Stabilization of Vibrating Beam with a Tip Mass Controlled by Combined Feedback Forces

Journal of Mathematical Analysis and Applications 256, 13᎐38 Ž2001. doi:10.1006rjmaa.2000.7217, available online at http:rrwww.idealibrary.com on

Stabilization of Vibrating Beam with a Tip Mass Controlled by Combined Feedback Forces1 Shengjia Li,2 Yiaoting Wang, and Zhandong Liang Department of Mathematics, Shanxi Uni¨ ersity, Taiyuan 030006, People’s Republic of China

Jingyuan Yu Institute of Information and Control, Beijing 100037, People’s Republic of China

and Guangtian Zhu Institute of Systems Science, The Chinese Academy of Sciences, Beijing 100080, People’s Republic of China Submitted by Konstantin A. Lurie Received January 11, 1999

A flexible structure consisting of a Euler᎐Bernoulli beam with a tip mass is considered. To stabilize this system we use a boundary control laws: yu x x x Ž1, t . q mu t t Ž1, t . s y␣ u t Ž1, t . q ␤ u x x x t Ž1, t . and u x x Ž1, t . s y␥ u x t Ž1, t .. A sensitivity asymptotic analysis of the system’s eigenvalues and eigenfunctions is set up. We prove that all of the generalized eigenfunctions of Ž2.9. form a Riesz basis of H . By a new method, we prove that the operator A generates a C0 contraction semigroup T Ž t ., t G 0. Furthermore T Ž t ., t G 0, is uniformly exponentially stable and the optimal exponential decay rate can be obtained from the spectrum of the system. 䊚 2001 Academic Press

Key Words: beam equation; boundary feedback control; Riesz basis; exponential stabilization; optimal exponential decay rate.

1 2

Project 69674011 supported by NSFC and Shanxi Foundation of Science. E-mail address: [email protected]. 13 0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.

14

LI ET AL.

1. INTRODUCTION In this paper, we consider stability of a vibrating beam system clamped at one end, controlled by combined forces, with a mass attached at the other end. This system is modeled by u t t q u x x x x s 0,

0 - x - 1,

u Ž 0, t . s u x Ž 0, t . s 0,

t G 0,

t G 0,

yu x x x Ž 1, t . q mu t t Ž 1, t . s y␣ u t Ž 1, t . q ␤ u x x x t Ž 1, t . , u x x Ž 1, t . s y␥ u x t Ž 1, t . ,

t G 0,

Ž 1.1.

t G 0,

where m ) 0 is the tip mass, and ␣ and ␤ are positive constants. Our problem is to prove that the solutions of the resulting closed-loop system decay uniformly to zero and the optimal decay rate can be determined by the spectrum of the closed-loop system. The model given by Ž1.1. is a variant of the SCOLE model, in the sense that u x x Ž1, t . s 0 and yu x x x Ž1, t . q mu t t Ž1, t . s 0, which has been studied in the past by many authors; see, e.g., w1x. It is known that for such types of models the feedback law u x x s 0, yu x x x Ž 1, t . q mu t t Ž 1, t . s y␣ u t Ž 1, t . ,

␣ ) 0,

t G 0,

Ž 1.2. is sufficient for strong Ži.e., asymptotic. stability, but not sufficient for uniform stability. It is well known that such compact perturbations are not sufficient to provide uniform stabilization. Hence, to obtain uniform stability one has to choose ‘‘stronger’’ feedback terms such as u x x x t ; see Rao w12x, where the lack of uniform stability for the SCOLE model with usual feedback laws was proven by using the compactness argument, and also uniform decay of the energy was obtained by means of higher order feedback for rather smooth initial data. In Bal w1x, a three dimensional model for the SCOLE system, including the moment of inertia at x s 1, was considered and then a feedback law similar to Ž1.2. and another feedback law based on optimal control techniques were studied. As stated above, these results only show the asymptotic or uniform decay of energy for the system considered, but do not prove the optimality of the decay rate. In w11x, Li considered the optimality decay rates for m s 0, ␣ G 0, and u x x Ž1, t . s yK 2 u x t Ž1, t ., respectively. In w9x, the optimality is studied for a cable with a mass. In this paper we investigate the uniform stability of the system given by Ž1.1. in a general case.

15

STABILIZATION OF VIBRATING BEAM

In order to study system Ž1.1., we rewrite Ž1.1. in the following equivalent form: u t t Ž x, t . q u x x x x Ž x, t . s 0,

0 - x - 1,

u Ž 0, t . s u x Ž 0, t . s 0,

t G 0,

t G 0,

yu x x x Ž 1, t . q mu t t Ž 1, t . s y␣ u t Ž 1, t . q ␤ u x x x t Ž 1, t . , u x t Ž 1, t . s y␥ u x x Ž 1, t . ,

t G 0,

Ž 1.3.

t G 0.

This paper is organized as follows. In the next section, we construct an assistant system, and we analyze the spectrum of system Ž1.1. or Ž1.3.. In Section 3, we get the representations of the eigenvalues and eigenfunctions of system Ž1.3. and the assistant system. In Section 4, we get the main result of this paper.

2. BASIC SPACES AND ANALYSIS OF THE SPECTRUM To study system Ž1.3., we usually define the auxiliary function ␩ as

␩ Ž t . s yu x x x Ž 1, t . q

m

␤

u t Ž 1, t . ,

t G 0.

Ž 2.1.

Inserting Ž2.1. into Ž1.3., we get that

␤␩ ˙Ž t. q ␩Ž t. q ␣ y

ž

m

␤

/

u t Ž 1, t . s 0,

t G 0,

Ž 2.2.

where a dot represents the time derivative. Let us introduce the spaces W s  u; u g H 2 Ž 0, t . , u Ž 0 . s u x Ž 0 . s 0 4 , T

H s  Ž u, ¨ , ␩ . ; u g W , ¨ g L2 Ž 0, 1 . , ␩ g R 4 ,

Ž 2.3. Ž 2.4.

where the superscript T stands for the transpose and the space H 2 Ž0, 1. stands for the Sobolev space. In H , we define the inner-product ² y, y :H s

1

H0

Ž u x x u x x q ¨¨ . dx q K␩␩ ,

Ž 2.5.

16

LI ET AL.

where y s Ž u, ¨ , ␩ .T g H , y s Ž u, ¨ , ␩ .T g H , and K ) 0 is chosen as

Ks

␤2 m q ␣␤

.

Ž 2.6.

Next we define the unbounded operator A, ¨

yu x x x x A ¨ s , 1 1 m ␩ y ␩y ␣y ¨ Ž 1. ␤ ␤ ␤ u

ž/



ž

/

0

T Ž u, ¨ , ␩ . g D Ž A . , Ž 2.7.

where T

D Ž A . s Ž u, ¨ , ␩ . ; u g H 4 Ž 0, 1 . l W , ¨ g W , ␩ g R;

½

¨ x Ž 1 . s y␥ u x x Ž 1 . , ␩ s yu x x x Ž 1 . q

m

␤

5

¨ Ž 1. .

Ž 2.8.

From the above notation, let u s u and ¨ s u t ; system Ž1.3. and Ž2.2. can be written formally as Y˙Ž t . s A Y Ž t . ,

t ) 0,

Y Ž 0 . s Y0 g H ,

Ž 2.9.

where y s Ž u, ¨ , ␩ .T , and ␩ is defined by Ž2.1.. In the rest of this section we investigate the spectrum of the operator A. To obtain this result, we compare the flexible beam with a tip mass to an assistant flexible beam without a tip mass, for the spectral properties. Let ␭ g C be an eigenvalue of A and let y s Ž u, ¨ , ␩ .T g DŽ A . be a corresponding eigenvector. The eigenproblem of Ž1.3. or Ž2.9. can be written as

␭2␾ Ž x . q ␾ Ž4. Ž x . s 0, 0 - x - 1, ␾ Ž 0 . s ␾ x Ž 0 . s 0, ␭␾ x Ž 1 . s y␥␾ x x Ž 1 . , ␭Ž ␣ q m ␭ . ␾ Ž 1 . y Ž ␤␭ q 1 . ␾ x x x Ž 1 . s 0.

Ž 2.10.

