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.
SetClearly, satisfies Thus, system (1) is equivalent to the following:
Set where is the usual Sobolev space of order . We take the state space asequipped with the following inner product, for any :Obviously is a Hilbert space.
We define an operator in bywith 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 calculateSince , 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 conditionshold, 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 thatholds.
Proof. Let be the solution of (1). Applying Young’s and Poincaré’s inequalities Taking , we getSince , we can set and ; thenThe desired inequality follows.
Let . We define a function bywhere 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 thatwhere and are defined as before.
SoIn what follows, we will calculate and .
Using integration by parts and the boundary condition, we havewhere we have used equalitiesSummarizing the above all, we haveSince we haveWe now estimate the integral terms with time delay. Applying Young’s and Poincaré’s inequalities, we haveThus,Taking , , we obtain The desired inequality follows.
Sincewe haveEmploying the estimate, we haveClearly, if the parameters , , , , and are such that the inequalitieshold, 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 inequalitythen there exists such that for inequality (41) holds true.
Proof. If (44) holds, then or equivalently Set Sincethere 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 equationand 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 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, letObviously, 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 .
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 .
Assume that , , (i.e., they are constants); we can obtain Figure 4. We get the best decay rate .
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).