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

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 C 0 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 α and damping parameters β, we describe (β, α)-region for which the system is exponentially stable. Furthermore, we obtain an estimation of the decay rate λ.


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][4][5][6][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][9][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][17][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:

Lemma 1.
Let A be defined as (8) and (9).Then A is a closed and densely defined linear operator in H.For any  > 0 and  > 0, 0 ∈ () and A −1 is compact on H. Hence (A) consists of all isolated eigenvalues of finite multiplicity.
Proof.It is easy to check that A is a closed and densely defined linear operator in H; the detail of the verification is omitted.

Theorem 2. Let
A and H be defined as before.Then A generates a  0 semigroup on H. Hence, system ( 10) is well posed.
Proof.For any real  = (, , ℎ)  ∈ D(A), we calculate Since  > 0, we have where  = 2 2 +1/2, which shows that A− is a dissipative operator.This together with Lemma 1 shows that A −  satisfies the conditions of Lumer-Phillips theorem [22].So A generates a  0 semigroup on H.

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 R + .If there exists a function () and some positive numbers  1 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 0 <  < 2.
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  2 and  3 such that holds.
Let  > 0. We define a function () by where Noting that  > 0, 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 V().For  1 () we have the following result.Lemma 6.Let  1 () 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 Ė () and Ġ().
Using integration by parts and the boundary condition, we have where we have used equalities Summarizing the above all, we have we have We now estimate the integral terms with time delay.Applying Young's and Poincaré's inequalities, we have Thus, Taking  1 =  − ,  2 =  − /, 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 V() ≤ 0.
Summarizing discussion above, we have the following result.
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  = 2/(1 + ), (41) is equivalent to Theorem 8. Set  = 2/(1+).If  and  satisfy the inequality then there exists  * > 0 such that for  ∈ (0,  * ] inequality (41) holds true.

So we have
(55) 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 /3(1 +  3 ) ≃ 0.246.Obviously, () and () both are monotonic function.

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  2 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.

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