17

STABILIZATION OF VIBRATING BEAM

Let ␭2 s ␳ 4 , and let ␻ k , k s 1, 2, 3, 4, represent the roots of equation ␻ 4 s y1. The eigendeterminant is ⌬U Ž ␭ .

s␳ 2

␻1

␻2

␻3

␻4

1

1

1

1

␦ 31

␦ 32

␦ 33

␦ 34

Ž ␳␥␻ 1 q ␻ 12 . e ␻

␳

1

Ž ␳␥␻ 2 q ␻ 22 . e ␻

2

␳

,

Ž ␳␥␻ 3 q ␻ 32 . e ␻ ␳ Ž ␳␥␻4 q ␻42 . e ␻ s

4

␳

Ž 2.11. where ␦ 3 j s Ž ␳ 2 Ž ␣ q m ␳ 2 . y Ž ␤␳ 2 q 1. ␻ k3 ␳ 3 . e ␻ k ␳ , k s 1, 2, 3, 4. From Ž2.7. and Ž2.10., by computing, we have the following lemmas. LEMMA 2.0. Re ␭ - 0.

Let ␭ be an eigen¨ alue of Ž2.10., and let im ␭ / 0. Then

Proof. Multiplying the first equation of Ž2.10. by ␾ Ž x . and integrating from 0 to 1 by part, we get that

␭2

1

H0

␾ Ž x . ␾ x x Ž x . dx q sy

␣␭ q ␭2 1 q ␤␭

1

H0

␾ x x Ž x . ␾ Ž x . dx

2

␾ Ž 1.

y

␭ ␥

2

␾ x Ž 1. .

Multiplying the first adjoint equation of Ž2.10. by ␾ Ž x . and integrating from 0 to 1 by part, we get that

␭2

1

H0

␾ Ž x . ␾ Ž x . dx q ␣␭ q ␭2

sy

1 q ␤␭

1

H0

␾ x x Ž x . ␾ Ž x . dx

␾ Ž 1.

2

y

␭ ␥

2

␾ x Ž 1. .

From the above equations, we get that 1

2

Ž ␭ q ␭ . Ž ␭ y ␭ . H ␾ Ž x . dx 0

s

ž

␣␭ q ␭2 1 q ␤␭

y

␣␭ q ␭2 1 q ␤␭

/

␾ Ž 1.

2

q

␭y␭ ␥

2

␾ x Ž 1. .

18

LI ET AL.

If Im ␭ / 0, by computing, we get

Ž ␭ q ␭.

1

žH

sy

0

␾ Ž x . ␾ Ž x . dx q

␣ q ␤ < ␭< 2 <1 q ␤␭ < 2

␾ Ž 1.

2

␾ Ž 1.

2

<1 q ␤␭ < 2 y

␾ x Ž 1.

/ 2

␥

.

From this, we get that Re ␭ - 0. The proof is complete. LEMMA 2.1. Function ␺␭Ž x . g H 4 Ž0, 1. is an eigenfunction of Ž2.10. corresponding to eigen¨ alue ␭ if and only if the ¨ ector Ž ␺␭, ␭␺␭ , ␩␭ .T is an eigen¨ ector of Ž2.9. corresponding to the eigen¨ alue ␭ Ž where ␩␭ s y␺␭Z Ž1. q m Ž .. ␤ ␭␺ 1 . The proof is obvious. We omit it. In order to investigate the eigenvalues and eigenfunctions of system Ž2.10., we introduce a special assistant system as the following: u t t Ž x, t . q u x x x x Ž x, t . s 0, u Ž 0, t . s u x Ž 0, t . s 0,

0 - x - 1,

u x t Ž 1, t . s u x x x t Ž 1, t . s 0,

t G 0.

Ž 2.12.

The eigenproblem of Ž2.12. is

␭2 u Ž x . q uŽ4. Ž x . s 0, 0 - x - 1, u Ž 0 . s uX Ž 0 . s 0, ␭ uX Ž 1 . s 0, ␭ uZ Ž 1 . s 0.

Ž 2.13.

The eigendeterminant is

␻1 1 ⌬ 0 Ž ␭ . s ␳ 3␻ e ␻ 1 ␳ 1

␻2 1 ␳ 3␻ 2 e ␻ 2 ␳

␻3 1 ␳ 3␻ 3 e ␻ 3 ␳

␻4 1 ␳ 3␻4 e ␻ 4 ␳ ,

␳ 5␻ 13 e ␻ 1 ␳

␳ 5␻ 23 e ␻ 2 ␳

␳ 5␻ 33 e ␻ 3 ␳

␳ 5␻43 e ␻ 4 ␳

Ž 2.14.

where ␻ k , k s 1, 2, 3, 4, are four roots of ␻ 4 q 1 s 0. Let

␰ Ž t . s u x x x Ž 1, t . ;

then ␰ t Ž t . s u x x x t Ž 1, t . s 0.

Ž 2.15.

STABILIZATION OF VIBRATING BEAM

19

We now define an assistant operator as ¨ u A0 ¨ s yu x x x x , ␰ 0

0 0

T Ž u, ¨ , ␰ . g D Ž A0 . .

Ž 2.16.

where DŽ A0 . s Ž u, ¨ , ␰ .; u g H 4 Ž0, 1. l W , ¨ g W , ␰ g R1 , ¨ X Ž1. s 0, ␰ s uZ Ž1.4 . let u s u, ¨ s u t . By operator A0 , systems Ž2.12. can be written as d dt

Y Ž t . s A0 Y Ž t . , Y Ž 0 . s Y0 ,

0 - t - q⬁,

Ž 2.17.

Y0 g H .

The eigenequation of Ž2.17. can be represented as

␭ u Ž x . y ¨ Ž x . s 0,

0 - x - 1,

␭¨ Ž x . q u Ž x . s 0, ␭␰ s 0,

0 - x - 1,

Ž4.

Ž 2.18.

with the boundary conditions u Ž 0 . s uX Ž 0 . s 0,

␭ uX Ž 1 . s 0,

␭ uZ Ž 1 . s 0.

Ž 2.19.

Obviously, ␭ s 0 is an eigenvalue of Ž2.18.. The corresponding eigenfunction are ⌽ 01 s Ž x 2 , 0, 0.T and ⌽ 02 s Ž x 3 , 0, 6.T . If ␭ / 0, the eigenvalue problem of Ž2.17. is equivalent to

␭ u Ž x . y ¨ Ž x . s 0,

0 - x - 1,

␭¨ Ž x . q u Ž x . s 0, 0 - x - 1, Z ␰ s 0, Ž ␰ s u Ž 1. . , Ž4.

Ž 2.20.

with the boundary condition u Ž 0 . s uX Ž 0 . s uX Ž 1 . s uZ Ž 1 . s 0,

Ž 2.21.

or equivalently A0

␾ ␾ s␭ , ␺ ␺

␾ g D Ž A0 . , ␺

ž/ ž/ ž/

Ž 2.22.

20

LI ET AL.

where A0 s



y

0 d4 dx

4

I 0

0

,

and D Ž A 0 . s  Ž ␾ , ␺ . ; ␾ g H 4 Ž 0, 1 . l W 0 , ␺ g W , ␾ Z Ž 1 . s 0 4 , T

where W 0 s  ␾ g W ; ␾X Ž 1. s 04 .

Ž 2.23.

That is, if ␭ / 0 is an eigenvalue of Ž2.17. or Ž2.22., the corresponding eigenfunction has the form ⌽ s Ž u, ¨ , 0.T . By the definition of A 0 , Ž2.22. is equivalent to the following quadratic eigenproblem:

␭2␾ Ž x . q ␾ Ž4. Ž x . s 0, 0 - x - 1, X X ␾ Ž 0 . s ␾ Ž 0 . s 0, ␾ Ž 1 . s ␾ Z Ž 1 . s 0.

Ž 2.24.

Because A 0 is a skew operator in the space W 0 = L2 Ž0, 1., all of the eigenfunctions of Ž2.22. or Ž2.20. form an orthogonal basis in space W 0 = L2 Ž0, 1.. From this, we get the following lemma. LEMMA 2.2. space H .

