- Research Article
- Open Access
A Converse Lyapunov Theorem and Robustness with Respect to Unbounded Perturbations for Exponential Dissipativity
Advances in Difference Equations volume 2010, Article number: 726347 (2010)
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 .
This paper is devoted to the following nonautonomous dynamical system:
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.
Recently Li and Kloeden  presented a converse Lyapunov theorem for exponential dissipativity of (1.1) in autonomous case with being globally Lipschitz in . This result can be seen as a generalization of some classical ones on global exponential asymptotic stability (see, for instance, , etc.), and was used by the authors to study robustness of exponential dissipativity with respect to small time delays. Here we give a nonautonomous analog of the result; moreover, instead of assuming to be globally Lipschitz in , we only impose on the following weaker condition.
(F1) There exists an such that(1.2)
where denotes the inner product in .
Note that if is globally Lipschitz in in a uniform manner with respect to , then (F1) is automatically satisfied. However, we emphasize that this condition also allows nonglobally Lipschitz functions. An easy example is the function , which is clearly not globally Lipschitz. One observes that
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 ).
Consider the scalar differential equation
It is easy to see that the equilibrium is globally asymptotically stable, and consequently the system is dissipative. However, since for any there exists an such that
we deduce that any solution of the perturbed system
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 . Let us first recall some basic definitions and facts.
The upper right Dini sup-derivative of a function is defined as
Let be an open interval, and let be an open subset of . Let . For and , define
We call the nonautonomous Dini sup-derivative of at along the vector . In case is differentiable at , it is easy to see that
Let be an open subset of . Assume that the continuous function is Lipschitz in uniformly in , that is, there exists an such that
and let . Then
This basic fact is actually contained in , and so forth. Here we give a simple proof for the reader's convenience.
We observe that
Since is Lipschitz in , one easily sees that
Therefore by definition we immediately deduce that
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 .
System (1.1) is said to be uniformly exponentially dissipative, if there exist positive numbers , , and such that
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.
Then there exists a function satisfying
for all , and , where , , , , , and are appropriate positive constants.
Moreover, if , namely, the system is uniformly exponentially asymptotically stable, then the constants , and vanish.
Since the ODE system (1.1) is exponentially dissipative, there exist positive constants , , and such that (2.9) holds. Let
By (2.9) it is clear that
Let , , and let . Then
Taking inner product of this equation with , by (F1) one finds that
from which it can be easily seen that
Thus we deduce that
where . Now for any and , we have
and it immediately follows by (2.9) that
where is independent of . This shows that satisfies (2.11).
by the choice of we have that
On the other hand, by Lemma 2.1 we find that
Setting , one obtains that
which indicates that satisfies (2.12).
Now let us define another Lyapunov function . For this purpose we take a nonnegative function as
where . Then
Indeed, if , then the estimate clearly holds true. So we may assume without loss of generality that with . We have
We claim that we actually have
Indeed, if , then by (2.9) we deduce that for all . On the other hand, by the definition of we have for all . Therefore in case , one trivially has
Now assume that . Then by the choice of we find that
Since and is nondecreasing in , one immediately deduces that
which completes the proof of (2.30).
By (2.9), (2.19), and (2.27) we have
Since and are arbitrary, we conclude that
By the definition of it is clear that
It immediately follows that
We also infer from (2.9) that
Therefore by definition of and the monotonicity property of , we have
In conclusion we have
Note that if , then
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.
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
Consider the following perturbed system:
where is a continuous function which corresponds to external perturbations.
Denote by the family of continuous functions that satisfies the following growth condition:
where is a continuous nonnegative function on with
for some . Our main result in this part is contained in the following theorem.
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.
Suppose that satisfies a sublinear growth condition
where , and is as in (3.3). Then one easily verifies that, for any , there exists a such that
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 simplicity in writing we set
Let be any solution of the perturbed system (3.1) with initial value . By Remark 2.4 we know that is locally Lipschitz in and hence has derivative almost everywhere. Note that at any point where has derivative we necessarily have
Now by Lemma 2.1 we find that
By (2.11) we have
Since , we find that
where are appropriate numbers (which are independent of the initial values). Thus
and and are the constants in (2.10). In particular, we have
and it follows that
for , where .
Integrating both sides of (3.12) from to , one finds that
Since is locally Lipschitz in , we find that
where . By the classical Gronwall lemma and (3.15) we obtain
Now for any fixed we integrate (3.14) from to and find that
Further integrating the above inequality in from to , it yields
where . By (2.10) one concludes that
where , and .
We also deduce by (2.10) and (3.15) that
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.
Assume that satisfies (F1) and the following sublinear growth condition
where , and is as in .
Then system (1.1) is necessarily not uniformly exponentially dissipative.
Suppose that (1.1) is uniformly exponentially dissipative. Then by Theorem 3.1, the perturbed system (3.1) is uniformly exponentially dissipative for any perturbation , provided is satisfied with sufficiently small. On the other hand, taking for any , by sublinear growth condition on one easily examine by using standard argument that the perturbed system
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
As another simple example of the application of Theorem 2.3, we consider the following Cohen-Grossberg neural networks with variable coefficients and multiple delays considered recently in :
where denote outside inputs and disturbances, , and
denote time delays, where . For the physical meaning of the coefficients we refer the reader to , and so forth. In case is bounded and independent of , the exponential dissipativity is actually considered in . Here we discuss the more general case. As in  we assume that
(H1) are bounded and locally Lipschitz,
(H2) each function belongs to ; moreover,(3.28)
(H3) , and are bounded continuous functions.
Assume (H1)–(H3). Then there exists an sufficiently small such that for any continuous functions satisfying
where is a function as in (3.3), system (3.26) is uniformly exponentially dissipative.
Consider the system
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.
For simplicity we write
Then system (3.26) can be reformulated as
We observe by (H1), (H3), and (3.29) that
where , and
where is a constant which only depends on the dimension of the phase space , and . Note that the function
satisfies (3.3) with therein replaced by another appropriate constant .
Now assume that , where and are the constants in (2.11) and (2.12). By repeating the same argument as in the proof of Theorem 3.1 with almost no modification, one can show that there exist constants and such that for any solution of system (3.26) with initial value
where , we have
Here denotes the usual norm of in . We omit the details.
The proof of the theorem is complete.
The above result contains Theorem 3.1 in  as a particular case.
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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).