Yubo Bai

Christophe Prieur

Zhiqiang Wang

STABILIZATION OF A RAYLEIGH BEAM WITH COLLOCATED PIEZOELECTRIC SENSOR/ACTUATOR

Keywords: 2020 Mathematics Subject Classification. Primary: 93C20, 93D15; Secondary: 35B40 stabilization, Rayleigh beam, piezoelectric actuator, polynomial decay, frequency domain approach

In this paper, stabilization of a Rayleigh beam with collocated piezoelectric sensor/actuator is studied. With a linear output feedback law, several stability results are achieved. The stability of the closed-loop system depends on the location of the actuator. The sufficient and necessary conditions for strong stability are provided. For almost all choices of the ends of the actuator, the energy of the system decreases to zero in a polynomial way and the explicit polynomial decay rate is obtained. Finally, the system is never exponentially stable for any choice of the ends of the actuator.

Introduction and main results.

1.1. History. The stabilization problem of flexible structures has attracted much attention in last few decades in the context of beam equation. Boundary control and interior control are two major topics in stabilization problem. Boundary stabilization of Rayleigh beam with only one control feedback was considered in [START_REF] Rao | A compact perturbation method for the boundary stabilization of the Rayleigh beam equation[END_REF] using a compact perturbation method. And [START_REF] Guo | Basis property of a Rayleigh beam with boundary stabilization[END_REF] investigated the Riesz basis property for the same equation. In [START_REF] Guo | The active disturbance rejection and sliding mode control approach to the stabilization of the Euler-Bernoulli beam equation with boundary input disturbance[END_REF], boundary stabilization of an Euler-Bernoulli beam with the external disturbance was solved. Dynamic boundary stabilization of an Euler-Bernoulli beam was considered in [START_REF] Guo | Dynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observation[END_REF]. Concerning interior stabilization, [START_REF] Ammari | Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force[END_REF] studied the pointwise feedback control of Euler-Bernoulli beam with two different boundary conditions and obtained the lack of exponential stability when the beam is hinged at both ends. In [START_REF] Gall | Output feedback stabilization of a clamped-free beam[END_REF][START_REF] Gall | Stabilization of a clamped-free beam with collocated piezoelectric sensor/actuator[END_REF], the output feedback stabilization of Euler-Bernoulli beam with a piezoelectric actuator was considered.

For pointwise feedback control problem of Rayleigh beam, one can refer to [START_REF] Ammari | Decay rates for a beam with pointwise force and moment feedback[END_REF] which considered the following equations.

w tt (x, t) -w xxtt (x, t) + w xxxx (x, t) + w t (ξ, t)δ ξ -w xt (ξ, t) d dx δ ξ = 0, 0 < x < π, w(0, t) = w(π, t) = w xx (0, t) = w xx (π, t) = 0.

(1)

It is showed that the exponential stability depends on the location of the control point. Indeed, the exponential stability holds if and only if ξ lies in a countable set. After this work, [START_REF] Weiss | Exponential stabilization of a Rayleigh beam using collocated control[END_REF] redesigned a collocated output feedback control that leads to the exponential stability and the result does not depend on the position of the control point. The control system in [START_REF] Weiss | Exponential stabilization of a Rayleigh beam using collocated control[END_REF] is

w tt (x, t) -αw xxtt (x, t) + w xxxx (x, t) = U(x, t), 0 < x < π, w(0, t) = w(π, t) = w xx (0, t) = w xx (π, t) = 0, (2)
where the control consists of a pointwise control and a distributed one,

U(x, t) = -u 0 (t) d dx δ ξ (x) -u 1,t (t)[αδ ξ (x) + b(x)].
Here, u 0 and u 1 are the two (scalar) input signals and

b(x) =          1 ξ + 1 π -ξ -1 x ξ , for x ≤ ξ, 1 ξ + 1 π -ξ -1 π -x π -ξ , for x > ξ.
The feedback control is designed according to [START_REF] Curtain | Exponential stabilization of well-posed systems by colocated feedback[END_REF] which focuses on the exponential stabilization of a linear abstract evolution system. Concerning stabilization problem for PDEs, Lyapunov method and frequency domain approach are common methods to use. In [START_REF] Bastin | Progress in Nonlinear Differential Equations and their Applications[END_REF], Lyapunov method was widely used to study boundary stabilization of one dimensional hyperbolic systems. As for frequency domain approach, we express the linear evolution PDEs as an abstract system and then apply some classical results on stability of linear C 0 -semigroups. A classical criteria for exponentially stable semigroup was established in [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF][START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF]. After that, [START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF] concerned strong stability of one-parameter semigroups. Several years later, [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF][START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF] were devoted to estimate the polynomial decay rate of solutions to linear evolution equation. A sufficient condition for polynomial stability of linear abstract evolution equation on Hilbert space was proposed in [START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF] and there were logarithm terms in the decay rate. Then [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF] improved the results in [START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF]. The criteria in [START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF] was proved to be sufficient and necessary for polynomial stability of linear evolution equation on Hilbert space and the logarithm terms could be removed. 1.2. Problem under consideration and main results. In this paper, we consider the control problem modelling the transverse deflection of a Rayleigh beam which is subject to the action of an attached piezoelectric actuator. If we suppose that the beam is hinged at both ends, the equation of Rayleigh beam can be written as (see, for instance, [START_REF] Crawley | Detailed models of piezoceramic actuation of beams[END_REF][START_REF] Ph | Theoretical, numerical and experimental discussion on the use of piezoelectric devices for control-structure interaction[END_REF])