All of the eigen¨ ectors of A0 or Ž2.17. form a Riesz basis in

Proof. Obviously, all of the eigenvectors ⌽nUj s Ž ␾n jr␭ n j , ␾n j .T , j s 1, 2, n s 1, 2, 3, . . . , of A 0 form an orthogonal basis in space W 0 = L2 Ž0, 1.. Because vectors ⌽ 01 s Ž x 2 , 0, 0.T and ⌽ 02 s Ž x 3 , 0, 6.T are two eigenvectors of A0 corresponding the eigenvalue ␭ 0 s 0, we get that ⌽ 01 s Ž x 2 , 0, 0.T , ⌽ 02 s Ž x 3, 0, 6.T , and ⌽n j s Ž ␾n jr␭ n j , ␾n j , 0.T s , j s 1, 2, n s 1, 2, 3, . . . , form a Riesz’s basis of space H . The proof is complete. Let ␭ s ␳ 2 / 0, and let ␻ j , j s 1, 2, 3, 4, be the four roots of ␻ 4 s y1. The eigendeterminant of Ž2.24. can be written as

␻1 1 ⌬U0 Ž ␭ . s ␳ 3␻ 3 e ␻ 1 ␳ 1

␻2 1 ␳ 3␻ 23 e ␻ 2 ␳

␻3 1 ␳ 3␻ 33 e ␻ 3 ␳

␻4 1 ␳ 3␻43 e ␻ 4 ␳

␳ 3␻ 1 e ␻ 1 ␳

␳ 3␻ 2 e ␻ 2 ␳

␳ 3␻ 3 e ␻ 3 ␳

␳ 3␻4 e ␻ 4 ␳

s y4'2 ␳ 6 Ž sinh '2 ␳ q sin '2 ␳ . .

Ž 2.25.

STABILIZATION OF VIBRATING BEAM

21

Taking ␻ 1 s e 3␲ i r4 , ␻ 2 s e 5␲ i r4 , ␻ 3 s e␲ i r4 , ␻4 s ey␲ i r4 , and ␭ s ␳ 2 , let

␻1 1 ⌬ 1Ž ␳ . s ␻ 3 e ␻ 1 ␳ 1

␻2 1 ␻ 23 e ␻ 2 ␳

␻3 1 ␻ 33 e ␻ 3 ␳

␻4 1 ␻43 e ␻ 4 ␳

␻ 12 e ␻ 1 ␳

␻ 22 e ␻ 2 ␳

␻ 32 e ␻ 3 ␳

␻42 e ␻ 4 ␳

s 4 Ž cosh '2 ␳ q cos '2 ␳ q 2 . ,

␻1 1 ⌬2Ž ␳ . s e ␻1 ␳ ␻ 12 e ␻ 1 ␳

␻2 1 e␻2 ␳ ␻ 22 e ␻ 2 ␳

␻3 1 e␻3 ␳ ␻ 32 e ␻ 3 ␳

Ž 2.26. ␻4 1 e␻4 ␳ ␻42 e ␻ 4 ␳

s 4'2 Ž sinh '2 ␳ y sin '2 ␳ . ,

␻1 1 ⌬3Ž ␳ . s 3 ␻1 ␳ ␻1e ␻1e␻1 ␳

␻2 1 ␻ 23 e ␻ 2 ␳ ␻2 e␻2 ␳

␻3 1 ␻ 33 e ␻ 3 ␳ ␻3 e␻3 ␳

Ž 2.27. ␻4 1 ␻43 e ␻ 4 ␳ ␻4 e ␻ 4 ␳

s y4'2 Ž sinh '2 ␳ q sin '2 ␳ . ,

Ž 2.28.

and

␻1 1 ⌬4Ž ␳ . s e␻1 ␳ ␻ 1 e ␻1 ␳

␻2 1 e␻2 ␳ ␻2 e␻2 ␳

␻3 1 e␻3 ␳ ␻3 e␻3 ␳

␻4 1 e␻4 ␳ ␻4 e ␻ 4 ␳

s 4 Ž cosh '2 ␳ q cos '2 ␳ y 2 . .

Ž 2.29.

Then the eigendeterminants of Ž2.11. can be represented as ⌬U Ž ␳ . s y␳ 3 Ž ␤␳ 2 q 1 . Ž ␳ 3⌬ 3 Ž ␳ . q ␥␳ 2⌬ 1 Ž ␳ . . q Ž ␣ q m ␳ 2 . ␳ 2 ␳ 3⌬ 4 Ž ␳ . q ␥␳ 2⌬ 2 Ž ␳ . s 4'2 ␤␳ 8 Ž sinh '2 ␳ q sin '2 ␳ . q 4'2 ␳ 6 Ž sinh '2 ␳ q sin '2 ␳ . q 4␥ Ž ␤␳ 2 q 1 . ␳ 5 Ž cosh '2 ␳ q cos '2 ␳ q 2 . q ␳ 2 Ž ␣ q m ␳ 2 . ␳ 34 Ž cosh '2 ␳ q cos '2 ␳ y 2 . q4'2 ␥␳ 2 Ž sinh '2 ␳ y sin '2 ␳ . ,

Ž 2.30.

22

LI ET AL.

and the eigendeterminant of Ž2.14. can be represented as ⌬ 0 Ž ␳ . s ␳ 2⌬U0 Ž ␳ . s ␳ 8⌬ 3 Ž ␳ . s 4'2 ␳ 8 Ž sinh '2 ␳ q sin '2 ␳ . . Ž 2.31. Let z s '2 ␳ , and let ⌫n s  z s x q iy g C; x q y s 2 n␲ ; y y x s 2 n␲ , 0 F < x < F 2 n␲ and 0 F y F 2 n␲ 4 , for n s 1, 2, 3, . . . . Let Cn s  z 2 ; z g ⌫n4 , n s 1, 2, 3, . . . . Obviously, Cn is a series of simple closed curves. We have the following lemmas. LEMMA 2.3. ha¨ e that

Let S s

'cosh

2

x q cosh 2 y ; then, for n large enough, we

1 2

S F
1 2

S F
for z s x q iy g ⌫n .

Ž 2.32.

Proof. Since cosh z q cos z s cos x cosh y q cos x cosh x q i Ž ysin x sinh x y sin x sinh y . , for z g ⌫n

and

x q y s 2 n␲ ,

we have
for z g ⌫n

and

x q y s 2 n␲ ,

Similarly, we get that
for z g ⌫n

and

y y x s 2 n␲ ,

Obviously, we have
for z g ⌫n .

'S 2 y 2 F

23

STABILIZATION OF VIBRATING BEAM

Since sin z " sinh z s sin x cosh y " cos x sinh x q i Ž cos x sinh y " sin y cosh x . , we have
for z g ⌫n

and

n large enough.

Obviously, we have
S 2 y 2 F
That is, 12 S F
␳y5⌬ Ž ␭ . y ␤␳y5⌬ 0 Ž ␭ . ␤ ␳y5⌬ 0 Ž ␭ . 2 z 2 Ž sin z q sinh z . q 2 z s

q - 1,

␤

z 2 q 1 Ž cos z q cosh z q 2 . 2 z 4 Ž sinh z q sin z .

ž

mz 2 2

q␣

/

ž

/

2 z Ž cosh z q cos z y 2 . q 2'2 ␥ Ž sinh z y sin z . zy1 z 4 Ž sinh z q sin z .

for ␭ g Cn

and

n large enough.

Ž 2.33.

By Rouche’s theorem, we see that ⌬ 0 Ž ␭. and ⌬U Ž ␭. have the same number of zeros in the interior of Cn for n large enough, and so do ␳y5⌬ 0 Ž ␭. and ␳y5⌬U Ž ␭..

24

LI ET AL.

