Journal of Difference Equations
Volume 2016 (2016), Article ID 3732176, 10 pages
http://dx.doi.org/10.1155/2016/3732176

Research Article

The Exponential Stability Result of an Euler-Bernoulli Beam Equation with Interior Delays and Boundary Damping

Peng-cheng Han, Yan-fang Li, Gen-qi Xu, and Dan-hong Liu

Department of Mathematics, Tianjin University, Tianjin 300350, China

Received 20 January 2016; Accepted 9 March 2016

Academic Editor: Honglei Xu

Copyright © 2016 Peng-cheng Han et al. 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.

Abstract.

We study the exponential stability of Euler-Bernoulli beam with interior time delays and boundary damping. At first, we prove the well-posedness of the system by the semigroup theory. Next we study the exponential stability of the system by constructing appropriate Lyapunov functionals. We transform the exponential stability issue into the solvability of inequality equations. By analyzing the relationship between delays parameters and damping parameters , we describe -region for which the system is exponentially stable. Furthermore, we obtain an estimation of the decay rate .

1. Introduction

It is well known that the time delay always exists in real system, which may be caused by acquisition of response and excitation data, online data processing, and computation of control forces. Since time delay may destroy stability [1, 2] even if it is very small, the stabilization problem of systems with time delays has been a hot topic in the mathematical control theory and engineering. In recent years, the systems described by PDEs with time delays have been an active area of research; see [3–7] and references therein. Generally speaking, there are mainly three kinds of time delay in the system, one is the interior time delay of the system (also called structural memory), one is the input delay (control delay), and the third is the output delay (measurement delay). Many scholars have made great efforts to minimize the negative effects of time delays although time delay cannot be eliminated due to its inherent nature, for example, [8–10] for boundary control with delays, [11, 12] for internal control delays, and [13] for output delays.

In past several years, the research on the Euler-Bernoulli beam with time delay has made great progress. For example, Park et al. [14] considered the stabilization problem of an Euler-Bernoulli beam with structural memory; Liang et al. [15] introduced the modified Smith predictor to Euler-Bernoulli beam with the boundary control and the delayed boundary measurement; Shang et al. [16–18] investigated the stabilization problem of the Euler-Bernoulli beam with boundary input delay; Yang et al. [19, 20] solved the stabilization problem of constant and variable coefficients Euler-Bernoulli beam with delayed observation and boundary control; at the same time, Jin and Guo [21] solved the output feedback stabilization of Euler-Bernoulli beam by Lyapunov approach. However, few people investigate the influence of an Euler-Bernoulli beam with interior delays and boundary damping on the system stability. In this paper we mainly study the exponential stability of a system described by the Euler-Bernoulli beam with interior delays and boundary damping. More precisely, we consider the following system, whose dynamic behavior is governed by the Euler-Bernoulli beam:

with , where , , and is the delay time. We mainly investigate its exponential stability.

The rest is organized as follows: In Section 2, we at first formulate problem (1) into an appropriate Hilbert space and then study the well-posedness of the system by the semigroup theory. In Section 3, we construct a Lyapunov functional for system (1) and prove the exponential stability under certain conditions. By optimization parameters we obtain a complicated relationship between the decay rate and delay time . Finally, in Section 4, we give a brief conclusion.

2. Well-Posedness of the System

In this section, we will discuss the well-posedness and some basic properties of system (1). For the purpose, firstly we formulate system (1) into an appropriate Hilbert space.

Set

Clearly, satisfies

Thus, system (1) is equivalent to the following:

Set

where is the usual Sobolev space of order . We take the state space as

equipped with the following inner product, for any :

Obviously is a Hilbert space.

We define an operator in by

with domain

With the assistance of operator , we can rewrite (4) as an evolution equation in :

where and .

For operator , we have the following result.

Lemma 1. Let be defined as (8) and (9). Then is a closed and densely defined linear operator in . For any and , and is compact on . Hence consists of all isolated eigenvalues of finite multiplicity.

Proof. It is easy to check that is a closed and densely defined linear operator in ; the detail of the verification is omitted.
Let , and for any we consider the equation where ; that is,

with boundary condition

By a complex calculation, we get the solution

Let , , be given as (13). Then we have and . The closed operator theorem asserts that , and is a bounded linear operator. Since , the Sobolev Embedding Theorem asserts that is a compact operator on . Hence, by the spectral theory of compact operator, consists of all isolated eigenvalues of finite multiplicity.

Theorem 2. Let and be defined as before. Then generates a semigroup on . Hence, system (10) is well posed.

