stabilization of an elastic plate with viscoelastic ...lsc.amss.ac.cn/~bzguo/papers/jota3.pdf ·...

journal of optimization theory and applications: Vol. 122, No. 3, pp. 669–690, September 2004 (© 2004)

Stabilization of an Elastic Plate withViscoelastic Boundary Conditions1

Q. Zhang2 and B. Z. Guo3

Communicated by F. L.Chernousko

Abstract. The boundary control problem of an elastic thin platewith boundary viscoelasticity is formulated in the standard form ofa linear infinite-dimensional system in the energy Hilbert space. Thefeedback control is designed so that the input and output are col-located. The frequency-domain approach is adopted in investigatingthe exponential stability of the closed-loop system. Finally, by consid-ering the boundary viscoelasticity as damping, we establish a strongstability result based on the LaSalle invariance principle and theHomander uniqueness theorem.

Key Words. Elastic plate, boundary control, viscoelasticity, semi-group, stability.

1. Introduction

Problems concerning elastic structures with viscoelastic boundaryconditions have been studied extensively by many authors; see e.g. Refs. 1–5 and references therein. It was proved in Ref. 1 that, when a dissipationboundary condition of the memory type is considered, the system can bestabilized asymptotically under nonlinear feedback control. In Ref. 2, theblow-up result was proved for a nonlinear one-dimensional wave equationwith memory boundary condition.

1This work was carried out with the support of the National Natural Science Foundationof China.

2Postdoctoral Fellow, Institute of Systems Science, Academy of Mathematics and SystemSciences, Academia Sinica, Beijing, China.

3Professor, Institute of Systems Science, Academy of Mathematics and System Sciences,Academia Sinica, Beijing, China.

6690022-3239/04/0900-0669/0 © 2004 Plenum Publishing Corporation

670 JOTA: VOL. 122, NO. 3, SEPTEMBER 2004

Global existence and asymptotic stability results for the solution of anonlinear one-dimensional wave equation with viscoelastic boundary con-dition were developed in Ref. 3. The existence, uniqueness, and strong sta-bility for the solution of a linear model of the heat equation in whichthe boundary can absorb heat and, due to hereditary factors, can retainpart of heat were developed in Ref. 4. In Ref. 5, a linear electromagneticmodel with boundary memory was investigated; the existence, uniqueness,and asymptotic stability for the solution of the system were presented.

Motivated by the work on the wave and heat equations mentionedabove, in this article we are concerned with an elastic thin plate whichoccupies a bounded domain �⊂R2 with C2-smooth boundary �. Assumethat �=�0∪�1, where �0 and �1 are relatively open subsets of �,�0 �=∅has positive boundary measure, and �0∩�1=∅. If �0 is clamped and thememory effect on �1 is taken into account, the vertical deflection y(x, t) ofthe thin elastic plate satisfies the following partial differential equation:

ytt (x, t)+�2y(x, t)=0, in �×R+, (1a)

y(x, t)= ∂νy(x, t)=0, on �0×R+, (1b)

B1y(x, t)−∫ ∞

0g′(s)∂ν [y(x, t)−y(x, t−s)]ds=0, on �1×R+, (1c)

B2y(x, t)+∫ ∞

0g′(s)[y(x, t)−y(x, t−s)]ds=u(x, t), on �1×R+, (1d)

y(x,0+)=y0(x), yt (x,0+)=y1(x), (1e)

y(x,−s)=ϑ(x, s), for 0<s <∞, (1f )

where g is the relaxation function, u is the boundary control, y0, y1, ϑ arethe given initial conditions; B1,B2 are the following boundary operators:

B1y=�y+ (1−µ)

(2ν1ν2

∂2y

∂x1∂x2−ν2

1∂2y

∂x22

−ν22∂2y

∂x21

),

B2y= ∂ν�y+ (1−µ)∂τ

[(ν2

1 −ν22

) ∂2y

∂x1∂x2+ν1ν2

(∂2y

∂x22

− ∂2y

∂x21

)];

ν= (ν1, ν2) is the unit outer normal vector, τ = (−ν2, ν1) is the unit tangentvector, and 0<µ<1/2 is the Poisson ratio.