Because ␭ s 0 is not an eigenvalue of A, ␴p Ž A . s  ␭ g C; ␭ / 0 and such that ⌬Ž ␭. s 04 . Obviously, ␭ s 0 is not a zero of ␳y5⌬U Ž ␭.. By computing, ␭ s 0 is an eigenvalue of A0 with eigenvectors ⌽ 0 j , j s 1, 2. By Ž2.18. ᎐ Ž2.21. and the symmetry of Cn with respect to the real axis, we see that there exist 2 m Žcounting by the multiple number of its eigenvectors. eigenvalues of A0 in the interior of Cn . Therefore there also exist 2 m generalized eigenvectors corresponding to all of the eigenvalues of A in the interior of Cn . The proof is complete. 3. ASYMPTOTIC REPRESENTATIONS OF EIGENVALUE AND EIGENFUNCTION In this section we give the asymptotic representations of the eigenvalues and eigenfunctions of assistant system Ž2.12. and system Ž1.3.. We first study the eigenvalues and eigenfunctions of the assistant system Ž2.12.. Because, for ␭ s 0, all of the solutions of Ž2.18. are ␾ Ž x . s ax 2 , and ␾ Ž x . s bx 3 , a, b g C, to study system Ž2.12., we only need to study system Ž2.20.. We will consider the quadratic eigenproblem Ž2.24.. Let ␻ 1 , ␻ 2 , ␻ 3 , and ␻4 be the four roots of ␻ 4 s y1, and let ␳ 4 s ␭2 . Then ␾ j Ž x . s e ␻ j ␳ x , j s 1, 2, 3, 4, are the basic solutions of ␾ YY Ž x . q ␭2␾ Ž x . s 0. For a complex number ␭ s ␳ 2 , the eigenfunction of Ž2.22. can be represented as 4

Ý cj e ␻ ␳ x ,

␾ Ž x, ␳ . s

j

js1

where c1 , c 2 , c 3 , and c 4 are arbitrary constants Žat least one of them not zero. and satisfy

␻ 1 c1 q ␻ 2 c 2 q ␻ 3 c 3 q ␻4 c 4 s 0, c1 q c 2 q c 3 q c 4 s 0, ␻ 13 e ␻ 1 ␳ c1 q ␻ 23 e ␻ 2 ␳ c 2 q ␻ 33 e ␻ 3 ␳ c3 q ␻43 e ␻ 4 ␳ c 4 s 0, ␻ 1 e ␻ 1 ␳ c1 q ␻ 2 e ␻ 2 ␳ c 2 q ␻ 3 e ␻ 3 ␳ c3 q ␻4 e ␻ 4 ␳ c 4 s 0.

Ž 3.1.

By linear algebraic theory, we have

␻1 1 ⌬ 3 Ž ␭. s 3 ␻ 1 ␳ ␻1 e ␻1 e␻1 ␳

␻2 1 3 ␻2 ␳ ␻2 e ␻2 e␻2 ␳

␻3 1 3 ␻3 ␳ ␻3 e ␻3 e␻3 ␳

␻4 1 3 ␻ 4 ␳ s 0. ␻4 e ␻4 e ␻ 4 ␳

Ž 3.2.

STABILIZATION OF VIBRATING BEAM

25

Obviously, ⌬ 3 Ž ␭. is the eigendeterminant of Ž2.24.. We now estimate the roots of the eigendeterminant ⌬ 3 Ž ␭.. LEMMA 3.1. The eigen¨ alues of Ž2.22. can be asymptotically represented as

␭ n1 s ey5 r4␲ i␲ yn q

ž ž

1

1

iqO

2

/ ž // / ž ž // ž ž / ž // ž / ž ž // ž

s ny

2

1

i␲ 1 q O

4

␭ n2 s ey3 r4␲ i␲ n y sy ny

4

1 4

1

1

2

Ž 3.3.

2

1

n

1

i␲ 1 q O

n s 1, 2, . . . ,

,

n2

iqO

4

n

,

n2

n s 1, 2, . . . ,

Ž 3.4.

and ␭ n j Ž j s 1, 2; n s 1, 2, . . . . is a simple root of ⌬ 0 Ž ␭.. Proof. Let ␻ 1 s e Ž3r4.␲ i, ␻ 2 s e Ž5r4.␲ i, ␻ 3 s e Ž1r4.␲ i, ␻4 s eyŽ1 r4.␲ i, and S0 s  ␳ ; ␲8 F arg ␳ F 38␲ 4 . Then ReŽ ␳␻ 1 . - 0, ReŽ ␳␻4 . ) 0, ␳␻ 3 s y␳␻ 2 , and e ␳␻ 1 ª 0, e ␻ 4 ␳ ª ⬁ Žexponentially, as ␳ ª ⬁. for ␳ g S0 . By computing, ⌬ 3 Ž ␭. s 0 can be written as e2 ␻ 2 ␳ y i q O

1

s 0.

Ž 3.5.

for ␳ g S0 ,

Ž 3.6.

ž / p

Because the equation e 2 ␻ 2 ␳ y i s 0, has roots

␳ˆn1 s ␻y1 2 ␲ Ž yn q

1 4

n s 1, 2, . . . ,

.i,

Ž 3.7.

and these roots ␳ˆn1 , n s 1, 2, . . . , on the line arg ␳ s ␲4 , by Rauche’s theorem, we see that the roots of Ž3.5. can be asymptotically represented as

␳n1 s ␻y1 2 ␲ yn q

ž

1 4

/

iqO

1

ž / n

,

n s 1, 2, . . . .

Ž 3.8.

From this we can get Ž3.3.. Let S1 s  ␳ ; y 38␲ F arg ␳ F y ␲8 4 . Substituting S0 for S1 and taking ␻ 1 s e Ž5r4.␲ i, ␻ 2 s e Ž3r4.␲ i, ␻ 3 s eyŽ␲ r4.i, and ␻4 s e Ž␲ r4.i in Ž3.5., simi-

26

LI ET AL.

larly, we get that e2 ␻ 2 ␳ q i q O

1

ž/ ␳

␳ g S1 .

s 0,

Ž 3.9.

Similarly, we can get that the roots of Eq. Ž3.9. can be asymptotically represented as

␳n2 s ␻y1 2 ␲ ny

ž

1 4

/

iqO

1

ž / n

,

n s 1, 2, . . . .

Ž 3.10.

From this, we can get Ž3.4.. By computation, in fields F1 s  ␳ ; 58␲ F arg ␳ F 78␲ 4 , F2 s  ␳ ; 98␲ F arg ␳ F 118␲ 4 , we can get same results as Ž3.8. and Ž3.12.. In the other fields there exist at most finite roots for ⌬ 3 Ž ␭.. The proof is complete. By Lemma 3.1, we will obtain the representations of eigenfunctions of eigenproblem Ž2.22.. 2 LEMMA 3.2. The eigenfunctions, corresponding to eigen¨ alue ␭ n1 s ␳n1 , of Ž2.22. can be represented as

␾n1 Ž x, ␳n1 . s 2 Ž 1 y i . cosh Ž ␻ 1 ␳n1 x q ␻ 2 ␳n1 . q Ž 1 q i . cosh Ž ␻ 1 ␳n1 x q ␻ 3 ␳n1 . ycosh Ž ␻ 1 ␳n1 x q ␻4 ␳n1 . q Ž 1 q i . cosh Ž ␻ 2 ␳n1 x q ␻ 1 ␳n1 . q Ž 1 y i . cosh Ž ␻ 2 ␳n1 x q ␻4 ␳n1 . ycosh Ž ␻ 2 ␳n1 x q ␻ 3 ␳n1 . ey␻ 4 ␳ n1 , x g w 0, 1 x ,

n s 1, 2, 3, . . . ,

Ž 3.11.

where ␻ 1 s e Ž3r4.␲ i, ␻ 2 s e Ž5r4.␲ i, ␻ 3 s e␲ i r4 , and ␻4 s ey␲ i r4 , and

␾n2 Ž x, ␳n2 . s 2 Ž 1 y i . cosh Ž ␻ 1 ␳n2 x q ␻ 2 ␳n2 . q Ž 1 q i . cosh Ž ␻ 1 ␳n2 x q ␻ 3 ␳n2 . y cosh Ž ␻ 1 q ␻4 ␳n2 . q Ž 1 q i . cosh Ž ␻ 2 ␳n2 x q ␻ 1 ␳n2 . q Ž 1 y i . cosh Ž ␻ 2 ␳n2 x q ␻4 ␳n2 . ycosh Ž ␻ 2 ␳n2 x q ␻ 3 ␳n2 . ey␻ 4 ␳ n2 , x g w 0, 1 x ,

n s 1, 2, 3, . . . ,

where ␻ 1 s e Ž5r4.␲ i, ␻ 2 s e Ž3r4.␲ i, ␻ 3 s ey␲ i r4 , and ␻4 s e␲ i r4 .