w tt (x, t) -αw xxtt (x, t) + w xxxx (x, t) = u(t) d dx [δ η (x) -δ ξ (x)], 0 < x < π,
w(0, t) = w(π, t) = w xx (0, t) = w xx (π, t) = 0, w(x, 0) = w 0 (x), w t (x, 0) = w 1 (x).

(3)

In the equations above w represents the transverse deflection of the beam, α > 0 is a physical constant, ξ and η stand for the ends of the actuator (0 < ξ < η < π), and δ y is the Dirac mass at the point y. The control input is given by u.

Here we are interested in the stabilization of system [START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF]. The control u is expressed as a function of the output w x (η) -w x (ξ). This corresponds to the situation where the output comes from a piezoelectric sensor located on the same interval (ξ, η) as the actuator. In order to choose the feedback control function u, we introduce a suitable Hilbert space and a energy function. Let inner product of V := H 2 (0, π) ∩ H 1 0 (0, π) be (w 1 , w 2 ) V = π 0 w 1 xx w 2 xx dx and inner product of

H := H 1 0 (0, π) be (v 1 , v 2 ) H = π 0 v 1 v 2 + αv 1 x v 2
x dx. Then H = V × H, endowed with the usual product norm, is a complex Hilbert space. Energy function corresponding to the Hilbert space is

E(w(t), w t (t)) = 1 2 π 0 (|w t (x, t)| 2 + α|w xt (x, t)| 2 + |w xx (x, t)| 2 )dx. ( 4
)
For every sufficiently smooth solution of (3), we have

Ė(w(t), w t (t)) = π 0 (w t w tt + αw xt w xtt + w xx w xxx )dx = π 0 (w t w tt -αw t w xxtt + w t w xxxx )dx = π 0 w t (x, t)u(t) d dx [δ η (x) -δ ξ (x)]dx = -u(t)(w xt (η, t) -w xt (ξ, t)).
Inspired by this calculation, we choose feedback control u(t)

:= κ(w xt (η, t) - w xt (ξ, t))
, where κ ≥ 0 is a constant. Then we focus on the following closed-loop system, for 0 < x < π,

w tt (x, t) -αw xxtt (x, t) + w xxxx (x, t) = κ(w xt (η, t) -w xt (ξ, t)) d dx [δ η (x) -δ ξ (x)], w(0, t) = w(π, t) = w xx (0, t) = w xx (π, t) = 0, w(x, 0) = w 0 (x), w t (x, 0) = w 1 (x). ( 5
)
Then we are interested in the asymptotic stability of closed-loop system [START_REF] Bastin | Progress in Nonlinear Differential Equations and their Applications[END_REF]. In order to state the main results, we give several definitions of stability. Definition 1.1. We say that ( 5) is strongly stable in H, if for any (w 0 , w 1 ) in H we have that

E(w(t), w t (t)) = 1 2 ∥(w(t), w t (t))∥ 2 H → 0, as t → +∞. (6)
In Section 2, we express system (5) as an abstract Cauchy problem for operator A, and the definitions ( 14) and ( 15) of the operator A and the domain D(A) of A are precisely given. Definition 1.2. We say that ( 5) is polynomially stable, if there exist two constants C > 0 and l > 0 such that for any (w 0 , w 1 ) in D(A) and any t ≥ 0,

∥(w(t), w t (t))∥ H ≤ Ct -1/l ∥(w 0 , w 1 )∥ D(A) . (7)
Definition 1.3. We say that ( 5) is exponentially stable in H, if there exist two constants C > 0 and δ > 0 such that for any (w 0 , w 1 ) in H and any t ≥ 0,

∥(w(t), w t (t))∥ H ≤ Ce -δt ∥(w 0 , w 1 )∥ H . (8)
Note that the norm on the right-hand of [START_REF] Cassels | An introduction to Diophantine approximation[END_REF] can not be the H-norm. Otherwise, by the semigroup properties, as mentioned in [START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF], [START_REF] Cassels | An introduction to Diophantine approximation[END_REF] implies [START_REF] Crawley | Detailed models of piezoceramic actuation of beams[END_REF]. Then we state the main results of this paper.

