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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 726794, 15 pages
http://dx.doi.org/10.1155/2013/726794
Research Article

Exponential Stabilization for Timoshenko Beam with Distributed Delay in the Boundary Control

Department of Mathematics, Tianjin University, Tianjin 300072, China

Received 23 October 2013; Accepted 18 November 2013

Academic Editor: Shen Yin

Copyright © 2013 Xiu Fang Liu and Gen Qi Xu. 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 consider the exponential stabilization for Timoshenko beam with distributed delay in the boundary control. Suppose that the controller outputs are of the form and ; where and are the inputs of boundary controllers. In the past, most stabilization results for wave equations and Euler-Bernoulli beam with delay are required . In the present paper, we will give the exponential stabilization about Timoshenko beam with distributed delay and demand to satisfy the lesser conditions for .

1. Introduction

Since the extensive applications of Timoshenko beam in high-Tech, the stabilization problem has been a hot topic in the mathematical control theory and engineering; for instance, see [15] and the references therein. In many literature, the control delay problem has been neglected. Due to extensive applications of the system with delay, more and more scholars devoted to study the stabilization of the system with controller delay. It is well known that time delay caused by controller memory usually takes the form, whereis a bounded variation function (or matrix-valued function) andis the control input. If the control is in the space, then the memory controller will take the form

Based on this reason, Xu et al. (see [6]) studied firstly stabilization of the 1-d wave systems with delay of the form. They proved that the system with control delay is exponential stable ifand unstable if. Nicaise and Pignotti in [7] studied the stability and instability of the wave equation with delay in boundary and internal distributed delay. Nicaise and Valein in [8] extend the 1-d wave equation to the networks of 1-d wave equations. Shang et al. in [9] studied Euler-Bernoulli beam and showed thatis not necessary, but the conditionis necessary. For the case of distributed delay, that is,and, Nicaise and Pignotti in [10] discussed a high dimensional wave equation. Under the condition that, they proved the velocity feedback control law also stabilizes exponentially the system.

From above we see that, andare determined by the controller. We cannot determine whether or notincludingin practice. Under the assumption of state being measurable, Shang and Xu in [11] designed a dynamic feedback controller for cantilever Euler-Bernoulli beam that stabilizes exponentially the system for any real. Recently Han and Xu in [12] extended this result to the case of output being measurable; they showed that a state observer can realize the state reconstruction from the output of the system. Xu and Wang in [13] discussed the Timoshenko beam with boundary control delay, and they also stabilized the system by a dynamic feedback controller. Note that the difference between [11, 13], one is a system of single input and single output, the other is a system of 2 inputs and 2 outputs. Such discussion will lead us to extend the method to a general system of multiinput and multioutput. So far, however, there is no result for any, andabout Timoshenko beams. In this paper, we still consider Timoshenko beam with boundary control distributed delay. We will seek for a dynamic feedback control law that exponentially stabilizes the Timoshenko beam with distributed delay under certain conditions.

The rest is organized as follows. In Section 2, we will describe the design process of controllers, including predict system and generation of signal, and then state the main results of this paper. In Section 3, we will give the representation of the transform system. In Section 4, we will prove our first result on the stabilization of the original system. In Section 5, we will prove the second result on the exponential stabilization of the induced system. In Section 6, we conclude the paper.

2. Design of Controllers and Main Results

Letbe the displacement andthe rotation angle of the beam. The motion of a cantilever beam is governed by the following partial differential equations:

whereandare the control force and torque from the controllers, respectively. If the controllers have no memory, namely, , whereare controller inputs, this model had been studied in [14]. If the controllers have memory, then the Timoshenko beam became

whereis the delay time,are the controller parameters, and, andare bounded measurable functions that are memory values of controllers. When,, (3) is just the model in [13].

We suppose that the state of (3) is measurable; that is,is measurable. We introduce an auxiliary system as follows:

Equation (4) is a partial state predictor.

Denote the state of (4) at the momentby

Using (3) we can verify that the functions group satisfy the following partial differential equations:

whereare measurable function andare bounded linear operators on; they are determined later.

Equation (6) is a system without delay, but the controls appear in the system interior and boundary. First we consider the stabilization problem of (6). Let us consider the energy functional of (6)

A direct calculation gives

Set

We take the feedback control law as

Then, the closed loop system associated with (6) is

We estimate the error of the system (3) with control (10) and the system (11).

Letbe the solution to (3) with control signals (10) and let function groupbe the solution to (11). Setand, and setand.

To discuss the stability, we consider the error both solutions in the energy space

In this paper, we will prove the following results.

Theorem 1. Letbe the solution to (3) with controls (10) and let be the solution to the closed-loop system (11). If the system (11) is asymptotically (exponentially) stable, then the system (3) also is asymptotically (exponentially) stable.

Theorem 2. Suppose that. Let be the eigenvalues of the free system (the system (2) without controls). Set Then the following assertions are true:(1)when the system (11) is exponentially stable;(2)if for all, but then the system (11) is asymptotically stable.

In the following sections, we will prove our results. In Section 3, we will determine functions. In Section 4, we will prove Theorem 1. In Section 5, we pay our attention to the proof of Theorem 2.

3. Representation of the System (6)

In this section, we will obtain the expressions for the functionsappearing in system (6) using (3) and (4).

We begin with introducing two useful lemmas.

Lemma 3 (see [13]). Define the differential operator inas follows: with domain Thenis a positive define operator with compact resolvent in; its eigenvalues are and the eigenfunctionscorresponding toare real functions and form a normalized orthogonal basis for.

Lemma 4 (see [13]). Letbe the normalized eigenfunction corresponding to the eigenvalueof. Then it holds that

Now let us return to (3). We write the equation in (3) into the vector form

and the boundary conditions are, and

The initial datum are

Setand. Definematrices

and define an operatorfromto, whereis dual space,

and define an operatorfromtoby

where.

With help of these notations, we can rewrite (3) into

and (4) into

where,.

We define two families of the bounded linear operators onby

Clearly, the following equalities hold, for any,

It is easy to know that the vector-valued function

is differentiable with respect toand

Further,satisfies (27).

Similarly, we know the vector-valued function

satisfy (28).

Set

Then we have

Thus,

Note that

So it holds that

Therefore, we have equations

and initial conditions

where.

Since all entries ofare meaningful as linear functional on, so for anyand,

Therefore, we have the following results.

Theorem 5. Let be the list of all eigenvalues of . Then the functions that appear in (6) are

and the linear operators are

4. The Proof of Theorem 1

In this section, we will prove Theorem 1. Here we mainly estimate the error:

According to the calculation in Section 3, we have

So,

Note thatis a Riesz basis sequence for. Thus, there exist positive constantssuch that

Let be the solution to (11), and be its energy functional; then we have

Therefore, we have

So, we can get

Ifis exponential stable, there exists a positive constantsuch thatWe can obtain the following result from above:

whereis a positive constant. Soalso decays exponentially.

5. The Proof of Theorem 2

In this section, we will discuss the stability of system (11). At first we considerwell posed of the system (6). For the sake of simplicity, we use the vector form of (6); that is,

The observation system corresponding to (55) is