Throughout the article, we assume always that the function g(·) satis-fies the following conditions:

JOTA: VOL. 122, NO. 3, SEPTEMBER 2004 671

(g1) g(·)∈C2[0,∞);(g2) g(t)>0, g′(t)<0, g′′(t)≥0, for t ≥0;(g3) g(∞)>0;(g4) g′(t)≥−Kg′′(t), for some K >0 and all t ≥0.

Condition (g2) implies that the boundary memory is strictly decreasingand the rate of memory loss is also decreasing. From (g2), we have alsothat both g(∞) and g′(∞) exist, g(∞)≥0. Condition (g3) means that thematerial behaves like an elastic solid at t=∞. Condition (g4) implies thatg′(t) decays exponentially, in particular, g′(∞)=0.

The energy corresponding to the system (1) is defined by

E(t)= (1/2)[a(y(·, t))+

∫�

|yt (x, t)|2dx

−∫ ∞

0

∫�1

g′(s)[|∂ν(y(x, t)−y(x, t− s))|2

+|y(x, t)−y(x, t− s)|2]d�ds], (2)

where a(w)=a(w,w) and

a(w1,w2)=∫

�

[∂2w1

∂x21

∂2w2

∂x21

+ ∂2w1

∂x22

∂2w2

∂x22

+µ

(∂2w1

∂x21

∂2w2

∂x22

+ ∂2w1

∂x22

∂2w2

∂x21

)

+2(1−µ)∂2w1

∂x1∂x2

∂2w2

∂x1∂x2

]dx, ∀w1,w2 ∈H 2(�). (3)

Surface damping treatments with viscoelastic material have been usedsuccessfully for many years to reduce vibration and noise of structuresespecially for plate and beam structures. Equations (1c) and (1d) describememory effects which can be caused, for example, by the interaction withanother viscoelastic body. Indeed, in the system (1), there are two kinds ofdissipation: one is represented as u(t), the artificial mechanism; the otheris due to the memory effect which works only over the boundary. We willprove that, under these two kinds of dissipation, the energy of the sys-tem (1) decays exponentially. The frequency domain approach developedin Ref. 6 is adopted in the investigation. We analyze also the asymptoticbehavior of the system (1) with only memory dissipation. By virtue of theLaSalle invariance principle and the Hormander uniqueness theorem, astrong stability result is reached.


In Section 2, we formulate first the system (1) into the standard formof an infinite-dimensional system

Y =AY +Bu;

hence, the feedback control u=−kB∗Y is designed so that the input andoutput are collocated. The well-posedness of the closed-loop system isproved. In Section 3, we show the exponential stability of the closed-loopsystem. Finally, in Section 4, the strong stability for the system (1) isestablished when u(t)=0.

2. Well-Posedness of the Closed-Loop System under Collocated OutputFeedback Control

For a long time, engineers have known that many mechanical systemslike elastic structures lead the passive/positive real system if the actuatorsand sensors are designed in a collocated fashion (Ref. 7). For a passivesystem, the output feedback control makes usually the closed-loop systemstable by the Lyapunov direct approach. In this section, we shall formulatethe system (1) into a standard linear infinite-dimensional control system inthe state Hilbert space and then design the output feedback control in theway of collocated input/output. To do this, we introduce first several func-tion spaces. Let

W ={w∈H 2(�)|w|�0 = ∂νw|�0 =0},‖w‖2W =a(w), ∀w∈W.

Define the boundary memory space by

Z=L2(0,∞;|g′(·)|;H 1(�1)),

‖z‖2Z=∫ ∞

0

∣∣g′(·)∣∣ [‖∂νz(s)‖2L2(�1)+‖z(s)‖2

L2(�1)

]ds, ∀z∈Z.

Set

H=W ×L2(�)×Z

equipped with the inner product induced norm

‖(w, v, z)‖2H=‖w‖2W +‖v‖2L2(�)+‖z‖2Z , ∀(w, v, z)∈H.

It is clear that H is a Hilbert space.


Remark 2.1. We have that a(·)1/2 is an equivalent norm on W, since�0 �=∅ has positive boundary measure; see e.g. Ref. 8. Moreover, it is obvi-

ous that(‖∂νz‖2L2(�1)

+‖z‖2L2(�1)

)1/2is an equivalent norm on H 1(�1). In

fact, if

‖∂νz‖2L2(�1)+‖z‖2

L2(�1)=0,

then

z= ∂νz=0, on �1.