Theorem 1.4. The system (5) is strongly stable in H if and only if κ > 0 and η+ξ 2π and η-ξ 2π are irrationals. Theorem 1.5. Let ε > 0. There exist a constant C > 0, an uncountable set A with Lebesgue measure 0 and a set B ε with Lebesgue measure 1 (A and B ε are defined in Section 2), such that for any (w 0 , w 1 ) in D(A) and for any t ≥ 0, the solution (w(t), w t (t)) of (5) satisfies 1. For any ξ and η with η+ξ 2π and η-ξ 2π belong to A, (7) holds for l = 4. 2. For any ξ and η with η+ξ 2π and η-ξ 2π belong to B ε , (7) holds for l = 4 + 4ε. Theorem 1.6. For any ξ and η in (0, π), the system (5) is not exponentially stable in H, and the polynomial stability [START_REF] Cassels | An introduction to Diophantine approximation[END_REF] does not hold for 0 < l < 1 2 . These three theorems are the main results of this paper which provide the full analysis for stability of the system (5). Theorem 1.4 gives the necessary and sufficient conditions for strong stability. Theorem 1.5 admits some sufficient conditions of polynomial stability. For the ends of the piezoelectric actuator in an uncountable zero measure set A, we have a decay rate of t -1/4 . For almost all choices of the ends of the piezoelectric actuator (in a set with Lebesgue measure 1), we have a slower decay rate than the rate of t -1/4 . Note that the polynomial stability holds for all initial data lying in more regular spaces. Theorem 1.6 shows that the system ( 5) is never exponentially stable and is not polynomial stable with high decay rate no matter where the piezoelectric actuator is located.

Remark 1.7. These three stability results all highly depend on the location of the piezoelectric actuator. This phenomenon is common in pointwise control and piezoelectric control. As for stabilization problem, one can find similar phenomenon in [START_REF] Ammari | Decay rates for a beam with pointwise force and moment feedback[END_REF][START_REF] Ammari | Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force[END_REF][START_REF] Gall | Output feedback stabilization of a clamped-free beam[END_REF]. As for controllability problem, the results in [START_REF] Bai | Exact controllability for a Rayleigh beam with piezoelectric actuator[END_REF][START_REF] Crépeau | Control of a clamped-free beam by a piezoelectric actuator[END_REF][START_REF] Tucsnak | Regularity and exact controllability for a beam with piezoelectric actuator[END_REF] also depend on the location of the piezoelectric actuator.

Remark 1.8. As we see above, different from the lack of exponential stability in Theorem 1.6, references [START_REF] Ammari | Decay rates for a beam with pointwise force and moment feedback[END_REF][START_REF] Weiss | Exponential stabilization of a Rayleigh beam using collocated control[END_REF] considered the pointwise or collocated feedback control for Rayleigh beam equation and obtained exponential stability. The reason they obtained the exponential stability is that they apply stronger feedback control. There are two feedback control in (1) but there is only one feedback control in (5). In control system [START_REF] Ammari | Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force[END_REF], there are also two input signals and the input signal u 1 acts on the whole beam by function b. However the control in (5) only acts on two points of the beam. Another work [START_REF] Curtain | Exponential stabilization of well-posed systems by colocated feedback[END_REF] considered the exponential stabilization of a linear abstract system. However the theory of [START_REF] Curtain | Exponential stabilization of well-posed systems by colocated feedback[END_REF] can not apply to our system, since the exact controllability in the natural energy space of the corresponding openloop system is necessary but the study in [START_REF] Bai | Exact controllability for a Rayleigh beam with piezoelectric actuator[END_REF] shows that system (3) is not exact controllable in the natural energy space H. 1.3. Sketch of the proof and outline. The methods to prove Theorems 1.4 and 1.5 are inspired by the methods used in [START_REF] Gall | Output feedback stabilization of a clamped-free beam[END_REF][START_REF] Gall | Stabilization of a clamped-free beam with collocated piezoelectric sensor/actuator[END_REF] which studied the output feedback stabilization of Euler-Bernoulli beam with a piezoelectric actuator. We use frequency domain approach to obtain some estimation of the resolvent on the imaginary axis. In the proof of polynomial stability, we use the multiplier method and we provide a careful computation of the first derivative value at x = π of the solution to a special linear system (40). Based on the calculation and some results from the theory of Diophantine approximation, we obtain an estimation of polynomial decay rate. As for Theorem 1.6, the lack of exponential stability and the lack of polynomial stability with high decay rate, we prove two results together. In detail, due to well-known results (see [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF][START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF][START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF]), we are necessary to prove the resolvent estimate sup

|β|≥1 |β| -l ∥(iβ -A) -1 ∥ = ∞ (9)
for 0 ≤ l < 1 2 . Indeed, to prove Theorem 1.6, we succeed to find an increasing sequence

{β n } n∈N * ⊆ R such that for 0 ≤ l < 1 2 |β n | -l ∥(iβ n -A) -1 ∥ → ∞, as n → ∞.
This fact implies the lack of exponential stability and the lack of polynomial stability with high decay rate.