Ž 3.12.

STABILIZATION OF VIBRATING BEAM

27

Furthermore, there exists a constant number M0 ) 0, such that e¨ ery eigenfunction ␾n j Ž x . holds the following estimation:

␾nX j Ž x, ␳n j . F M0 n,

␾n j Ž x, ␳n j . F M0 ,

j s 1, 2,

n s 1, 2, 3, . . . .

Ž 3.13.

Proof. Let ␻ 1 s e Ž3r4.␲ i, ␻ 2 s e Ž5r4.␲ i, ␻ 3 s e Ž␲ r4.i, ␻4 s eyŽ ␲ r4.i, and ␳ s ␭2 . We consider Lemma 2.2 in the field: S0 s  ␳ ; ␲8 F arg ␳ F 38␲ 4 . Because eigenfunction ␾n1Ž x, ␳n1 . of Ž1.3. corresponding to eigenvalue 2 ␭ n1 s ␳n1 can be represented as ␾n1Ž x, ␳n1 . s Ý4js1 c j e ␻ j ␳ x , where c1 , c 2 , c 3 , and c 4 satisfy Ž3.1., by linear algebraic theory, we have 4

␾n1 Ž x, ␳n1 . s

␻1 1

␻2 1

␻3 1

e ␻ 1 ␳ n1 x ␻ 1 e ␻ 1 ␳ n1

e ␻ 2 ␳ n1 x ␻ 2 e ␻ 2 ␳ n1

e ␻ 3 ␳ n1 x ␻ 3 e ␻ 3 ␳ n1

␻4 ey␻ 4 ␳ n1 ey ␻ 4 ␳ n1 . Ž 3.14. e ␻ 4 ␳ n1Ž xy1. ␻4

By computing, we get Ž3.11.. Taking ␻ 1 s e Ž5r4.␲ i, ␻ 2 s e Ž3r4.␲ i, ␻ 3 s eyŽ ␲ r4.i, and ␻4 s e Ž␲ r4.i, in Ž3.14., and substituting S0 for S1 s  ␳ ; y 38␲ F arg ␳ F y ␲8 4 , we can get Ž3.12.. The proof is complete. We now discuss the asymptotic representations of the eigenvalues and the eigenfunctions of Ž2.9. or Ž1.3.. We first study the distribution of eigenvalues Ž2.10.. Let ␻ 1 , ␻ 2 , ␻ 3 , and ␻4 be the roots of ␻ 4 s y1. Let ␳ 4 s ␭2 / 0. By the method of Lemma 3.1, we get the following lemma. LEMMA 3.3. The eigen¨ alues of Ž2.10. can be represented as

␯n1 s ␭ n1 1 q O

1

2

s

ž ž // ž ž ž // ž

␯n2 s ␭ n2 1 q O

2 ␳n1

1

2 ␳n2

␳n1 q O

2

s

␳n2 q O

1

ž // ž //

2

␳n1 1

␳n2

,

n s 1, 2, . . . , Ž 3.15.

,

n s 1, 2, . . . , Ž 3.16.

2

where ␭ n1 , ␭ n2 , ␳n1 , and ␳n2 are determined by Ž3.3., Ž3.4., Ž3.9., and Ž3.13., respecti¨ ely. Furthermore, ␯n k Ž k s 1, 2. is simple for n large enough.

28

LI ET AL.

Proof. Let ␻ 1 s e Ž3r4.␲ i, ␻ 2 s e Ž5r4.␲ i, ␻ 3 s e Ž1r4.␲ i, ␻4 s eyŽ1 r4.␲ i, and S0 s  ␳ ; ␲8 F arg ␳ F 38␲ 4 ; then ReŽ ␳␻ 1 . - 0, ReŽ ␳␻4 . ) 0, ␳␻ 3 s y␳␻ 2 ; and e ␳␻ 1 ª 0, e ␻ 4 ␳ ª ⬁ Žexponentially, as ␳ ª ⬁. for ␳ g S0 . Therefore, for ␭ / 0, the roots of ⌬U Ž ␭. s 0 satisfy

1

␳8

⌬U Ž ␭ . s ␳y4

␻1 1 ␦ 31 ␦41

␻2 1 ␦ 32 ␦42

s ␤⌬ 3 Ž ␳ . q

1

␳2

␻3 1 ␦ 33 ␦43

␻4 1 ␦ 34 ␦44

⌬ 3 Ž ␳ . y ␳y3 Ž ␣ q m ␳ 2 . ⌬ 4 Ž ␳ .

q Ž ␤␳ 2 q 1 . ␳y3␥ ⌬ 1 Ž ␳ . y Ž ␣ q m ␳ 2 . ␳y4␥ ⌬ 2 Ž ␳ . s 0.

Ž 3.17. By Lemma 3.1 and Rauche’s theorem, we can see that the roots of Ž3.17. can be represented as

␮ n1 s ␳n1 q O

1

ž / ␳n1

,

n s 1, 2, . . . .

Ž 3.18.

From this, we can get Ž3.15.. Because ␳n1 is simple, we can see that ␮ n1 is also simple for n large enough. Let S1 s  ␳ ; y 38␲ F arg ␳ F y ␲8 4 . Replacing S0 by S1 , and taking ␻ 1 s e Ž5r4.␲ i, ␻ 2 s e Ž3r4.␲ i, ␻ 3 s eyŽ␲ r4.i, and ␻4 s e Ž␲ r4.i in Ž3.17., by Rauche’s we can get that

␮ n2 s ␳n2 q O

1

ž / ␳n2

,

n s 1, 2, . . . .

Ž 3.19.

From this, we get Ž3.16., and we can see that ␮ n2 is simple for n large enough. In fields F1 s  ␳ ; 58␲ F arg ␳ F 78␲ 4 and F2 s  ␳ ; 98␲ F arg ␳ F 118␲ 4 , we can get the same results as Ž3.18. and Ž3.19.. In the other fields, there at most exist finite roots of ⌬U Ž ␭.. The proof is complete.

29

STABILIZATION OF VIBRATING BEAM

We now discuss the representation of the eigenfunctions Ž1.3. or Ž2.10.. Let

␺ Ž x, ␮ . s 2'2 sinh Ž ␻ 3 ␮ x q ␻4 ␮ . q i sinh Ž ␻ 3 ␮ x q ␻ 1 ␮ . q Ž 1 q i . sinh Ž ␻ 3 ␮ x q ␻ 2 ␮ . y sinh Ž ␻ 1 ␮ x q ␻ 2 ␮ . qi sinh Ž ␻ 1 ␮ x q ␻ 3 ␮ . y Ž 1 y i . sinh Ž ␻ 1 ␮ x q ␻ 4 ␮ . ey␻ 4 ␮ ,

Ž 3.20.

where ␻ j , j s 1, 2, 3, 4, are the four roots of equation ␻ 4 s y1. By Lemmas 3.1᎐3.3, we can get the following result. LEMMA 3.4. The eigenfunctions corresponding to eigen¨ alue ␯ s ␮2 of Ž2.10. can be asymptotically represented as f n1 Ž x, ␮ n1 . s ␾n1 Ž x, ␮ n1 . q

1

␯n1

␺n1 Ž x, ␮ n1 . ,

x g w 0, 1 x ,

n s 1, 2, . . . ,

Ž 3.21.

where ␻ 1 s e Ž3r4.␲ i, ␻ 2 s e Ž5r4.␲ i, ␻ 3 s e␲ i r4 , and ␻4 s ey␲ i r4 , and f n2 Ž x, ␮ n2 . s ␾n2 Ž x, ␮ n2 . q

1

␯n2

␺n2 Ž x, ␮ n2 . ,

x g w 0, 1 x ,

n s 1, 2, . . . ,

Ž 3.22.

where ␻ 1 s e Ž5r4.␲ i, ␻ 2 s e Ž3r4.␲ i, ␻ 3 s ey␲ i r4 , ␻4 s e␲ i r4 , and ␯n j s ␮2n j . Furthermore, there exists a constant M0 ) 0 such that, for e¨ ery eigenfunction f n j Ž x ., the following estimations holds: f n j Ž x, ␮ n j . F M0 ,

f nX j Ž x, ␮ n j . F M0 n,

j s 1, 2,

n s 1, 2, . . . .