It follows that

∇z=ν∂νz=0, on �1.

Therefore,

z=0, in H 1(�1).

Next, we introduce some operators (Ref. 9) as follows.

(i) We set

Lz(s)=∫ ∞

0g′(s)z(s)ds,

A0 =�2,

D(A0)={w∈H 4(�)∩W |B1w|�1=B2w|�1 =0}.

It is easy to know that A0 is a positive self-adjoint operator onL2(�).

(ii) The Green operators N1 and N2 are introduced to describe theboundary conditions,

N1g=h⇔

�2h=0, in �,

h= ∂νh=0, on �0,

B1h=g, on �1,

B2h=0, on �1,

N2g=h⇔

�2h=0, in �,

h= ∂νh=0, on �0,

B1h=0, on �1,

B2h=g, on �1.


With the help of the regularity theory for elliptic equations (Ref. 10),we know that

N1 :L2(�1)→H 5/2(�) is continuous,N2 :L2(�1)→H 7/2(�) is continuous.

With the operators defined above, we may rewrite the system (1) as

ytt (·, t)+A0[y(·, t)−N1Lz(·, t, s)+N2Lz(·, t, s)−N2u(·, t, s)]=0,

(4)

where

z(·, t, s)=y(x, t)−y(x, t− s), x ∈�1.

Considering L2(�) as the pivot space, [D(A0)] ⊂ L2(�) ⊂ [D(A0)]′ andextending A0 to be A0 :L2(�)→ [D(A0)]′, we can rewrite (4) as

ytt (·, t)=−A0y(·, t)+ A0N1Lz(·, t)− A0N2Lz(·, t)+ A0N2u(·, t)∈ [D(A0)]

′. (5)

Thus, we can put the system (1) into the standard form of a linearinfinite-dimensional system in H,

Y (t)=AY (t)+Bu,

where

Y (t)=y(·, t)

yt (·, t)z(·, t, s)

, z(·, t, s)=y(x, t)−y(x, t− s), X∈�1

A=

0 I 0

−A0 0 A0N1L− A0N2L

0 I − ∂

∂s

, D(A)={Y ∈H|AY ∈H},

Bu= 0

A0N2u

0

, B :L2(�1)→ [D(A∗)]′ is continuous.

Finally, a direct computation gives


(N∗2 A0f, g)L2(�1)= (A0f,N2g)L2(�)= (�2f,N2g)L2(�)

=∫

�

f �2(N2g)dx−∫

�1

[f B2(N2g)− ∂νf B1(N2g)

]d�

+∫

�1

[B2f N2g−B1f ∂ν(N2g)

]d�

=−∫

�1

f gd�,

for all f ∈D(A0) and g∈L2(�1). Therefore,

N∗2 (A0)∗f =N∗2 A0f =−f |�1 , f ∈D(A0).

If follows that

B∗w

v

z

=−ν|�1 , ∀w

v

z

∈D(A∗). (6)

Now, we are in a position to design our feedback control so that the inputand output are collocated (Ref. 11),

u=−kB∗(y, yt , z)T =kyt |�1 , k≥0. (7)

Then, under this output feedback, the closed-loop system becomes

ytt (x, t)+�2y(x, t)=0, in �×R+, (8a)

y(x, t)= ∂νy(x, t)=0, on �0×R+, (8b)

B1y(x, t)−∫ ∞

0g′(s)∂ν [y(x, t)−y(x, t− s)]ds=0, on �1×R+, (8c)

B2y(x, t)+∫ ∞

0g′(s)[y(x, t)−y(x, t−s)]ds=kyt (x, t), on �1×R+, (8d)

y(x,0+)=y0(x), yt (x,0+)=y1(x), (8e)

y(x,−s)=ϑ(x, s), for 0<s <∞. (8f )

The initial boundary problem (8) can be written as an evolutionaryequation in H,

Y (t)=AY(t), Y (0)=Y0,

where

Y = (y, yt , z), Y0= (y0, y1, y0−ϑ),


A= 0 I 0−�2 0 0

0 I −∂/∂s

,

with the domain

D(A)={(w, v, z)∈H|�2w∈L2(�), v∈W,z(·)∈H 1(0,∞;|g′(·)|;H 1(�1)),

z(0)=0,[B1w−

∫ ∞0

g′(s)∂νz(s)ds]�1=0,[

B2w+∫ ∞

0g′(s)z(s)ds

]�1=kv|�1

},

where

H 1(0,∞;|g′(·)|;H 1(�1))={z(s)∈Z|(∂/∂s)z(s)∈Z}.The following theorem ensures that the system (8) is well-posed in H.