To summarize, the complete asymptotic behavior of the system (5) is obtained. The location of the piezoelectric actuator dominates the asymptotic behavior of the system. Especially, under this choice of control function, the exponential stability and the polynomial stability with high decay rate do not hold no matter where the piezoelectric actuator is located.

There is a gap in polynomial stability. The polynomial decay rate (7) with The paper is organized as follows. In Section 2 we give some preliminaries on Diophantine approximation and the well-posedness of the closed-loop system. Theorems 1.4, 1.5 and 1.6 are proved in Sections 3, 4 and 5 respectively.

Preliminaries.

2.1. Preliminaries on Diophantine approximation. We need some results from the theory of Diophantine approximation (see [START_REF] Cassels | An introduction to Diophantine approximation[END_REF][START_REF] Lang | Introduction to Diophantine approximations[END_REF]). For a real number ρ, we denote by ∥ρ∥ Z the difference, taken positively, between ρ and the nearest integer, i.e., ∥ρ∥ Z = min n∈Z |ρ -n|.

Let us denote by A the set of all irrationals ρ in (0, 1) such that if [0, a 1 , . . . , a n . . . ] is the expansion of ρ as a continued fraction, then {a n } ∞ n=1 is bounded. Its Lebesgue measure is equal to zero (see [START_REF] Cassels | An introduction to Diophantine approximation[END_REF]). The following property of this set is essentially useful in our work (see [START_REF] Lang | Introduction to Diophantine approximations[END_REF]). Proposition 2.1. A number ρ is in A if and only if there exists a constant C > 0 such that for all strictly positive integer Q,

∥Qρ∥ Z ≥ C Q . ( 10
)
The next proposition, which is proved in [START_REF] Cassels | An introduction to Diophantine approximation[END_REF], shows that an inequality slightly weaker than [START_REF] Curtain | Exponential stabilization of well-posed systems by colocated feedback[END_REF] holds for almost all points in (0, 1). This proposition is the definition of set B ε . Proposition 2.2. For any ε > 0 there exists a set B ε ⊆ (0, 1) having Lebesgue measure equal to 1 and a constant C > 0, such that for any ρ in B ε and for all strictly positive integer Q,

∥Qρ∥ Z ≥ C Q 1+ε . (11)
The following proposition on simultaneous approximation (see [START_REF] Cassels | An introduction to Diophantine approximation[END_REF]) used in [START_REF] Tucsnak | Regularity and exact controllability for a beam with piezoelectric actuator[END_REF] is quite important to prove Theorem 1.6. Proposition 2.3. Let ρ 1 , . . . , ρ k be k irrationals in (0, 1). Then there exists a strictly increasing sequence of natural numbers Q n such that

Q 1 k n max i=1,...,k (∥Q n ρ 1 ∥ Z , . . . , ∥Q n ρ i ∥ Z , . . . , ∥Q n ρ k ∥ Z ) ≤ k k + 1 ∀n ≥ 1.
2.2. Well-posedness of (5). Now we express system (5) as an abstract Cauchy problem. Let us introduce some notions which are also used in [START_REF] Gall | Output feedback stabilization of a clamped-free beam[END_REF]. If w is a function in

H 1 (0, b) ∩ H 1 (b, π), we define {w x } in L 2 (0, π) by {w x }(x) := w D ′ (0,b) x (x) if x ∈ (0, b), w D ′ (b,π) x (x) if x ∈ (b, π),
where w

D ′ (0,b) x (respectively w D ′ (b,π) x ) denotes the distributional derivative ∂w ∂x in D ′ (0, b) (respectively in D ′ (b, π)). Let [w] b := w(b + ) -w(b -). It follows that w x = {w x } + [w] b δ b in D ′ (0, π).
Denote R := (I -α∂ xx ) -1 . It is well-known that operator R is an isomorphism from V ′ to L 2 (0, π) and an isomorphism from H -1 (0, π) to H by Lax-Milgram Theorem. Applying R to both sides of the first equation of ( 5) to obtain

w tt + Rw xxxx = κ(w xt (η, t) -w xt (ξ, t))R d dx [δ η (x) -δ ξ (x)]. (12)
If we introduce v := w t , then we can formally introduce an operator A on H with

A(w, v) = v, R -w xxxx + κ(v x (η) -v x (ξ)) d dx [δ η -δ ξ ] . (13)
In order to define the domain D(A) of the operator A, inspired by [START_REF] Gall | Output feedback stabilization of a clamped-free beam[END_REF], let us note