Ž 2.23. Proof. Let ␻ 1 s e 3r4␲ i, ␻ 2 s e 5r4␲ i, ␻ 3 s e␲ r4 i, ␻4 s ey␲ r4 i, and ␳ 4 s ␭2 . We consider Lemma 3.4 on field S0 s  ␳ ; ␲8 F arg ␳ F 38␲ 4 . Because the eigenfunction corresponding to eigenvalue ␯n1 s ␮ 2n1 can be represented as f n1Ž x, ␮ n1 . s Ý4js1 c j e ␻ j ␮ n1 x , where c1 , c 2 , c 3 , and c 4 are determined by the conditions of Ž2.10., and because



␻1 1 ␦41

␻2 1 ␦42

␻3 1 ␦43

␻4 1 ␦44

0

30

LI ET AL.

has rank 3, by the theory of linear algebra and Ž2.11., we can get

f n1 Ž x, ␮ n1 . s ey␻ 4 ␮ n1

s

␻1 1

␻2 1

␻3 1

␻4 1

e ␻ 1 ␮ n1 x ␦41

e ␻ 2 ␮ n1 x ␦42

e ␻ 3 ␮ n1 x ␦43

e ␻ 4 ␮ n1Ž xy1. ␦44

␻1 1

␻1 1

␻1 1

e ␻ 1 ␮ n1 x ␻ 1 e ␻ 1 ␮ n1

e ␻ 2 ␮ n1 x ␻ 2 e ␻ 2 ␮ n1

e ␻ 3 ␮ n1 x ␻ 3 e ␻ 3 ␮ n1

q

␥ ␯n1

␻ 1 ey␻ 4 ␮ n1 ey ␻ 4 ␮ n1 ␻ 4 ␮ n1Ž xy1. e ␻4

␻1 1

␻2 1

␻3 1

e ␻ 1 ␮ n1 x ␻ 12 e ␻ 1 ␮ n1

e ␻ 2 ␮ n1 x ␻ 22 e ␻ 2 ␮ n1

e ␻ 3 ␮ n1 x ␻ 32 e ␻ 3 ␮ n1

␻4 ey␻ 4 ␮ n1 ey ␻ 4 ␮ n1 ␻ 4 ␮ n1Ž xy1. , e ␻42

where ␦4 j s Ž K 2 ␻ j2r␯n1 q ␻ j . e ␻ j ␮ n1 , j s 1, 2, 3, 4. By computing, we can get f n1 Ž x, ␮ n1 . s ␾n1 Ž x, ␮ n1 . q

␥ ␯n1

␺n1 Ž x, ␮ n1 . .

Taking ␻ 1 s e Ž5r4.␲ i, ␻ 2 s e Ž3r4.␲ i, ␻ 3 s eyŽ ␲ r4.i, ␻4 s e Ž␲ r4.i, and S1 s  ␳ ; y 38␲ F arg ␳ F y ␲8 4 , similarly, we can get Ž3.22.. The proof is complete.

4. RIESZ BASIS AND STABILIZATION In this section, we first study the construction of the eigenvectors of Ž2.9., and we prove that all of the eigenvectors of A constitute a Riesz basis of the space H . Second, we study the stability of system Ž2.9.. We now begin the first part of this section. LEMMA 4.1. There exists a integer N0 such that f n1 Ž x, ␮ n1 . y ␾n1 Ž x, ␳n1 . - Mr< ␳n1 < ,

for n G N0 ,

Ž 4.1 .

where ␻ 1 s e Ž3r4.␲ i, ␻ 2 s e Ž5r4.␲ i, ␻ 3 s e Ž␲ r4.i, and ␻4 s eyŽ ␲ r4.i, and f n2 Ž x, ␮ n2 . y ␾n2 Ž x, ␳n2 . - Mr< ␳n2 < ,

for n G N0 ,

where ␻ 1 s e Ž5r4.␲ i, ␻ 2 s e Ž3r4.␲ i, ␻ 3 s eyŽ ␲ r4.i, and ␻4 s e Ž␲ r4.i.

Ž 4.2.

31

STABILIZATION OF VIBRATING BEAM

Proof. We first prove Ž4.1.. By Lemmas 3.1᎐3.4, we see that the eigenfunction f n1 of Ž1.3. corresponding to eigenvalue ␯n1 can be represented as f n1 Ž x, ␮ n1 . y ␾n1 Ž x, ␳n1 . s ␾ Ž x, ␮ ny 1 . y ␾n1 Ž x, ␳n1 . q

␥ ␮ n1

␺ Ž x, ␮ n1 . ey␻ 4 ␳ n1 . Ž 4.3.

Noting that Re ␻ 1 ␮ n1 - 0 and Re ␻ 1 ␳n1 - 0, and that Re ␻ 2 ␳n1 , Re ␻ 2 ␮ n1 , Re ␻ 3 ␳n1 , and Re ␻ 3␶n1 are bounded, we have that

␾ Ž x, ␮ n1 . y ␾ Ž x, ␳n1 . ␻ 3 xO



F 4 sinh ␻ 3 ␳n1 x q ␻4 ␳n1 q

=sinh



␻ 3 xO



␻ 1 xO



␳n1

1

ž / ␳n1

q ␻4 O

ž

1

␳n1

1

ž / ␳n1

q ␻3O

ž

1

␳n1

1

ž / ␳n1

q ␻3O

ž

1

␳n1

2

q sinh ␻ 3 ␳n1 x q ␻ 2 ␳n1 q

=sinh

ž /

2

␻ 3 xO



␳n1

1

2

q sinh ␻ 1 ␳n1 x q ␻ 3 ␳n1 q

=sinh

ž /

q ␻4 O 2

␻ 1 xO



1

1

ž / ␳n1

q ␻2O

ž

1

␳n1

2

␻ 3 xO

1

ž / ␳n1

q ␻2O 2

ž

1

␳n1

0 / 0 / 0 / 0 / 0 / 0

32

LI ET AL.

␻ 1 xO



q sinh ␻ 1 ␳n1 x q ␻4 ␳n1 q

=sinh



␻ 1 xO



␻ 3 xO



␻ 1 xO

FM

x < ␳n1 <

e ␻ 4 ␳ n1Ž xy1. .



1

ž / ␳n1

1

ž / ␳n1

1

ž / ␳n1

1

ž / ␳n1

␻ 3 xO

1

ž / ␳n1

q ␻2O

q ␻2O

q ␻2O

q ␻ 1O 2

1

ž / ␳n1

q ␻ 1O 2

␳n1

0 ž / 0 ž / 0 ž / 0 ž / 0 ž / 0 ž / 0 ž / 0

q ␻2O

2

q sinh ␻ 3 ␳n1 x q ␻ 1 ␳n1 q

=sinh

␳n1

2

␻ 3 xO



ž /

1

ž /

q ␻4 O

2

q sinh ␻ 1 ␳n1 x q ␻ 3 ␳n1 q

=sinh

1

2

␻ 1 xO



␳n1

2

q sinh ␻ 3 ␳n1 x q ␻ 2 ␳n1 q

=sinh

ž /

q ␻4 O 2

␻ 3 xO



1

1

␳n1

1

␳n1

1

␳n1

1

␳n1

1

␳n1

1

␳n1

1

␳n1

ey ␻ 4 ␳ n1

Ž 4.4 .

STABILIZATION OF VIBRATING BEAM

33

So f n1 y ␾ Ž x, ␳n1 . F Mr< ␳n1 < ,

for n large enough.

Ž 4.5.

