global attractors under perturbations [4]. It is readily known that if a nonlinear system with a global attractor is perturbed, then the perturbed one also has an attractor which is near , provided the perturbation is sufficiently small; see, for instance, [5, 6], and so forth. However, in general we only know that is a local attractor. Whether (or under what circumstances) the global feature can be preserved is an interesting but, to the authors' knowledge, still open problem. (For concrete systems there is the hope that one may check the existence of global attractors by using the structure of the systems.) Since the dissipativity of a system usually implies the existence of the global attractor, in many cases the key point to answer the above problem is then reduced to examine the robustness of dissipativity under perturbations.
Such a problem has obvious practical sense. Unfortunately the answer might be negative even if in some simple cases which seem to be very nice at a first glance, as indicated in Example 1.1 below (from which it is seen that dissipativity can be quite sensitive to perturbations).
Example 1.1 (see [7]).
with goes to as .
Note that is bounded on ; hence, is globally Lipschitz.
In this present work we demonstrate that exponential dissipativity has nice robustness properties. Actually we will show that it is robust under some types of even unbounded perturbations.
This paper is organized as follows. In Section 2 we give a converse Lyapunov theorem mentioned above, and in Section 3 we prove robustness of exponential dissipativity.
2. A Converse Lyapunov Theorem
In this section we give a converse Lyapunov theorem which generalizes a recent result in [2]. Let us first recall some basic definitions and facts.
Lemma 2.1.
Proof.
This basic fact is actually contained in [3], and so forth. Here we give a simple proof for the reader's convenience.
The proof is complete.
We will denote by the solution operator of (1.1), that is, for each , is the unique solution of the system with initial value .
Definition 2.2.
The main result in this section is the following theorem.
Theorem 2.3 (Converse Lyapunov Theorem).
Suppose that satisfies the structure condition (F1). Assume that system (1.1) is uniformly exponentially dissipative.
for all , and , where , , , , , and are appropriate positive constants.
Moreover, if , namely, the system is uniformly exponentially asymptotically stable, then the constants , and vanish.
Proof.
where is independent of . This shows that satisfies (2.11).
which indicates that satisfies (2.12).
which completes the proof of (2.30).
This implies that is nonincreasing in .
Invoking (2.15), (2.21), and (2.25), we find that is a Lyapunov function satisfying all the required properties in the theorem.
In case , it can be easily seen from the above argument that .
The proof is complete.
Remark 2.4.
If we assume that is also locally Lipschitz in , then is locally Lipschitz in as well. Now assume that is locally Lipschitz in . Then by the construction of and one easily verifies that is locally Lipschitz in . Consequently has derivative in almost everywhere.
3. Robustness of Exponential Dissipativity to Perturbations
As for the applications of the converse Lyapunov theorem given in Section 2, we consider in this section the robustness of exponential dissipativity to perturbations.
3.1. Robustness to External Perturbations
where is a continuous function which corresponds to external perturbations.
for some . Our main result in this part is contained in the following theorem.
Theorem 3.1.
Assume that is locally Lipschitz in and satisfies (F1). Suppose that the system (1.1) is uniformly exponentially dissipative.
Then there exists an sufficiently small such that, for any , the perturbed system (3.1) is uniformly exponentially dissipative.
Remark 3.2.
namely, . Hence the conclusion of the theorem naturally holds.
Proof of Theorem 3.1.
Let be the Lyapunov function of the unperturbed system given in Theorem 2.3, and take , where and are the constants in (2.11) and (2.12). We show that for any the perturbed system (3.1) is uniformly exponentially dissipative.
for , where .
where , and .
Therefore, (3.22) and (3.23) complete the proof of what we desired.
As a direct consequence of Theorem 3.1, we have the following interesting result.
Corollary 3.3.
where , and is as in .
Then system (1.1) is necessarily not uniformly exponentially dissipative.
Proof.
is not dissipative. This leads to a contradiction and proves the conclusion.
3.2. The CohenGrossberg Neural Networks with Unbounded External Inputs and Disturbances

(H_{1}) are bounded and locally Lipschitz,

(H_{2}) each function belongs to ; moreover,(3.28)

(H_{3}) , and are bounded continuous functions.
Theorem 3.4.
where is a function as in (3.3), system (3.26) is uniformly exponentially dissipative.
Proof.
with . By (H2) one easily verifies that satisfies (F1); moreover, system (3.30) is exponentially dissipative. Let be the Lyapunov function of the system given by Theorem 2.3. We show that if is sufficiently small, then (3.26) is uniformly exponentially dissipative, provided (3.29) is fulfilled.
satisfies (3.3) with therein replaced by another appropriate constant .
Here denotes the usual norm of in . We omit the details.
The proof of the theorem is complete.
Remark 3.5.
The above result contains Theorem 3.1 in [9] as a particular case.
Declarations
Acknowledgments
The authors highly appreciate the work of the anonymous referees whose comments and suggestions helped them greatly improve the quality of the paper in many aspects. This paper is supported by NNSF of China (10771159).
Authors’ Affiliations
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