Theorem 2.1. Assume that the function g satisfies (g1) through (g3)

and k≥0. Then, the operator A generates a C0-semigroup eAt of contrac-tion on H.

Proof. We prove first that

R(I −A)=H.

Namely, we need to show that the system

w−v=f, (9a)

v+�2w=g, (9b)

z(s)−v+ (∂/∂s)z(s)=h(s), (9c)

possesses a solution (w, v, z)∈D(A) for every (f, g, h)∈H. In fact, it fol-lows from (9) that

v=w−f ∈W, (10a)

w+�2w=f +g∈L2(�), (10b)

z(s)= (1− e−s)w− (1− e−s)f +∫ s

0eτ−sh(τ )dτ ∈Z. (10c)

Therefore,

v∈W and z(·)∈H 1(0,∞; ∣∣g′(·)∣∣ ;H 1(�1)), z(0)=0.


Furthermore, for any w∈W satisfying (10b)–(10c),

�2w∈L2(�),

B1w−∫ ∞

0g′(s)∂νz(s)ds=0,

B2w+∫ ∞

0g′(s)z(s)ds=kv,

we have that, for all φ ∈W,∫�

wφdx+a(w,φ)+∫

�1

[(kw+Xw)φ+X∂νwφ]d�

=∫

�

(f +g)φdx+∫

�1

[(kf +Xf +)φ+ (X∂νf + ∂ν)∂νφ]d�, (11)

where

X=−∫ ∞

0g′(s)(1− e−s)ds ≥ 0,

=∫ ∞

0g′(s)

∫ s

0eτ−sh(τ )dτds.

Thanks to the Lax-Milgram theorem (Ref. 12), equation (11) admits aunique solution w∈W. Combining this with (10a) and (10c), we see that(w, v, z)∈D(A) solves the equation

(I −A)(w, v, z)= (f, g, h).

Next, for any Y = (w, v, z)∈D(A), we have

Re(AY,Y )H=−k

∫�1

|v|2 d�− (1/2)

∫ ∞0

∫�1

g′′(s)(|z(s)|2

+|∂νz(s)|2)d�ds ≤0. (12)

Hence, A is dissipative. Finally, by Theorem 1.4.6 of Ref. 13, D(A) isdense in H. Then, it follows from the Lumer-Phillips theorem that A gen-erates a C0-semigroup of contraction on H. The proof of Theorem 2.1 iscomplete.


3. Exponential Stability

This section is devoted to the exponential stability of the closed-loopsystem (8) for the positive feedback gain k >0. Our approach is based onthe Huang frequency domain criterion for the exponential stability of C0-semigroups on the Hilbert spaces (Ref. 6): eAt is exponentially stable ifand only if the following conditions are fulfilled:

sup{Re(η)|η∈σ(A), spectrum ofA} ≤σ0 <0, (13)

sup{∥∥∥(ηI −A)−1

∥∥∥L(H)|Re(η)≥0

}.=γ <+∞. (14)

To do this, we need the following lemma.

Lemma 3.1. Assume a function G(·)∈C[0,∞). Then, for any ε > 0,

there exists a constant δ >0 such that

inf{β∈R||β|≥ε>0}

∫ ∞0|G(s)‖ e−iβs −1|2ds≥ δ. (15)

Proof. Direct computation shows that

�(β).=∫ ∞

0|G(s)‖ e−iβs −1|2ds

=2∫ ∞

0|G(s)| (1− cosβs)ds

=2∫ ∞

0|G(s)|ds−

∫ ∞0|G(s)|

(e−iβs + eiβs

)ds. (16)

Applying the Riemann-Lebesgue lemma (Ref. 14), we have that thereexists κ >ε such that∣∣∣∫ ∞

0|G(s)|(e−iβs + eiβs)ds

∣∣∣<∫ ∞0|G(s)|ds, |β|>κ.