A(w, v) ∈ H ⇔ v ∈ V and -w xxx + κ(v x (η) -v x (ξ))[δ η -δ ξ ] ∈ L 2 (0, π).
Therefore, the restriction of w xxx to each of the intervals (0, ξ), (ξ, η) and (η, π) has to be a L 2 (0, π) function which means that w belongs to H 3 (0, ξ) ∩ H 3 (ξ, η) ∩ H 3 (η, π). Using the notion introduced above, we have

w xxx = {w xxx } + [w xx ] ξ δ ξ + [w xx ] η δ η and -w xxx + κ(v x (η) -v x (ξ))[δ η -δ ξ ] = -{w xxx } -(κ(v x (η) -v x (ξ)) + [w xx ] ξ )δ ξ + (κ(v x (η) -v x (ξ)) -[w xx ] η )δ η . Then -w xxx + κ(v x (η) -v x (ξ))[δ η -δ ξ ] belongs to L 2 (0, π)
provided that all the coefficients in front of the Dirac measures vanish, i.e.,

[w xx ] η = κ(v x (η) -v x (ξ)) = -[w xx ] ξ .

So the definition of D(A) is

D(A) = {(w, v)| w, v ∈ V, w ∈ H 3 (0, ξ) ∩ H 3 (ξ, η) ∩ H 3 (η, π), w xx (0) = w xx (π) = 0, [w xx ] η = κ(v x (η) -v x (ξ)) = -[w xx ] ξ }. (14)
And now we can define the operator A on H with

A(w, v) = (v, -R{w xxx } x ), ∀(w, v) ∈ D(A). (15)
Then system (5) can be written as the abstract Cauchy problem

   dz dt = Az, t > 0, z(0) = (w 0 , w 1 ). ( 16
)
The well-posedness of ( 5) follows an application of the classical semigroup theory.

Theorem 2.4. If κ > 0, then A generates a C 0 -semigroup e tA of contractions on H.

According to a classical result of semigroup theory, Theorem 2.4 is proved if we show that D(A) is dense in H and that A is dissipative and that zero belongs to ρ(A), the resolvent set of A. First notice that D(A) is dense in H obviously. Then we show that A is dissipative.

Lemma 2.5. For any z = (w, v) in D(A) we have that

(Az, z) H = 2iIm π 0 v xx w xx dx -κ|v x (η) -v x (ξ)| 2 . ( 17
)
In particular, Re(Az, z)

H = -κ|v x (η) -v x (ξ)| 2 ≤ 0, i.e., A is dissipative. Proof. For any z = (w, v) in D(A), we have (Az, z) H = π 0 v xx w xx dx - π 0 [R{w xxx } x v + αR{w xxx } xx v x ]dx. Notice that Rf xx = (Rf -f )/α, then (Az, z) H = π 0 v xx w xx dx - π 0 [R{w xxx } x v + (R{w xxx } -{w xxx })v x ]dx = π 0 v xx w xx dx + π 0 {w xxx }v x dx.
And we have

π 0 {w xxx }v x dx = ξ 0 w xxx v x dx + η ξ w xxx v x dx + π η w xxx v x dx = - π 0 w xx v xx dx + w xx (ξ -)v x (ξ) + w xx (η -)v x (η) -w xx (ξ + )v x (ξ) -w xx (η + )v x (η) = - π 0 w xx v xx dx -[w xx ] ξ v x (ξ) -[w xx ] η v x (η) = - π 0 w xx v xx dx -κ|v x (η) -v x (ξ)| 2 .
Finally, we have

(Az, z) H = π 0 v xx w xx dx - π 0 w xx v xx dx -κ|v x (η) -v x (ξ)| 2 = 2iIm π 0 v xx w xx dx -κ|v x (η) -v x (ξ)| 2 .
Next proposition shows that zero belongs to ρ(A) and finishes the proof of Theorem 2.4. Proposition 2.6. 0 belongs to ρ(A).

Proof. We have to prove that the operator A : D(A) → H is one-to-one and onto, and that its inverse A -1 : H → H is continuous. Fix any (f, g) in H, and let us investigate the equation A(w, v) = (f, g), where (w, v) belongs to D(A). We have to solve

v = f, -R{w xxx } x = g,
which is equipped by boundary conditions. Denote λ = κ(f x (η) -f x (ξ)), and introduce the solution w in H 3 (0, π) of the following equation

-R w xxxx = g, w(0) = w(π) = w xx (0) = w xx (π) = 0. Since R is an isomorphism from H -1 (0, π) to H, w exists uniquely. Set w = w + w, we need to solve      R{ w xxx } x = 0, w(0) = w(π) = w xx (0) = w xx (π) = 0, [ w xx ] η = λ = -[ w xx ] ξ , (18)
where w is to be found in

