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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 361970, 8 pages
Robustness of Exponential Dissipation with respect to Small Time Delay
School of Science, Tianjin Polytechnic University, Tianjin 300387, China
Received 22 March 2013; Revised 20 May 2013; Accepted 24 May 2013
Academic Editor: Yuri V. Rogovchenko
Copyright © 2013 Xunwu Yin. 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.
We investigate robustness of exponential dissipation for the following general nonlinear evolutionary equation with small time delay: We firstly obtain a converse Lyapunov theorem. With the help of it, we establish an important result on robustness of exponential dissipation to small time delay assuming that the nonlinearity is globally Lipschitz.
As is well known, time delays are usually encountered in practical control systems. The stability analysis has received attentions over the last several decades. Mathematically, it is also very important to understand the sensitivity of the dynamical behavior of the system to the introduction of small time delays. For linear systems, we well understand this problem, including both finite dimensional and infinite dimensional situations, see [1–5]. However, for nonlinear systems, the problem is much more difficult, but there are some very nice results in [6–10].
This paper is devoted to the following general nonlinear evolutionary equation with small delay: where the nonlinearity is assumed to be globally Lipschitz. Here, is a sectorial operator on a Banach space . is a fractional power space. Here, we investigate the effects of small time delay on the exponential dissipation of the corresponding evolutionary equation without delay: where .
In , Lyapunov introduced his famous sufficient conditions for asymptotic stability of the following nonautonomous dynamical system: There, we can also find the first contribution to the converse question, known as converse Lyapunov theorems. In recent years, the answers have proved instrumental in establishing robustness of various stability notions and have served as the starting point for many nonlinear control systems design concepts.
In 2005, Li and Kloeden  presented a converse Lyapunov theorem for exponential dissipation of the following general nonlinear differential equations with multiple small time delays: where is assumed to be globally Lipschitz. They also prove that exponential dissipation remains under small time delays. This result can be seen as a generalization of some classical ones on global exponential asymptotic stability (e.g., ) and was used by the authors to study robustness of exponential dissipation with respect to small time delays.
Recently, Guo and Li  gave a nonautonomous analog of the result. They not only present a converse Lyapunov theorem but also prove robustness of the uniform exponential dissipation with respect to unbounded external perturbations.
In the dynamical theory, a basic problem concerns the robustness of global attractors under perturbations . It is known that if a nonlinear system with a global attractor is perturbed, then the perturbed one also has an attractor near , provided that the perturbation is sufficiently small [7, 15]. However, in general, we only know that is a local attractor. Whether the global feature can be preserved is still an open problem. To our great joy, a dissipative system usually implies the existence of the global attractor. So, if one wants to settle the above problem, he only needs to examine the robustness of dissipation under perturbations. In this present work, we will investigate the infinite dimensional situations which are more difficult than the finite ones. With the nonlinearity being globally Lipschitz, we obtain a converse Lyapunov theorem and prove that exponential dissipation has nice robustness properties under small time delay.
In this paper, we study the following delayed initial value problem: For simplicity, we use and to denote the norm on and , respectively. We write with the norm defined by
Next, we will recall some basic definitions and facts.
The upper right Dini derivative of a function is defined as
Let and be an open neighborhood of . For and , we define
We will denote by the solution of (2), where .
Definition 1. The system (2) is said to be exponentially dissipative, if there exist positive numbers , and such that
Lemma 2. Let be an open subset of . Assume that the function is Lipschitz; that is, there exists a such that Let be a solution of (2). Then,
Proof. The detailed proof is contained in [12, 16]. Here, we give a simple proof for the reader’s convenience. Making use of Taylor formula, we observe that Since is Lipschitz, one easily sees that Therefore, by definition (7), we immediately deduce that The proof is finished.
At last, we come to the main theorem on analytic semigroup which is extremely important in the study of the dynamics of nonlinear evolutionary equations .