Hence,

�(β)≥∫ ∞

0|G(s)|ds, |β|>κ. (17)

Next, since � is a positive continuous function on the compact subset ε≤|β|≤κ, there exists δ0 >0 such that

�(β)≥ δ0, ε≤|β|≤κ. (18)


Set

δ.=min{δ0,

∫ ∞0|G(s)|ds}.

We then deduce from (17) and (18) that

�(β)≥ δ >0, ε≤|β|≤∞.

Our main result is the following theorem.

Theorem 3.1. Suppose that there exists a point x0 ∈ R2 such that,when setting m(x)=x−x0, we have

�0={x ∈�|m ·ν≤0}, �1={x ∈�|m ·ν >0}. (19)

Let the function g(·) satisfy (g1)–(g4) and the gain constant k > 0. Then,eAt is exponentially stable; i.e., there exists M,δ >0 such that

E(t)≤Me−δtE(0), ∀t ≥0.

The proof of Theorem 3.1 requires several lemmas. Let us explain firstthe idea of the proof. According to the Huang result, if eAt is not expo-nentially stable, then either (13) or (14) will not be valid. Assume that (14)fails. Then, there exist a sequence of complex numbers

ηn= ξn+ ζni, with Re(ηn)= ξn≥0,

and a sequence of functions

Yn= (wn, vn, zn)∈D(A), with ‖Yn‖H=1,

such that

‖(ηnI −A)Yn‖H→0, (20)

i.e.,

fn.=ηnwn−vn→0, in W, (21a)

gn.=ηnvn+�2wn→0, in L2(�), (21b)

hn(s).=ηnzn(s)−vn+ (∂/∂s)zn(s)→0, in Z. (21c)

Moreover, it follows from (20) that

limn→∞Re((ηnI −A)Yn,Yn)H=0.


Thus, by (12), we have that ξn→0 and

k limn→∞

∫�1

|vn|2 d�=0, (22)

limn→∞

∫�1

∫ ∞0

g′′(s)(|zn(s)|2+|∂νz(s)|2)dsd�=0. (23)

Lemma 3.2. The following assertions hold true:

(i) limn→∞‖zn(·)‖2Z=0,

(ii) limn→∞

∫�

|vn|2 dx= limn→∞a(wn)=1/2,

(iii) limn→∞

∫�1

|wn|2 d�= limn→∞

∫�1

|∂νwn|2 d�=0.

Proof. Using (g4) and (23) yields

limn→∞‖zn(·)‖2Z≤K lim

n→∞

∫ ∞0

∫�1

g′′(s)(|zn(s)|2+|∂νzn(s)|2)d�ds=0.

(24)

Next, taking the inner product of (21a) with vn in L2(�), the inner prod-uct of (21b) with wn in L2(�), adding them up, and noticing the fact thatξn→0, we obtain

limn→∞

(a(wn)−‖vn‖2L2(�)

)=− lim

n→∞

∫�1

vnwnd�+∫ ∞

0

∫�1

g′(s)(zn(s)wn+ ∂νzn(s)∂νwn)d�ds.

(25)

By (24),∣∣∣∣∫ ∞0

∫�1

g′(s)(zn(s)wn+ ∂νzn(s)∂νwn)d�ds

∣∣∣∣≤2‖wn‖W (g(0)−g(∞))1/2 ‖zn(·)‖Z→0. (26)


Substituting (22) and (26) into (25) yields

limn→∞(a(wn)−‖vn‖2L2(�)

)=0.

This, together with (i) and the fact that ‖Yn‖H=1, gives (ii).On the other hand, substituting (21a) into (21c) and solving the

differential equation, we obtain

zn(s)= (1− e−ηns)wn− (1/ηn)(1− e−ηns)fn

+∫ s

0eηn(τ−s)hn(τ )dτ, in Z. (27)

We claim that

|ηn|≥ ε >0.

