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This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

Exponential and polynomial stabilization of the Kirchhoff string by nonlinear boundary control

Yuhu Wu1* and Jianmin Wang2

Author Affiliations

1 Department of Mathematics, Harbin University of Science and Technology, Harbin, 150080, P.R. China

2 Department of Electronics Science and Technology, Harbin University of Science and Technology, Harbin, 150080, P.R. China

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Boundary Value Problems 2013, 2013:215  doi:10.1186/1687-2770-2013-215


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/215


Received:22 April 2013
Accepted:6 August 2013
Published:21 October 2013

© 2013 Wu and Wang; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper addresses the stabilization problem of the nonlinear Kirchhoff string using nonlinear boundary control. Nonlinear boundary control is the negative feedback of the transverse velocity of the string at one end, which satisfies a polynomial-type constraint. Employing the multiplier method, we establish explicit exponential and polynomial stability for the Kirchhoff string. The theoretical results are assured by numerical results of the asymptotic behavior for the system.

1 Introduction

Stabilization and vibration controllability of string or beam systems arising from different engineering backgrounds has attracted attention of many researchers [1-4]. In particular, boundary feedback stabilization of string and beam systems has become an important research area [5-7]. This is because, in a practice system, vibration is more easily controlled through a boundary point than using point sensors or actuators away from the boundaries [8,9].

There are several nonlinear mathematical models that describe the transversal vibration of stretched strings. One such model is presented in the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M1">View MathML</a>

(1)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M5">View MathML</a> are two constants. Obviously, the above equation is a simple prototype of the classical equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M6">View MathML</a>

which was proposed by Kirchhoff [10]. Here l is the length of the string; E is Young’s modulus of the material; ρ is density; h is the area of the cross section; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M7">View MathML</a> is the transversal displacement of the point x of the string at time t. This model has been studied by researchers from the physical and mathematical points of view; see, e.g., references [11-13] and the references therein.

In this paper, we consider Kirchhoff string (1) with the following boundary conditions (see Figure 1):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M8">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M9">View MathML</a>

(3)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M11">View MathML</a> denotes the tension in the string at time t. The boundary condition in equation (2) implies that the string is fixed at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M12">View MathML</a>. The boundary condition in equation (3) represents the balance of the transversal component of the tension in the string and the control input u which is applied transversally at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M13">View MathML</a>. Because the tension in the string represented by equation (1) is not constant and is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M14">View MathML</a>

(4)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3">View MathML</a> (see [14]), the boundary condition in equation (3) can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M16">View MathML</a>

(5)

thumbnailFigure 1. Schematic of the nonlinear Kirchhoff string with boundary control.

Shahruz and Krishna [13] investigated the stabilization of Kirchhoff string (1) with a linear negative velocity control, which means the boundary control u has a linear negative velocity feedback form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M17">View MathML</a>

(6)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3">View MathML</a>, where L is a positive constant. They established exponential stability. In [15], the absolute stability of the Kirchhoff string (1) with linear sector boundary control was considered. It is well known that linear strings represented by equation (1), for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M19">View MathML</a>, can be stabilized by the control law in equation (6); see e.g., references [16-18]. Moreover, Shahruz [19], Fung et al.[5], and Li and Hou [20] developed linear boundary control laws for axially moving strings. It is worth mentioning that Kobayashi [21] designed a linear parallel compensator based on boundary displacement observer and proved the string (1) can be stabilized by parallel compensator control.

In the literature mentioned above, such as [13,16] and [20], the exponential stabilization result for various string systems by linear boundary control mainly relies on the Lyapunov direct method. In this work, we investigate the stabilization of string (1) with a more general and ‘flexible’ boundary control (see hypothesis (H) in Section 2). The feedback function u is not required to satisfy a strict control law such as (6), but just satisfies some appropriate polynomial-type constraint. In this general boundary control case, it seems that the Lyapunov direct method is no more applicable. So, we need to use a more meticulous method to deal with the stabilization problem. Applying the multiplier method, we establish not only exponential stability result but also polynomial stability result for Kirchhoff string (1).