H 2 (0, π)∩H 3 (0, ξ)∩H 3 (ξ, η)∩H 3 (η, π). Since R{ w xxx } x = 0 implies that { w xxx } = 0 in L 2 (0, π), from (18)
we have

w(x) =      a 1 x, 0 < x < ξ, b 0 + b 1 x + b 2 x 2 , ξ < x < η, c 1 (x -π), η < x < π, where a 1 , b 0 , b 1 , b 2 , c 1 are constants. Since H 2 (0, π) is included in C 1 (0, π) because of the Sobolev Embedding Theorem and [ w xx ] η = λ = -[ w xx ] ξ , we have                a 1 ξ = b 0 + b 1 ξ + b 2 ξ 2 , c 1 (η -π) = b 0 + b 1 η + b 2 η 2 , a 1 = b 1 + 2b 2 ξ, c 1 = b 1 + 2b 2 η, -2b 2 = λ.
Since the determinant of the coefficients matrix of the above 5 × 5 linear equation is 2π, constants a 1 , b 0 , b 1 , b 2 , c 1 are uniquely determined by the linear equation. This proves the existence and uniqueness of w and the existence and uniqueness of (w, v) in D(A) such that A(w, v) = (f, g). It remains to prove that the map

A -1 : H → H is continuous. Let (w, v) = A -1 (f, g). Then ∥v∥ H = ∥f ∥ H ≤ C∥f ∥ V because of the Sobolev Embedding Theorem. And we have ∥ w∥ V ≤ C∥g∥ H because R is an isomorphism from H -1 (0, π) to H.
And from the linear equation we get

∥ w∥ V ≤ C|λ| = Cκ|f x (η) -f x (ξ)| ≤ C∥f ∥ V . Therefore, we have ∥(w, v)∥ H ≤ C∥(f, g)∥ H
, which shows that the map A -1 : H → H is continuous. Then we obtain that zero belongs to ρ(A) and it concludes the proof of Proposition 2.6 and of Theorem 2.4.

3.

Strong stability (Proof of Theorem 1.4). In this section we use Arendt-Batty Theorem (see [START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF]) to prove strong stability. First we deal with the only if part. If system ( 5) is strongly stable in H, it follows from Lemma 2.5 that for any (w 0 , w 1 ) in D(A) and for any T ≥ 0,

E(w(T ), w t (T )) -E(w 0 , w 1 ) = -κ T 0 |w tx (η, t) -w tx (ξ, t)| 2 dt.
Thus, the condition κ > 0 is necessary for the energy to decrease. On the other hand, if η+ξ 2π or η-ξ 2π is a rational number, there exists a k ≥ 1 such that

sin k(η + ξ) 2 sin k(η -ξ) 2 = 0.
Then the state of the form (w 0 , w 1 ) = (w 0 k sin(kx), w 1 k sin(kx)) gives rise to a solution of (5) whose energy is constant and does not tend to zero. Then κ > 0 together with η+ξ 2π and η-ξ 2π are irrationals constitutes a necessary condition for the strong stability of system [START_REF] Bastin | Progress in Nonlinear Differential Equations and their Applications[END_REF].

Next we check that this condition is also sufficient. Here we need a property of operator A.

Lemma 3.1. The operator A -1 : H → H is compact. Proof. Choose any bounded sequence {(f n , g n )} n≥1 included in H, our aim is to prove that {(w n , v n )} n≥1 := {A -1 (f n , g n )} n≥1 is precompact in H. Since ∥A -1 (f n , g n )∥ D(A) = ∥A -1 (f n , g n )∥ H + ∥(f n , g n )∥ H and A -1 : H → H is continuous, we have A -1 : H → D(A) is continuous. Then for any bounded sequence {(f n , g n )} n≥1 included in H, {A -1 (f n , g n )} n≥1 is bounded in D(A).
Then it is sufficient to prove the embedding D(A) → H is compact. Let {(w n , v n )} n≥1 is any bounded sequence in D(A), we need to prove there exists a subsequence {(w n k , v n k )} k≥1 which converges in H. Assume that ∥(w n , v n )∥ D(A) ≤ C for some constant C > 0 and then for any n ≥ 1, we have

∥w n ∥ V ≤ C, ∥v n ∥ V ≤ C and ∥R{w n,xxx } x ∥ H ≤ C for some constant C > 0. Since embedding H 2 (0, π) → H 1 (0, π) is compact, there exist some v in H and a subsequence {v n k } k≥1 such that v n k → v in H as k → ∞. ( 19
) Notice that ∥R{w n,xxx } x ∥ H ≤ C implies ∥{w n,xxx }∥ L 2 (0,π) ≤ C. Then we have ∥w n ∥ V + ∥w n ∥ H 3 (0,ξ) + ∥w n ∥ H 3 (ξ,η) + ∥w n ∥ H 3 (η,π) ≤ C
for some constant C > 0, which implies that there exist some w in V and a subsequence {w n k } k≥1 such that

w n k ⇀ w in V ∩ H 3 (0, ξ) ∩ H 3 (ξ, η) ∩ H 3 (η, π) as k → ∞,
where the notation ⇀ is used to denote weak convergence. So we have

w n k → w in H 2 (0, ξ) ∩ H 2 (ξ, η) ∩ H 2 (η, π) and therefore w n k → w in V as k → ∞. (20)
Equations ( 19) and [START_REF] Lions | Non-homogeneous boundary value problems and applications[END_REF] show that the embedding D(A) → H is compact.

