A Converse Lyapunov Theorem and Robustness with Respect to Unbounded Perturbations for Exponential Dissipativity
© X. Li and Y. Guo. 2010
Received: 18 December 2009
Accepted: 17 April 2010
Published: 23 May 2010
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  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.
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 . 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 ).
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 . Let us first recall some basic definitions and facts.
This basic fact is actually contained in , and so forth. Here we give a simple proof for the reader's convenience.
The proof is complete.
The main result in this section is the following theorem.
Theorem 2.3 (Converse Lyapunov Theorem).
which completes the proof of (2.30).
The proof is complete.
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
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.
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.
Then system (1.1) is necessarily not uniformly exponentially dissipative.
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
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.
The proof of the theorem is complete.
The above result contains Theorem 3.1 in  as a particular case.
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).
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