The remainder of this technical paper is arranged as follows. Section 2 describes the model of the Kirchhoff nonlinear string and introduces the control assumption. The problem of exponential and polynomial stability is addressed in Section 3. Finally, a numerical example is demonstrated where the nonlinear distributed parameter infinite-dimensional equation is solved by applying the finite element method in Section 4.

2 Problem formulation

Consider the nonlinear Kirchhoff string model as shown in Figure 1. For the sake of easy reading and later referring, the governing equation, the boundary conditions and the initial functions are put together as

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M24">View MathML</a> in equation (7d) are the initial displacement and velocity of the string, respectively. We assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M26">View MathML</a> and that at least one of the functions f or g is not identically zero over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M27">View MathML</a>.

To obtain a precise stabilization result, we make the following hypothesis on the continuous control feedback <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M28">View MathML</a>:

(H) There exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M30">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M31">View MathML</a>

(8)

Remark 2.1 Obviously, condition (H) is equivalent to, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M32">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M33">View MathML</a>

Obviously, hypothesis (H) is a ‘flexible’ and ‘robust’ condition, which allows the feedback function u to vary in an appropriate geometric region given by a polynomial-type constraint. For example, Figure 2 illustrates a feedback control u satisfying a linear sector constraint (H) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M36">View MathML</a>, and Figure 3 shows a feedback control u satisfying a nonlinear constraint (H) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M36">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M40">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3">View MathML</a>, the boundary control (7c) is the negative feedback of transversal velocity of the string at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M13">View MathML</a>.

thumbnailFigure 2. Linear sector constraint.

thumbnailFigure 3. Nonlinear sector constraint.

For the existence and uniqueness of the solution of the general Kirchhoff equation, we refer to [11,12] and references therein. In this work, we study the stabilization of the string in (7a) by this negative feedback boundary control u, which provides a dissipative effect.

Remark 2.2 According to boundary condition (7b) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M12">View MathML</a>, we easily get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M44">View MathML</a>

(9)

We define the natural energy function of time for system (7a)-(7d) and denote it by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M45">View MathML</a>. The scalar-valued function E is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M46">View MathML</a>

(10)

Especially, from the initial displacement and velocity condition (7d), we obtain the initial energy as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M47">View MathML</a>

Since at least one of the functions f and g is not identically zero over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M48">View MathML</a> we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M49">View MathML</a>.

3 Stabilization by boundary control

In this section we state and prove our main result. For this purpose we establish several lemmas.

Lemma 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M50">View MathML</a>be the solution for system (7a)-(7d). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M51">View MathML</a>

(11)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M52">View MathML</a>

(12)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M53">View MathML</a>

(13)

Proof See the Appendix. □

Now, we give a property of the energy function E.

Proposition 3.1The time-derivative of the energy functionEin equation (10), along the solution of system (7a)-(7d) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M54">View MathML</a>

(14)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3">View MathML</a>.

Proof Differentiating the energy function (10) with respect to t, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M56">View MathML</a>

(15)

According to equation (11) and boundary control (7c), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M57">View MathML</a>

(16)

Substituting equation (16) into equation (15) and observing (7a), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M58">View MathML</a>

(17)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3">View MathML</a>. We obtain equation (14). □

Remark 3.1 From Proposition 3.1, we obtain the energy identity for system (7a)-(7d),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M60">View MathML</a>

Therefore, the energy E is a decreasing function of time.

During the subsequent stability analysis, we utilize the following inequality.

Lemma 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M50">View MathML</a>be the solution for system (7a)-(7d). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M62">View MathML</a>

(18)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3">View MathML</a>.

Proof Applying the Cauchy-Schwarz inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M64">View MathML</a>

(19)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3">View MathML</a>. On the other hand, the definition of energy function (10) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M66">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3">View MathML</a>. It follows from the above inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M68">View MathML</a>

(20)

Together with (19) and (20), we get equation (18). Hence we complete the proof of Lemma 3.2. □

Now, we present a Gronwall-type lemma (see Komornik [22], pp.124), which will play an essential role when establishing the stabilization result.

Lemma 3.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M69">View MathML</a>be a non-increasing function. Assume that there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M70">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M71">View MathML</a>

Then the following estimation is true, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M73">View MathML</a>

We give a priori estimation for the energy function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M74">View MathML</a>, which was established in [15]. For the sake of completeness, we give the proof here.

