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A Converse Lyapunov Theorem and Robustness with Respect to Unbounded Perturbations for Exponential Dissipativity
Advances in Difference Equations volume 2010, Article number: 726347 (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
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 [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.
Recently Li and Kloeden [2] 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, [3], 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 [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]).
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 [2]. 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

Lemma 2.1.
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

Proof.
This basic fact is actually contained in [3], 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
.
Definition 2.2.
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.
Proof.
Since the ODE system (1.1) is exponentially dissipative, there exist positive constants ,
, and
such that (2.9) holds. Let

We first define a function as follows: (The techniques used here are adopted from [2, 8], etc.)

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).
Since

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

Let

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

Therefore

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
.
Now set

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
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.
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.
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

Therefore

Since , we find that

where are appropriate numbers (which are independent of the initial values). Thus

where

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

Hence

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.
Corollary 3.3.
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.
Proof.
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 [9]:

where denote outside inputs and disturbances,
, and

denote time delays, where . For the physical meaning of the coefficients we refer the reader to [9], and so forth. In case
is bounded and independent of
, the exponential dissipativity is actually considered in [9]. Here we discuss the more general case. As in [9] we assume that
-
(H1)
are bounded and locally Lipschitz,
-
(H2) each function
belongs to
; moreover,
(3.28) -
(H3)
, and
are bounded continuous functions.
Theorem 3.4.
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.
Proof.
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

where

Then system (3.26) can be reformulated as

where

We observe by (H1), (H3), and (3.29) that

where , and

Therefore

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.
Remark 3.5.
The above result contains Theorem 3.1 in [9] as a particular case.
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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).
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Li, X., Guo, Y. A Converse Lyapunov Theorem and Robustness with Respect to Unbounded Perturbations for Exponential Dissipativity. Adv Differ Equ 2010, 726347 (2010). https://doi.org/10.1155/2010/726347
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DOI: https://doi.org/10.1155/2010/726347