Otherwise, there exists a subsequence of ηn, still denoted by ηn, whichconverges of zero. However, it follows from (21a) that vn converges to zeroin L2(�), which contradicts (ii). Since ξn→0, we have |ζn|≥ ε >0. There-fore, (27) becomes

limn→∞ zn(s)= lim

n→∞(1− e−iζns)wn, in Z. (28)

Using Lemma 3.1 and (i) in (28) yields

limn→∞

∫�1

(|wn|2+|∂νwn|2)d�=0.

Lemma 3.3. There exists a positive constant κ depending on thedomain � such that

limn→∞

[∫�

|vn|2dx+a(wn)

]≤ lim

n→∞

[(1/2)

∫�1

(m ·ν) |vn|2 d�+ (1/2κ)

∫�1

|∂νwn|2 d�

−∫

�1

B2wn(m ·∇wn)d�+κ

∫�1

|B1wn|2 d�

]. (29)


Proof. We need to prove (29) only for w ∈H 4(�)∩W by a densityargument. It follows from (21b) (see e.g. Ref. 8) that

0←∫

�

(ηnvn+�2wn)m ·∇wndx

=ηn

∫�

vnm ·∇wndx+a(wn)− (1/2)

∫�0

(m ·ν) |�wn|2 d�

+∫

�1

[B2wn(m ·∇wn)−B1wn∂ν(m ·∇wn)]d�

+(1/2)

∫�1

(m ·ν)(∣∣∣∂2wn

∂x21

∣∣∣2+ ∣∣∣∂2wn

∂x22

∣∣∣2+2µRe

(∂2wn

∂x21

∂2wn

∂x22

)

+2(1−µ)

∣∣∣ ∂2wn

∂x1∂x2

∣∣∣2)d�. (30)

In addition, by (21a) and the fact that Re(ηn)→0, we have

limn→∞ηn

∫�

vnm ·∇wndx

= limn→∞(ηn/ηn)

∫�

vnm ·∇vndx

= limn→∞

(∫�

|vn|2 dx− (1/2)

∫�1

m ·ν |vn|2 d�

). (31)

Substituting (31) into (30) yields

limn→∞

[∫�

|vn|2 dx+a(wn)

]= lim

n→∞

[(1/2)

∫�1

(m ·ν) |vn|2 d�+ (1/2)

∫�0

(m ·ν) |�wn|2 d�

−∫

�1

(B2wn(m ·∇wn)−B1u∂ν(m ·∇wn))d�

−(1/2)

∫�1

(m ·ν)

(∣∣∣∣∂2wn

∂x21

∣∣∣∣2+ ∣∣∣∣∂2wn

∂x22

∣∣∣∣2+2µRe

(∂2wn

∂x21

∂2un

∂x22

)

+2(1−µ)| ∂2wn

∂x1∂x2|2)

d�

]. (32)


Moreover, it is clear that, on �1,

−(m ·ν)

∣∣∣∣∣∂2wn

∂x21

∣∣∣∣∣2

+∣∣∣∣∣∂2wn

∂x22

∣∣∣∣∣2

+2µRe

(∂2wn

∂x21

∂2wn

∂x22

)+2(1−µ)

∣∣∣∣∣ ∂2wn

∂x1∂x2

∣∣∣∣∣2

≤−r(1−µ)

∣∣∣∣∣∂2wn

∂x21

∣∣∣∣∣2

+∣∣∣∣∣∂2wn

∂x22

∣∣∣∣∣2

+2

∣∣∣∣∣ ∂2wn

∂x1∂x2

∣∣∣∣∣2 , (33)

|∂ν(m ·∇wn)|2≤2|∂νwn|2+2‖m‖2L∞(�1)

∣∣∣∣∣∂2wn

∂x21

∣∣∣∣∣2

+∣∣∣∣∣∂2wn

∂x22

∣∣∣∣∣2

+2

∣∣∣∣∣ ∂2wn

∂x1∂x2

∣∣∣∣∣2 ,

(34)

where r is a positive constant such that

m ·ν≥ r, for all x ∈�1.