Proof. For any real , we calculate

Since , we have

where , which shows that is a dissipative operator. This together with Lemma 1 shows that satisfies the conditions of Lumer-Phillips theorem [22]. So generates a semigroup on .

3. Exponential Stability of the System

In this section, we consider the exponential stability issue of system (1) based on Lyapunov method.

The energy function of system (1) is defined as

In what follows, we will give some lemmas that are the foundation of our method.

Lemma 3 (see [23]). Let be a nonnegative function on . If there exists a function and some positive numbers and such that the conditions

hold, then decays exponentially at rate .

In order to construct a function satisfying the conditions in Lemma 3, we set

where is a constant and satisfies

We can establish an equivalence relation between and via the following Lemma.

Lemma 4. Let and be defined as before. Then there exist positive constants and such that

holds.

Proof. Let be the solution of (1). Applying Young’s and Poincaré’s inequalities

Taking , we get

Since , we can set and ; then

The desired inequality follows.

Let . We define a function by

where

Noting that , according to Lemma 4 we can see that the following result is true.

Lemma 5. Let defined as before. Then satisfies condition (17); that is,

In what follows, we calculate . For we have the following result.

Lemma 6. Let be defined as before and let be the solution of (1). Then

Proof. By definition, we see that

where and are defined as before.
So

In what follows, we will calculate and .
Using integration by parts and the boundary condition, we have

where we have used equalities

Summarizing the above all, we have

Since

we have

We now estimate the integral terms with time delay. Applying Young’s and Poincaré’s inequalities, we have

Thus,

Taking , , we obtain

The desired inequality follows.

Since

we have

Employing the estimate, we have

Clearly, if the parameters , , , , and are such that the inequalities

hold, then we have .

Summarizing discussion above, we have the following result.

Theorem 7. Let be the solution of (1), and let and . If inequalities (41) hold, then the energy function decays exponentially at rate .

We now are in a proposition to study the solvability of inequalities (41). Noting that is not a system parameter, it is only a middle parameter which is introduced in the multiplier term. From the third inequality of (41) we see that and have a relationship:

Taking , (41) is equivalent to

Theorem 8. Set . If and satisfy the inequality

then there exists such that for inequality (41) holds true.

Proof. If (44) holds, then

or equivalently

Set

Since

there exist and such that and .
Set

and . Clearly, when , we have and , so (43) holds. Hence (41) holds true.

In what follows, we discuss the property of the function

We consider equation

and it is equivalent to

This equation has three real roots . So we have

3.1. -Region of the Exponential Stability

According to (52) we determine . And according to (53) we draw the -region.

The picture of is given as Figure 1. -region is given by

Figure 1: The graph of function , which gives the relationship between and .

Figure 1 gives the graph of function that gives the relationship between and with which system (1) is exponentially stable. From this picture we see that if is larger, we cannot stabilize it by the boundary damping. has upper bound .

3.2. The Best Decay Rate

Suppose that with . According to (43) we determine the best decay rate . Note that , where and solve the following equation, respectively:

Firstly, let

Obviously, and both are monotonic function. For comparing to the two equations of (56), we define the function :

Since , we have . When , we get ; that is, . Therefore, when , .

An example is given: Assume that , , (i.e., they are constants). We obtain the approximative zero points of and from Figure 2. That is, and . Therefore, we get the best decay rate .

Figure 2: , , and .

Next, we consider . By researching (58), we can obtain and . We have .

Let

We have . According to -region of the exponential stability, we obtain . After comparison,

To summarize, , when , we can get . Therefore, the best decay .

Then, we give two examples.

Assume that , , (i.e., they are constants); we can obtain Figure 3. We get the best decay rate .

Figure 3: , , and .

Assume that , , (i.e., they are constants); we can obtain Figure 4. We get the best decay rate .

Figure 4: , , and .

Finally, we consider . We can also obtain , when . Here, . After comparison,

Therefore, Not all were always correct.

Summarizing the above all, the best decay is easy to determine, when . The best decay is not easy to determine, when . We have two conclusion:

(i), the best decay ;(ii), when , the best decay .

4. Conclusions

In this paper, using the Lyapunov functional approach we discussed the exponential stabilization of an Euler-Bernoulli beam equation with interior delays and boundary damping. Different from the earlier papers, we added a multiplier term to the Lyapunov function so as to transform the exponential stability. By solving the inequality equations, we give the exponential stability region of the system.

We note that the method used in this paper also can apply to the investigation of the exponential stability of other model. In the future, we will study the boundary feedback control anti-interior time delay for other models.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

Acknowledgments

This research is supported by the Natural Science Foundation of China (Grant nos. NSFC-61174080, 61573252, and 61503275).

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