Since A -1 is compact, operator A is a discrete operator and then σ(A), the spectrum of A, consists of isolated eigenvalues. Applying Arendt-Batty Theorem, we only need to prove iR belongs to ρ(A). Lemma 3.2. If κ > 0 and η+ξ 2π and η-ξ 2π are irrationals, then iR belongs to ρ(A).

Proof. Since σ(A) consists of isolated eigenvalues, it is sufficient to check that for any β in R \ {0} the equation

(iβ -A)(w, v) = (0, 0), (w, v) ∈ D(A) (21)
admits only the trivial solution (w, v) = (0, 0). Using ( 17) and ( 21), we obtain that

0 = ((iβ -A)(w, v), (w, v)) H = i β∥(w, v)∥ 2 H -2Im π 0 v xx w xx dx + κ|v x (η) -v x (ξ)| 2 . Hence [w xx ] η = -[w xx ] ξ = κ(v x (η) -v x (ξ)) = 0 and w belongs to V ∩ H 3 (0, π). So (21) yields -β 2 w + Rw xxxx = 0, w(0) = w(π) = w xx (0) = w xx (π) = 0.
Solving this equation we obtain that when

β 2 = λ k = k 4 1+αk 2 for some k ≥ 1, this equation admits an non-zero solution w(x) = C sin(kx) where C is a complex constant. Therefore, v(x) = iβw(x) = iβC sin(kx). It follows that 0 =κ(v x (η) -v x (ξ)) = iβCκk(cos(kη) -cos(kξ)) = -2iβCκk sin k(η + ξ) 2 sin k(η -ξ) 2 .
Since κ > 0 and η+ξ 2π and η-ξ 2π are irrationals, it is necessary that C = 0. Consequently w = v = 0. This concludes the proof of Lemma 3.2 and Theorem 1.4.

4.

Polynomial stability (Proof of Theorem 1.5). This section is devoted to prove polynomial stability and estimate the decay rate. The methods of proof are inspired by the methods used in [START_REF] Gall | Output feedback stabilization of a clamped-free beam[END_REF][START_REF] Gall | Stabilization of a clamped-free beam with collocated piezoelectric sensor/actuator[END_REF].

According to Borichev-Tomilov criteria (see [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF]), polynomial decay rate ( 7) is equivalent to the resolvent estimate

sup |β|≥1 |β| -l ∥(iβ -A) -1 ∥ < ∞. (22)
We prove two different cases ( η±ξ 2π belongs to A and η±ξ 2π belongs to B ε ) of ( 22) simultaneously, so we set

l =      4, if η ± ξ 2π ∈ A, 4 + 4ε, if η ± ξ 2π ∈ B ε . (23)
We argue by contradiction. If ( 22) is false, then there exist

β n in R and (w n , v n ) in D(A) for n ≥ 1 such that ∥(w n , v n )∥ H = 1, |β n | → ∞ (24)
and

|β n | l (f n , g n ) → (0, 0) in H, (25)
where

(f n , g n ) = (iβ n -A)(w n , v n ), namely, |β n | l (iβ n w n -v n ) = |β n | l f n → 0 in V, ( 26
)
|β n | l (iβ n v n + R{w n,xxx } x ) = |β n | l g n → 0 in H. ( 27
)
Note that putting conjugate on (26) and ( 27) is equivalent to replace iβ n , w n and v n by -iβ n , w n and v n , respectively. So we assume without loss of generality that β n > 0. We obtain a contradiction through two lemmas. The proofs of these two lemmas are given after the proof of Theorem 1.5. with l ≥ 0. We have

lim n→∞ ∥w n ∥ 2 V = lim n→∞ ∥v n ∥ 2 H = lim n→∞ ∥β n w n ∥ 2 H = 1 2 , ( 28
)
and

3 π 0 |w n,xx | 2 dx + β 2 n π 0 [|w n | 2 + α|w n,x | 2 ]dx = 2β 2 n π 0 α|w n,x | 2 dx + O(|β n w n,x (π)| 2 ) + o(1), (29)
where and in what follows, o(1) and O(1) are the infinitesimal and the bounded term respectively as n → ∞.

Remark 4.2. Notice that Lemma 4.1 (and the following Lemma 4.3) holds not only for l ≥ 4 but also for 0 ≤ l < 4. We only use the expression ( 23) of l at the end of the proof of Theorem 1.5.

Therefore, we obtain a contradiction to (28) if β n w n,x (π) → 0 as n → ∞, because the left-hand side of (29) has a limit 2 but the right-hand side is smaller than 3/2 as n → ∞.