Furthermore, for any κ >0, we have∫�1

B1wn∂ν(m ·∇wn)d�

≤κ

∫�1

|B1wn|2 d�+ (1/2κ)

∫�1

|∂νwn|2d�

+(‖m‖2L∞(�1)

2κ

)∫�1

(∣∣∣∣∂2wn

∂x21

∣∣∣∣2+ ∣∣∣∣∂2wn

∂x22

∣∣∣∣2+2

∣∣∣∣ ∂2wn

∂x1∂x2

∣∣∣∣2)d�. (35)

Selecting κ >0 satisfying

κ≥‖m‖2L∞(�1)/r(1−µ) (36)

and applying (19) and (33)–(35) to (32), we obtain (29).

Proof of Theorem 3.1. Replacing (22) and Lemma 3.2 (iii) into (29)yields

limn→∞

[∫�

|vn|2 dx+a(wn)

]≤ lim

n→∞

[−∫

�1

B2wn(m ·∇wn)d�+κ

∫�1

|B1wn|2 d�

]. (37)


By the definition of D(A), we have

limn→∞

[∫�

|vn|2dx+a(wn)

]≤∫

�1

(∫ ∞�0

g′(s)zn(s)ds−kvn

)(m ·∇wn)d�

+κ

∫�1

∣∣∣∣ ∫ ∞0

[g′(s)∂νzn(s)ds

∣∣∣∣2d�

]. (38)

Using (22) and the same argument as (26), we deduce that

limn→∞

[∫�

|vn|2 dx+a(wn)

]=0,

which contradicts Lemma 3.2 (ii). Then, (14) follows.Finally, we show (13). Since eAt is a C0-semigroup of contraction,

ρ(A) contains the set {η|Re(η)>0}. For any ξ ∈ (0,1/2γ ) and ζ ∈R, it fol-lows from (14) that

(−ξ + iζ −A)−1= (1/4γ + iζ −A)−1[1− (ξ +1/4γ )(1/4γ + iζ −A)−1]−1.

Hence,

sup{Re(η)|η∈σ(A)}<−1/4γ,

proving Theorem 3.1.

4. Strong Stability

In Section 3, we show that the energy of the system (8) decays expo-nentially when k > 0. A normal question is: how about the free system?That it to say, when there is no artificial control imposed on the boundary(k=0), is the system (8) stable or not? In this section, we will answer thisquestion positively in the sense of strong stability. Unfortunately, exponen-tial stability still remains an open question.

To begin with the strong stability discussion, let us recall some classi-cal results on the strong stability of system associated with a C0-semigrouptrajectory (see e.g. Refs.15–17).

Denote by

�(Y)=∪{etAY |t ≥0}


the orbit through Y for any Y ∈H. The ω-limit set of Y ∈H is defined as

ω(Y )=∩{�(etAY )|t ≥0}.It was proved in Refs. 15 and 16 that, if �(Y) is precompact in H for anyY ∈H, then ω(Y ) is nonempty, connected, compact in H and

limt→∞ etAY =ω(Y ).

Moreover,

ω(Y )={0}is equivalent to

σp(A)∩ iR=∅,where σp(A) denotes the point spectrum of the operator A. Therefore, inorder to arrive at the strong stability of the semigroup etA, one needs toshow only that (Ref. 15):

(i) �(Y) is precompact in H for any Y ∈H;(ii) σp(A)∩ iR=∅.

Lemma 4.1. See Ref. 18, Theorem 5.3.3. Let Q1,Q2 be two openconvex sets in R

n, such that Q1⊂�2. Let P(D) be a differential operatorwith constant coefficients, such that every hyperplane that is characteristicwith respect to P(D) and intersects Q2 meets also Q1. Then every distri-bution solution w∈D′(Q2) satisfying P(D)w=0 and vanishing in Q1 mustvanish in Q2.

Lemma 4.2. Suppose that ξ ∈R. The boundary-value problem

−ξ2w+�2w=0, in �, (39a)

w= ∂νw=0, on �0, (39b)

w= ∂νw= ∂ννw= ∂νννw=0, on �1 (39c)

admits a unique solution w=0 for all x ∈ �.

Proof. Let P(D) be the differential operator

P(D)=−ξ2+�2, in �.

It is clear that P(D) has no real characteristic vector. By the Holmgrenlocal uniqueness theorem, we know that the solution to (39) vanishes in aneighborhood of �1 in �. Then, w=0 in � follows from Lemma 4.1.


Our last result is on the strong stability of (8) when k=0. We removecondition (g4) on the function g and condition (19) on the region inTheorem 3.1.

