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

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:

(1.1)

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

(1.3)

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

(1.4)

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

(1.5)

we deduce that any solution of the perturbed system

(1.6)

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

(2.1)

Let be an open interval, and let be an open subset of . Let . For and , define

(2.2)

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

(2.3)

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

(2.4)

and let . Then

(2.5)

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

(2.6)

Since is Lipschitz in , one easily sees that

(2.7)

Therefore by definition we immediately deduce that

(2.8)

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

(2.9)

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

(2.10)

(2.11)

(2.12)

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

(2.13)

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

(2.14)

By (2.9) it is clear that

(2.15)

Let , , and let . Then

(2.16)

Taking inner product of this equation with , by (F1) one finds that

(2.17)

from which it can be easily seen that

(2.18)

Thus we deduce that

(2.19)

where . Now for any and , we have

(2.20)

and it immediately follows by (2.9) that

(2.21)

where is independent of . This shows that satisfies (2.11).

Since

(2.22)

by the choice of we have that

(2.23)

On the other hand, by Lemma 2.1 we find that

(2.24)

Setting , one obtains that

(2.25)

which indicates that satisfies (2.12).

Now let us define another Lyapunov function . For this purpose we take a nonnegative function as

(2.26)

where . Then

(2.27)

Indeed, if , then the estimate clearly holds true. So we may assume without loss of generality that with . We have

(2.28)

Let

(2.29)

We claim that we actually have

(2.30)

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

(2.31)

Now assume that . Then by the choice of we find that

(2.32)

Since and is nondecreasing in , one immediately deduces that

(2.33)

which completes the proof of (2.30).

By (2.9), (2.19), and (2.27) we have

(2.34)

Therefore

(2.35)

Since and are arbitrary, we conclude that

(2.36)

By the definition of it is clear that

(2.37)

It immediately follows that

(2.38)

We also infer from (2.9) that

(2.39)

Therefore by definition of and the monotonicity property of , we have

(2.40)

In conclusion we have

(2.41)

Note that if , then

(2.42)

This implies that is nonincreasing in .

Now set

(2.43)

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:

(3.1)

where is a continuous function which corresponds to external perturbations.

Denote by the family of continuous functions that satisfies the following growth condition:

(3.2)

where is a continuous nonnegative function on with

(3.3)

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

(3.4)

where , and is as in (3.3). Then one easily verifies that, for any , there exists a such that

(3.5)

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

(3.6)

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

(3.7)

Now by Lemma 2.1 we find that

(3.8)

By (2.11) we have

(3.9)

Therefore

(3.10)

Since , we find that

(3.11)

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

(3.12)

where

(3.13)

and and are the constants in (2.10). In particular, we have

(3.14)

and it follows that

(3.15)

for , where .

Integrating both sides of (3.12) from to , one finds that

(3.16)

Since is locally Lipschitz in , we find that

(3.17)

Hence

(3.18)

where . By the classical Gronwall lemma and (3.15) we obtain

(3.19)

Now for any fixed we integrate (3.14) from to and find that

(3.20)

Further integrating the above inequality in from to , it yields

(3.21)

where . By (2.10) one concludes that

(3.22)

where , and .

We also deduce by (2.10) and (3.15) that

(3.23)

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

(3.24)

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

(3.25)

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]:

(3.26)

where denote outside inputs and disturbances, , and

(3.27)

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

(3.29)

where is a function as in (3.3), system (3.26) is uniformly exponentially dissipative.

Proof.

Consider the system

(3.30)

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

(3.31)

where

(3.32)

Then system (3.26) can be reformulated as

(3.33)

where

(3.34)

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

(3.35)

where , and

(3.36)

Therefore

(3.37)

where is a constant which only depends on the dimension of the phase space , and . Note that the function

(3.38)

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

(3.39)

where , we have

(3.40)

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.