Similarly, we can prove the case j s 2. The proof is complete. By computing, we get that Y 2 ␾n1 yi Ž Ž 1 y i . cosh Ž ␻ 1 ␳n1 x q ␻ 2 ␳n1 . . Ž x, ␳n1 . s 2 ␳n1

q Ž 1 q i . cosh Ž ␻ 1 ␳n1 x q ␻ 3 ␳n1 . y Ž cosh Ž ␻ 1 ␳n1 x q ␻4 ␳n1 . . q␻ i Ž Ž 1 q i . cosh Ž ␻ 2 ␳n1 x q ␻ 1 ␳n1 . . q Ž 1 y i . cosh Ž ␻ 2 ␳n1 x q ␻4 ␳n1 . q Ž cosh Ž ␻ 2 ␳n1 x q ␻ 3 ␳n1 . . ey␻ 4 ␳ n1 ,

Ž 4.6.

and similarly, we have that Y ␺n1 Ž x, ␮ n1 . s ␮2n1 i Ž sinh Ž ␻ 3 ␮ n1 x q ␻4 ␮ n1 . q i sinh Ž ␻ 3 ␮ n1 x q ␻ 1 ␮ n1 .

q Ž 1 q i . sinh Ž ␻ 3 ␮ n1 x q ␻ 2 ␮ n1 . . qi Ž sinh Ž ␻ 1 ␮ n1 x q ␻ 2 ␮ n1 . y Ž sinh Ž ␻ 1 ␮ n1 x q ␻ 3␶n1 . q Ž 1 y i . sinh Ž ␻ 1 ␮ n1 x q ␻4 ␮ n1 . . . ey␻ 4 ␳ n1 . Ž 4.7. We have the following result. q⬁ LEMMA 4.2.  ⌽n1 s Ž ␾n1Ž x .r␭ n1 , ␾n1Ž x ., 0.T 4ns1 and  Fn1 s Ž f n1Ž x .r T q⬁ ␮ n1 , f n1Ž x ., ␩n1r␮ n1 . 4ns1 are the eigenfunctions of Ž2.18. and Ž2.9., respecti¨ ely. Furthermore, there exists a constant N0 ) 0 such that

q⬁

Ý nsN0

5 Fn1 y ⌽n1 5 2H F M

q⬁

Ý nsN0

1 n2

.

Ž 4.8.

34

LI ET AL.

Proof. Because 5 Fn1 y

⌽n1 5 2H

y ␻ 4 ␳ n1 2

s Ž f n1 y ␾n1 . e

␩

q s

1

q

ž

H0

Y f n1

␯n1

< ey ␻ 4 ␳ n1 < 2 < ␮ n1 < 2 q

1

H0

q

q

Y ␾n1

y

␳n1 1

H0

y

␳n1

␩ ␯n1

␳ n1 2

/

e

dx

2

/

y ␻ 4 ␳ n1

dx q

e

␺ Ž x, ␮ n1 .

1

H0

< ␮ n1 < 2 1

4

2

␩

dx

ž

␺ Y Ž x, ␮ n1 .

␯n1

y

4

␳ n1 2

dx

2

␯n1

␾ Y Ž x, ␮ n1 .

2

ey ␻ 4 ␳ n1

␯n1

Ž ␾ Ž x, ␮ n1 . y ␾ Ž x, ␳n1 . . ey ␻

< ey ␻ 4 ␳ n1 < 2

H0

q

␯n1

2 y ␻ 4 ␳ n1

ey␻ 4 ␳ n1

Ž f n1Ž x . y ␾n1Ž x . . ey ␻

H0

F2

ž

Y ␾n1

2

␯n1

1

q

Y f n1

dx

␾ Y Ž x, ␭ n1 . ␭ n1

2

/

y ␻ 4 ␳ n1

e

dx

2

ey␻ 4 ␳ n1

,

Ž 4.9.

from Ž4.4. ᎐ Ž4.7. and Ž4.9., we have that 5 Fn1 y ⌽n1 5 2H F Mrn2 .

Ž 4.10.

From Ž4.10., we have Ž4.8.. The proof is complete. Remark. Let ␻ 1 s e Ž5r4.␲ i, ␻ 2 s e Ž3r4.␲ i, ␻ 3 s eyŽ1 r4.␲ i, and ␻ 1 s e ; we can prove that the above lemmas hold for ␾ and ␺ defined in Lemmas 2.2 and 2.4. So we have that there exists an integer N1 such that Ž1r4.␲ i

q⬁

2

Ý Ý 5 Fn j y ⌽n j 5 2H - q⬁.

nsN1 js1

Ž 4.11.

STABILIZATION OF VIBRATING BEAM

35

By the above lemmas, we have the following result. THEOREM 4.3. Let ␣ ) 0, ␤ ) 0, m ) 0 and 0 - ␥ - ⬁. All of the generalized eigen¨ ectors, Fn j , j s 1, 2, n s 0, 1, 2. . . . , of system Ž2.9. form a Riesz basis of space H s V = L2 Ž0, 1.. Proof. Let Ý⬁Ž A *. s  ␾ ; Ž ␭ I y A .y1␾ is an entire function of ␭4 Žsee Lemma 6 on p. 2296 of Ref. w8x., where A * is the adjoint of A. Ž A *.y1 is compact because Ay1 is compact. By Lemma 5 of Ref. w8x we know that dim Ý⬁Ž A *. s 0 or ⬁. Let SpŽ A . denote the closed linear subspace spanned by all the generalized eigenvectors of A. By Lemma 5 of Ref. w8x, we have SpŽ A . H s Ý⬁Ž A *.. If SpŽ A . H /  04 , then dim Ý⬁Ž A *. s ⬁. Take linear independent vectors  Fˆn k ; k s 1, 2; n s 0, 1, 2, . . . , N1 y 14 ; Ý⬁Ž A *. s SpŽ A . H . From Ž4.11. we have N1y1

⬁

2

2

Ý Ý 5 Fˆn k y ⌽n k 5 2 q Ý Ý 5 Fn k y ⌽n k 5 2 - q⬁. Ž 4.12.

ns0 ks1

nsN1 ks1

Obviously,  Fˆn k ; k s 1, 2; n s 0, 1, 2, . . . , N14 j  Fn k ; k s 1, 2; n s N1 , N1 q 1, . . . 4 is ␻-linear independent. By Bair’s theorem ŽTheorem 11.3 of Ref. w6x, we see that  Fˆn k ; k s 1, 2; n s 0, 1, 2, . . . , N1 4 j  Fn k ; k s 1, 2; n s N1 , N1 q 1, ⭈⭈⭈ 4 forms a Riesz basis of H . That is, dim SpŽ A .. H s dim Ý⬁Ž A *. s 2 N1 y 1. This is a contradiction with dim Ý⬁Ž A *. s ⬁. Therefore dim Ý⬁Ž A *. s  04 . Let  Fˆj ; j s 1, 2, . . . , m4 j  Fn k ; k s 1, 2; n s N1 , N1 q 1, ⭈⭈⭈ 4 be all the generalized eigenvectors of A. By Bair’s theorem and Ž4.11., m F 2 N1. From w6, Corollary 11.4x, we see that there exists a positive integer n 0 G N1 such that the sequence  ⌽n k ; k s 1, 2; n s 0, 1, 2, . . . , n 0 4 j  Fn k ; k s 1, 2; n s n 0 q 1, n 0 q 2, ⭈⭈⭈ 4 is a Riesz basis of H . From this, we get H s Sp   ⌽n k ; k s 1, 2; n s 1, 2, . . . , n 0 4 4 [Sp Fn k ; k s 1, 2; n s n 0 q 1, n 0 q 2, ⭈⭈⭈ 4 .

Ž 4.13.

From Ž4.12., we see that  Fn k ; k s 1, 2; n s n 0 q 1, n 0 q 2, ⭈⭈⭈ 4 is a Riesz basis of Sp Fn k ; k s 1, 2; n s n 0 q 1, n 0 q 2, ⭈⭈⭈ 4 . Therefore  Fˆj ; j s 1, 2, . . . , m4 j  Fn k ; k s 1, 2; n s N1 , N1 q 1, ⭈⭈⭈ 4 form a Riesz basis of subspace SpŽ A .. Because SpŽ A . H s  04 , SpŽ A . H s H . So  Fj ; j s 1, 2, . . . , m4 j  Fn k , k s 1, 2; n s N1 , N1 q 1, ⭈⭈⭈ 4 form a Riesz basis of H . The proof is complete. In the following, we will prove the stability of system Ž2.9.. By the standard approach, we can get the following result.

