- Research Article
- Open Access

# A Converse Lyapunov Theorem and Robustness with Respect to Unbounded Perturbations for Exponential Dissipativity

- Xudong Li
^{1}and - Yuan Guo
^{1}Email author

**2010**:726347

https://doi.org/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 .

## Keywords

- Asymptotic Stability
- Global Attractor
- Nonlinear Control System
- Robustness Property
- Gronwall Lemma

## 1. Introduction

where is always assumed to be a continuous vector field which is locally Lipschitz in space variable . Our main aim is two-fold: 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.

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.

*nonautonomous Dini sup-derivative*of at along the vector . In case is differentiable at , it is easy to see that

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 Cohen-Grossberg Neural Networks with Unbounded External Inputs and Disturbances

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

## References

- Lyapunov AM:
**The general problem of the stability of motion.***International Journal of Control*1992,**55**(3):531-534. 10.1080/00207179208934253MathSciNetView ArticleGoogle Scholar - Li D, Kloeden PE:
**Robustness of asymptotic stability to small time delays.***Discrete and Continuous Dynamical Systems. Series A*2005,**13**(4):1007-1034. 10.3934/dcds.2005.13.1007MATHMathSciNetView ArticleGoogle Scholar - Yoshizawa T:
*Stability Theory by Lyapunov's Second Method*. The Mathematical Society of Japan, Tokyo, Japan; 1996.Google Scholar - Hale JK:
*Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs*.*Volume 25*. American Mathematical Society, Providence, RI, USA; 1988.Google Scholar - Li D:
**Morse decompositions for general dynamical systems and differential inclusions with applications to control systems.***SIAM Journal on Control and Optimization*2007,**46**(1):35-60. 10.1137/060662101MathSciNetView ArticleGoogle Scholar - Robinson JC:
*Infinite-Dimensional Dynamical Systems, Cambridge Texts in Applied Mathematics*. Cambridge University Press, Cambridge, UK; 2001:xviii+461.View ArticleGoogle Scholar - Kloeden PE, Siegmund S:
**Bifurcations and continuous transitions of attractors in autonomous and nonautonomous systems.***International Journal of Bifurcation and Chaos*2005,**15**(3):743-762. 10.1142/S0218127405012454MATHMathSciNetView ArticleGoogle Scholar - Hafstein SF:
**A constructive converse Lyapunov theorem on exponential stability.***Discrete and Continuous Dynamical Systems. Series A*2004,**10**(3):657-678. 10.3934/dcds.2004.10.657MATHMathSciNetView ArticleGoogle Scholar - Jiang M, Shen Y, Liao X:
**Boundedness and global exponential stability for generalized Cohen-Grossberg neural networks with variable delay.***Applied Mathematics and Computation*2006,**172**(1):379-393. 10.1016/j.amc.2005.02.009MATHMathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.