Theorem 3 (fundamental theorem on sectorial operators). Let be a positive, sectorial operator on a Banach space and let be the analytic semigroup generated by . Then, the following statements hold.(1)For any , there is a constant such that for all (2)For , there is a constant such that for and (3)For every , there is a constant such that for all and
3. Main Results
In this section, we will prove our two main results: one is converse Lyapunov theorem, and the other is robustness of exponential dissipation to small time delay.
Theorem 4 (converse Lyapunov theorem). Suppose that in (2) is globally Lipschitz with Lipschitz constant . Suppose that the system without delay (2) is exponentially dissipative. Then, there exists a function satisfying for all , where , , and are appropriate positive constants.
Proof. Since the system (2) is exponentially dissipative, there exist positive constants , , and such that
Let , and define as follows:
By (21) and the elementary inequality, it is easy to check that
So, satisfies the right inequality of (18).
Next, by the Lipschitz continuity of , it is easy to verify that there exists a constant such that Considering (21) and (23), for any , we have So, satisfies (20).
Since by the choice of and (21), we have that Consequently, by Lemma 2, In particular, setting , one obtains that which indicates that satisfies (19).
Now, let us define another Lyapunov function . We firstly take a nonnegative function as where . It is easy to check that satisfies Now, we let We firstly verify the following fact: Indeed, if , then by (21) According to the definition of , we know that . Therefore, in case of , one trivially has If , then by the choice of we find that Since and is nondecreasing in , one can deduce the correctness of (33).
Next, we will check that also satisfies (20). By (33), (31), (24), and (21)
Next, we will check that for arbitrary , is bounded by Firstly, according to the definition of , it is obvious to see that So, it follows that Recalling (21), we infer Frequently, by the definition of and the monotonicity property of , we get So, we verify the correctness of (38).
Lastly we need to check that is nonincreasing in . Note that It is easy to see the validity of our checking.
Now, let Considering (23), (38), (25), (37), and (31), we can get the validity of (18), (19), and (20). The proof is complete.
In order to prove the second result, we need to verify the following lemma.
Lemma 5. Suppose that is globally Lipschitz with Lipschitz constant , that is, for any , and that the system (2) is exponentially dissipative. Then, there exist and such that when , any solution of (1) with initial value satisfies
Proof. According to (45), it is easy to see that there is an such that
Firstly, we prove that for arbitrary , there exists such that any solution of (1) with initial value satisfies
because can be expressed as follows:
By (47) and (15), we can obtain
According to the Gronwall inequality, one easily sees that
Now, we choose and fix and with Let . We will show that where .
We argue by contradiction and suppose that for some solution of (1), it holds that for some . Observing that , we deduce that there exists a such that
Thanks to Theorem 4, there is a Lyapunov function satisfying (18)–(20). By Lemma 2, we find that By (47) and (55), we see that for Denote by the Lipschitz constant of on . Then, we infer from (20) that At the same time, from Lemma 3.3.2 of , we can show that is locally Hölder. That is to say, Therefore, on we have that because According to (18) and (61), we find that If we denote that then by the Gronwall inequality Utilizing (18) again, we conclude that for , By the choice of and , one easily checks that Hence, in particular, for , we find that This contradicts (56).
Now, the conclusion of the theorem follows immediately from (48) and (53). And the proof is complete.
Proof. Let and be the solution of (1). According to Lemma 5, for all , repeating the same argument as in (65), one easily sees that the first inequality in (65) remains valid for all . Furthermore, making use of (18), we deduce that
By (46), it can be easily seen that
where () are appropriate positive constants independent of and . For , we have by (46) that
Therefore, taking , one concludes that
Now, we fix a and independent of such that So, Setting , we find that
Next, we will use mathematical induction to prove that where .
Indeed, let . If , then (77) clearly holds true. Suppose that (77) holds for ; that is, Then, in particular, From (76), we know that If we consider as , then the above can be rewritten as From (79), Choosing , we conclude that Thus, we see that (77) holds for .
By (77), we know that Furthermore, by Lemma 5, we see that where .
We observe that when , so we infer from (85) that where . So, we easily see that This completes the proof of the theorem.
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