36

LI ET AL.

THEOREM 4.4. The operator A defined by Ž2.7. and Ž2.8. generates a C0 contraction semigroup T Ž t ., t G 0, and there exist constants M G 1 and ␦ ) 0 such that T Ž t.

y␦ t

H F Me

t G 0.

,

Ž 4.14.

Proof. First, for any y s Ž u, ¨ , ␩ .T g DŽ A ., integrating by parts, we can get the following: ² A y, y :H s

1

H0

sy

Ž u x x¨ x x y ¨ u x x x x . dx y K

␤

u 2x x x Ž 1 . y

Km ␣

␤3

K

␤

␩ ␩q ␣y

ž ž

m

␤

¨ 1

/ Ž ./

¨ 2 Ž 1 . y ␥y1 ¨ x2 Ž 1 . .

Ž 4.15.

Note that due to the particular choice of K given by Ž2.6., the term multiplying ¨ Ž1. u x x x Ž1. in Ž4.15. vanishes. It follows from Ž4.15. that the operator A is dissipative. By Lemma 2.0, noting that ␭ s 0 is a regular point of A, we see that if a complex ␭ is an eigenvalue of A, then Re ␭ - 0. Because all of the generalized eigenvectors of A form a Riesz basis of the space V = L2 Ž0, 1., we see that A generates a C0 contraction semigroup T Ž t ., t G 0. From Lemma 3.4, we see that f n1Ž x, ␮ n1 . s ␾n1Ž x, ␮ n1 . q Ž1r␯n1 . ␺n1Ž x, ␮ ., n s 1, 2, . . . , are the eigenfunctions of Ž2.10.. Let h n11 Ž x . s Ž 1 y i . cosh Ž ␻ 1 ␳n1 x q ␻ 2 ␳n1 . q Ž 1 q i . cosh Ž ␻ 1 ␳n1 x q ␻ 3 ␳n1 . ycosh Ž ␻ 1 ␳n1 x q ␻4 ␳n1 . ey␻ 4 ␳ n1

Ž 4.16.

and h n12 Ž x . s Ž 1 q i . cosh Ž ␻ 2 ␳n1 x q ␻ 1 ␳n1 . q Ž 1 y i . cosh Ž ␻ 2 ␳n1 x q ␻4 ␳n1 . qcosh Ž ␻ 2 ␳n1 x q ␻ 3 ␳n1 . ey␻ 4 ␳ n1 .

Ž 4.17.

By computing, we get that 1

H0

h n11 Ž x .

2

dx s O

1

ž / n

,

␾n1 Ž 1, ␮ n1 .

1

H0 2

h n12 Ž x .

s 16 q O

2

dx s 4 q O 1

ž / n

.

1

ž / n

. Ž 4.18.

Ž 4.19.

37

STABILIZATION OF VIBRATING BEAM

Y Ž x, ␮ n1 . From Ž4.6. ᎐ Ž4.19. we get ␾n1Ž x, ␮ n1 . s h n11Ž x . q h n12 Ž x . and ␾n1 2 2 Ž s yi ␮ n1 h n12 Ž x . y h n12 Ž x ... Multiplying the eigenequation ␯n1 f n1Ž x, ␮ n1 . Ž4. Ž qf n1 x, ␮ n1 . s 0 by f n1 Ž x, ␮ n1 . and integrating by parts, we get that

␯n12

1

H0

f n1 Ž x, ␮ n1 .

2

dx s y y

␣␯n1 q m␯n12 ␯n1 ␥

2

f n1 Ž 1, ␮ n1 .

1 q ␤␯n1

f n1 Ž 1, ␮ n1 .

2

y

1

f n1 Ž x, ␮ n1 .

H0

2

dx ,

Ž 4.20. ␯n12

1

H0

2

f n1 Ž x, ␮ n1 .

sy

dx

␣␯n1 q m␯n12 1 q ␤␯n1

␾n1 Ž 1, ␮ n1 .

2

␯n1

y

␥

2

␾n1 Ž x, ␮ n1 .

qO

1

ž / n

.

Ž 4.21. By computing, Ž4.21. becomes

␯n1

1

H0

2

h n11 Ž x . q h n12 Ž x . dx q ␯n1

sy

␣ q m␯n1 1 q ␤␯n1

␾n1 Ž 1, ␮ n1 .

2

1

h n11 Ž x . y h n12 Ž x .

H0

qO

1

ž / n2

2

dx

.

Ž 4.22.

Let ␯n1 s a n1 q ibn1. Taking the real part of Ž4.21., we get 2 a n1

1

H0 ž h sy

2

n11

Ž x . q h n12 Ž x .

␣ q m␯n1 1 q ␤␯n1

2

/ dx

␾n1 Ž 1, ␮ n1 .

2

qO

1

ž / n2

.

Ž 4.23.

So, from Ž4.18., Ž4.19., we have that lim a n1 s y lim

nªq⬁

nªq⬁

2 Ž ␣ q man1 . Ž 1 q ␤ an1 . q m ␤ bn1 2 2 Ž bn1 q Ž 1 q ␤ a n1 .

= ␾n1 Ž 1, ␮ n1 . sy

2m

␤

,

2

1

. H0 ž

h n1 Ž x .

2

q h n1 Ž x .

2

/ dx

2

Ž 4.24.

38

LI ET AL.

where we have used the fact lim nªq⬁ < bn1 < s q⬁. Because < ␯n1 < ª q⬁ Žas n ª ⬁., from Lemma 3.3, we can see that lim nªq⬁ < bn1 < s q⬁. Similarly, let ␯n2 s a n2 q ibn2 ; we can get that lim a n2 s y

nªq⬁

2m

␤

.

Ž 4.25.

By the theory of basis and the operator semigroup theory, we see that there exist constants M G 1 and ␦ ) 0 such that T Ž t . F Mey␦ t ,

for t G 0.

Ž 4.26.

The proof is complete. From the above theorems, we get the following result. THEOREM 4.5. The optimal decay rate of C0 semigroup T Ž t ., t G 0, generated by A, can be obtained from the following formula:

␻ 0 s Sup  Re ␭ ; ␭ g ␴ Ž A . 4 .

Ž 4.27.

REFERENCES 1. A. V. Balakrishnan, Compensator design for stability enhancement with collocated controllers, IEEE Trans. Automat. Control 36, No. 9 Ž1991., 994᎐1008. 2. G. Chen, M. C. Delfour, A. M. Krall, and G. Payre, Modeling, stabilization and control of serially connected beam, SIAM J. Control Optim. 25 Ž1987., 526᎐546. 3. S. J. Li, Time dependent elastic system, Acta Math. Sci. Supp. 15 Ž1995., 39᎐59. 4. J. L. Lions, Exact controllability, stabilization and perturbation for distributed systems, SIAM Re¨ . 30, No. 1 Ž1988., 1᎐68. 5. A. Pazy, ‘‘Semigroups of Linear Operators and Applications to Partial Differential Equations,’’ Springer-Verlag, New YorkrBerlinrHeidelbergrTokyo, 1983. 6. I. Singer, ‘‘Bases in Banach Spaces,’’ Springer-Verlag Berlin-Heidelberg, New York, 1970. 7. Robert M. Young, ‘‘An Introduction to Non-harmonic Fourier Series,’’ Academic Press, 1980. 8. N. Dunford and J. T. Schwartz, ‘‘Linear Operators,’’ Vol. 3, Interscience, New York, 1971. 9. O. Morgul, B. P. Rao, and F. Conrad, On the stabilization of a cable with a tip mass, IEEE Trans. Automat. Control 39, No. 10 Ž1994.. 10. D. L. Russel, Decay rates for weakly damped systems in Hilbert space obtained with control theoretic methods, J. Differential Equations 19 Ž1975., 344᎐370. 11. J. Y. Yu, S. J. Li, Y. T. Wang, and Z. D. Liang, Optimal decay rate of vibrating beam equations controlled by combined boundary feedback forces, Science in China Ž Series E . 42, No. 4 Ž1999., 354᎐364. 12. B. P. Rao, Decay estimates of solutions for a hybrid system of flexible structures, European J. Appl. Math. 4 Ž1993., 303᎐319.