Lemma 3.4The energy functionEin equation (10), along the solution of system (7a)-(7d), satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M75">View MathML</a>

(21)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3">View MathML</a>.

Proof We multiply equation (7a) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M77">View MathML</a> and do integration over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M27">View MathML</a>, with respect to x. We obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M79">View MathML</a>

(22)

using equations (13) and (12) in Lemma 3.1. It follows from (10) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M80">View MathML</a>

(23)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M81">View MathML</a>, according to boundary control (7c), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M82">View MathML</a>

(24)

Hence, substituting equation (23) into equation (22) and using equation (24), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M83">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M84">View MathML</a>, we complete the proof of Lemma 3.4. □

Lemma 3.5For any constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M85">View MathML</a>, the energy functionEalong the solution of system (7a)-(7d) satisfies the following estimation, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M86">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M87">View MathML</a>

(25)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M88">View MathML</a>.

Proof According to inequality (21) in Lemma 3.4, we have, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M89">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M90">View MathML</a>

(26)

Moreover, using integration by parts, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M91">View MathML</a>

Hence, inequality (26) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M92">View MathML</a>

(27)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M93">View MathML</a>

Firstly, we estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M94">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M95">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M96">View MathML</a>

(28)

where the last inequality follows from the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M74">View MathML</a> is a decreasing function. On the other hand,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M98">View MathML</a>

(29)

Finally, inserting the two inequalities, (28) and (29), in (27), we get inequality (25). This completes the proof of Lemma 3.5. □

We now state the main stabilization result for system (7a)-(7d).

Theorem 3.1Assume that assumption (H) holds. Then there exist three constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M99">View MathML</a>such that, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M101">View MathML</a>

(30)

Remark 3.2 It is worth to mention that Theorem 2.4 in [13] can be viewed as a special cases of Theorem 3.1. Indeed, in the linear control case (6), the exponential stability in Theorem 3.1 coincides with the result in [13].

Proof of Theorem 3.1 We distinguish two cases related to the parameter r to establish the energy decay rate.

Case (I): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M34">View MathML</a>;

Case (II): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M103">View MathML</a>.

In Case (I), we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M104">View MathML</a>. According to hypothesis (H), we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M105">View MathML</a>

Hence, from inequality (25) and equation (14), we deduce that, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M86">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M107">View MathML</a>

(31)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M108">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M109">View MathML</a> is given in Lemma 3.5.

Now we deal with Case (II). In this case, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M110">View MathML</a>. We first admit the following fact (the proof is given in the Appendix).

Claim 1For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M111">View MathML</a>, we have the following estimates, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M86">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M113">View MathML</a>

(32)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M114">View MathML</a>

(33)

Now, inserting inequalities (32) and (33) into (25), we obtain, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M86">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M116">View MathML</a>

(34)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M117">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M118">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M109">View MathML</a> is given in Lemma 3.5. Now we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M120">View MathML</a>. Then it is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M121">View MathML</a>. Hence, inequality (34) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M122">View MathML</a>

Recalling <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M123">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M124">View MathML</a>. Hence, the above inequality is rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M125">View MathML</a>

(35)

where the last inequality follows from Remark 3.1. Finally, by letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M126">View MathML</a> in (31), (35) and using Lemma 3.3 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M127">View MathML</a>, we complete the proof of Theorem 3.1.  □

Remark 3.3 According to the proof of Theorem 3.1, it is easy to see that the constants σ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M128">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M129">View MathML</a> in Theorem 3.1 can be chosen as, respectively, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M131">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M132">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M133">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M134">View MathML</a>. This means that the coefficients of the exponential or polynomial decay rate are exactly determined only by the initial tension a, the initial energy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M135">View MathML</a> and the feedback control u. However, in the polynomial decay case, the order of decay rate is determined only by the feedback control u.

Finally, it is shown that the boundary control u stabilizes the nonlinear Kirchhoff string.

Theorem 3.2Assume that assumption (H) holds. Then there exist two constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M136">View MathML</a>such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M2">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M139">View MathML</a>

(36)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M141">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M128">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M129">View MathML</a>, σare given in Theorem 3.1.