Theorem 4.1. Assume that the function g(·) satisfies (g1) through(g3). Then the semigroup eAt is strongly stable; i.e., for all k ≥ 0, theenergy of the system (8) satisfies

limt→∞E(t)=0.

Proof. For a fixed Y = (w, v, z)∈D(A), let

(w(t), v(t), z(t))= etA(w, v, z).

Then, for any t ≥0,

(w(t), v(t), z(t))∈C([0,∞);D(A)).

Hence, w(t) solves the following elliptic problem:

�2w∈L2(�), (40a)

B1w=∫ ∞

0g′(s)∂νz(s)ds ∈L2(�1), (40b)

B2w=−∫ ∞

0g′(s)z(s)ds+kv∈H 1(�1). (40c)

By the regularity theory for elliptic equations (Ref. 10), problem (40)admits a unique solution w∈H

72 (�). Therefore,

w(t)∈C(0,∞;H 12 (�)), (41a)

v(t)∈C(0,∞;W), (41b)

z(t)∈C(0,∞;H 1(0,∞; ∣∣g′(·)∣∣ ;H 1(�1))). (41c)

This shows that the orbit �(Y ) is precompact in H for all Y ∈ D(A).

Moreover, since etA is a contraction semigroup, the set of points whichgenerate precompact orbits is closed in H (Refs.15, 19). Therefore, �(Y )

is precompact in H for every Y ∈H.

It remains to show that

σp(A)∩ iR=∅.


To do this, it suffices to show that

Y = (w, v, z)=0

is the only solution of the equation

(iξ −A)(w, v, z)=0, ∀ξ ∈R. (42)

Note that (42) is the same as

iξw−v=0, (43a)

iξv+�2w=0, (43b)

iξz(s)−v+ (∂/∂s)z(s)=0, s >0, z(0)=0. (43c)

It follows that

v= iξw, in W, (44a)

− ξ2w+�2w=0, in L2(�), (44b)

z(s)= (1− eiξs)w, s >0, in Z. (44c)

If ξ =0, (44a) implies that v=0 in W and z=0 in Z; hence, w solves thefollowing equations:

�2w=0, in �, (45a)

w= ∂νw=0, on �0, (45b)

B1w=∫ ∞

0g′(s)∂νz(s)ds=0, on �1, (45c)

B2w=−∫ ∞

0g′(s)z(s)ds+kv=0, on �1. (45d)

It is clear that (45) possesses a unique solution w=0.Suppose that ξ �=0. By (42) and (12), we have

k

∫�1

|v|2 d�+∫ ∞

0

∫�1

g′′(s)(|z(s)|2+|∂νz(s)|2)d�ds

=Re((iξ −A)(w, v, z(·)), (w, v, z(·)))H=0. (46)

Since k≥0, combining (44c) with (46), we obtain∫ ∞0

g′′(s)∣∣∣1− eiξs

∣∣∣2ds

[‖w‖2

L2(�1)+‖∂νw‖2L2(�1)

]=0.

Then, it follows from Lemma 3.1 that

‖w‖L2(�1)=‖∂νw‖L2(�1)

=0. (47)


By (44a), we have also

v∣∣�1 = iξw

∣∣�1=0. (48)

Finally, substituting (47) into (44c) yields

z=0, in Z. (49)

Therefore, by (44b), (47)–(49), and the definition of D(A), we see that w

satisfies the following equations:

− ξ2w+�2w=0, in �, (50a)

w= ∂νw=0, on �0, (50b)

w= ∂νw=0, on �1, (50c)

B1w=∫ ∞

0g′(s)∂νz(s)ds=0, on �1, (50d)

B2w=−∫ ∞

0g′(s)z(s)ds+kv=0, on �1. (50e)

It is known from Ref. 9, pages 306–309 that, on �1,

B1w= ∂ννw,

B2w= ∂νννw.

Hence, w satisfies

−ξ2w+�2w=0, in �,

w= ∂νw=0, on �0,

w= ∂νw= ∂ννw= ∂νννw=0, on �1.

(51a)

By Lemma 4.2, we see that (51) admits a unique solution w=0. Togetherwith (44a) and (49), this completes the proof of Theorem 4.1.

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