Proof According to the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M144">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M146">View MathML</a>

(37)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3">View MathML</a>. By combining (37) with (30) in Theorem 3.1, we complete the proof of Theorem 3.2. □

4 Numerical results

In this section we consider a computational example for the closed-loop system (7a)-(7d). To illustrate the control performance of the boundary control law satisfying condition (H), numerical simulations by using the finite element method (FEM) are performed. We use Lagrange ‘hat’ basis with FEM equidistant meshes. The system parameters used in the simulations are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M150">View MathML</a>. The initial conditions are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M151">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M152">View MathML</a>. That is we consider the following Kirchhoff system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M153">View MathML</a>

(38)

The dynamic responses of the controlled Kirchhoff string were simulated under two feedback control laws:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M154">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M155">View MathML</a>

Obviously, the feedback control function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M156">View MathML</a> satisfies the constraint condition (H) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M159">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M160">View MathML</a> satisfies the constraint condition (H) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M162">View MathML</a>. Then, according to Theorem 3.2, the asymptotic behavior of the transverse vibration of system (38) under control law <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M156">View MathML</a> (or control law <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M160">View MathML</a>) possesses exponential decay (or polynomial decay with degree −1, because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M37">View MathML</a>). The string response <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M7">View MathML</a> of closed-loop system (38) with control law <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M156">View MathML</a> and control law <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M160">View MathML</a> are shown in Figure 4 and Figure 5, respectively. The corresponding transversal displacement at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M169">View MathML</a> is shown in Figures 6 and 7, respectively.

thumbnailFigure 4. String response of closed-loop system (38) under control law<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M170">View MathML</a>.

thumbnailFigure 5. String response of closed-loop system (38) under control law<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M171">View MathML</a>.

thumbnailFigure 6. Transverse displacement of the string at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M172">View MathML</a>under control law<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M170">View MathML</a>.

thumbnailFigure 7. Transverse displacement of the string at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M172">View MathML</a>under control law<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M171">View MathML</a>.

From Figures 4 and 5, it can bee seen that, in the case of control law <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M160">View MathML</a>, the decay of the transverse vibration relatively slow compared to the case of control law <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M156">View MathML</a>. Indeed, from Figure 6 and Figure 7, we know that in the case of control law <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M156">View MathML</a>, the transverse vibration has been suppressed exponentially (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M179">View MathML</a>), whereas in the case of control law <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M160">View MathML</a>, the transverse vibration has been suppressed polynomially (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M181">View MathML</a>). It is coincident with the results of Theorem 3.2.

Appendix

A.1 Proof of Lemma 3.1

Integrating by parts and (9) we get, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M183">View MathML</a>

So we obtain equation (11). Next, integrating by parts we compute

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M184">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3">View MathML</a>. That is, equation (12) holds. Finally, we write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M186">View MathML</a>

(39)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M187">View MathML</a>

(40)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M3">View MathML</a>. Substituting equation (40) into equation (39), we obtain equation (13). Thus the proof of Lemma 3.1 is complete.

A.2 Proof of Claim 1

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M103">View MathML</a>, hypothesis (H) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M190">View MathML</a>

Hence, it is true that, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M191">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M192">View MathML</a>

(41)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M193">View MathML</a>

(42)

Then, from (41) and Proposition 3.1 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M194">View MathML</a>

(43)

From Young’s inequality, we have, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M195">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M196">View MathML</a>

(44)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M197">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M198">View MathML</a>. Applying the above inequality with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M199">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M200">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M201">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M202">View MathML</a>, we obtain, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M111">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M204">View MathML</a>

(45)

By inserting (45) into (43), we deduce inequality (32) in Claim 1. Similarly, from (42) and Proposition 3.1 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M205">View MathML</a>

(46)

Applying again inequality (44) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M199">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M207">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M201">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M202">View MathML</a>, we obtain, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M111">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/215/mathml/M211">View MathML</a>

(47)

Inserting (47) into (46), we get inequality (33) in Claim 1.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant 11226128) and the Natural Science Foundation of Heilongjiang Province of China (Grant F201113).

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