In next lemma, we give the upper bound of β n w n,x (π). In what follows, the letters C, C ′ , C ′′ , . . . denote positive constants which may vary from line to line.

|β n w n,x (π)| ≤ C ∥g n ∥ H + ∥f n ∥ V sin(b n π) 2 + (h(b n )) 2 , ( 30
)
where

b n = αβ 2 n + β n α 2 β 2 n + 4 2 1 2
, and the function h : R → R is defined by

h(b) = 2 cos b(π -η + ξ) + 2 cos b(π -η -ξ) -2 cos(bπ) -cos b(π -2ξ) -cos b(π -2η).
Therefore, it is sufficient to estimate the upper bound (30) of

|β n w n,x (π)|. Since b n → ∞ as β n → ∞, we need to estimate the lower bound of sin(b n π) 2 + (h(b n )) 2 . For b in N * , we have h(b) =2 cos(bπ)(1 -cos b(η + ξ))(cos b(η -ξ) -1) = -8 cos(bπ) sin b η + ξ 2 sin b η -ξ 2 2 . ( 31
)
Recall the conditions of Theorem 1.5, if η±ξ 2π belong to A, from [START_REF] Curtain | Exponential stabilization of well-posed systems by colocated feedback[END_REF] in Section 2 we see that there exists a small constant C > 0 such that for all b in N * ,

sin b(η ± ξ) 2 = sin π b(η ± ξ) 2π -p ≥ sin πC b ≥ C b . (32)
And if η±ξ 2π belong to B ε , from [START_REF] Ph | Theoretical, numerical and experimental discussion on the use of piezoelectric devices for control-structure interaction[END_REF] in Section 2 we have there exists a small constant

C > 0 such that for all b in N * , sin b(η ± ξ) 2 = sin π b(η ± ξ) 2π -p ≥ sin πC b ≥ C b 1+ε . ( 33
)
Then by the expressions ( 23) and (31) for l and h(b), we have

|h(b)| ≥ C b l
for b in N * . Since the function h is uniformly Lipschitz continuous on R, the same inequalities hold (with different constants) for b large enough in

k∈N * k - C ′ k l , k + C ′ k l ,
if the constant C ′ > 0 is large enough. On the other hand we may associate with, that constant

C ′ > 0, a constant C ′′ > 0 such that | sin(bπ)| ≥ C ′′ b l for b large enough satisfying |b -k| ≥ C ′ /k l for all k in N * . It follows that sin(bπ) 2 + (h(b)) 2 ≥ C b l (34)
for b large enough. Gathering together (30) amd (34) we have

|β n w n,x (π)| ≤ Cb l n (∥g n ∥ H + ∥f n ∥ V ) ≤ C ′ β l n (∥g n ∥ H + ∥f n ∥ V ) → 0, as n → ∞.
(35) Therefore, the proof of Theorem 1.5 is achieved.

The remaining thing is to prove Lemmas 4.1 and 4.3.

4.1. Proof of Lemma 4.1. We prove this lemma by the multiplier method. From (17), we deduce that

κ|v n,x (η) -v n,x (ξ)| 2 = Re((iβ n -A)(w n , v n ), (w n , v n )) H = Re((f n , g n ), (w n , v n )) H ,
hence using ( 24) and ( 25), we have

|v n,x (η) -v n,x (ξ)| 2 → 0 as n → ∞. ( 36
)
On the other hand, ( 23), ( 24), ( 26) and ( 27) imply that as n → ∞

iβ n ∥w n ∥ 2 V -(v n , w n ) V → 0, ( 37
)
iβ n ∥v n ∥ 2 H + (R{w n,xxx } x , v n ) H → 0. ( 38
)
Taking the difference of (37) and (38) we deduce

iβ n (∥w n ∥ 2 V -∥v n ∥ 2 H ) - π 0 v n,xx w n,xx dx - π 0 [R{w n,xxx } x v n + αR{w n,xxx } xx v n,x ]dx → 0.
However, by [START_REF] Lang | Introduction to Diophantine approximations[END_REF], we deduce

Im - π 0 [R{w n,xxx } x v n + αR{w n,xxx } xx v n,x ]dx + π 0 v n,xx w n,xx dx = 2Im π 0 v n,xx w n,xx dx, which implies that Im π 0 [R{w n,xxx } x v n + αR{w n,xxx } xx v n,x ]dx + π 0 v n,xx w n,xx dx = 0. Hence β n (∥w n ∥ 2 V -∥v n ∥ 2 H ) → 0. ( 39
)
Therefore, using ( 24) and ( 26) we obtain (28). Now we prove (29). We eliminate v n in (27) by using ( 26), and we have

-β 2 n w n + R{w n,xxx } x = g n + iβ n f n in H, (40)
and hence

(-β 2 n w n + R{w n,xxx } x , φw n,x ) H = (g n + iβ n f n , φw n,x ) H (41)
for any real function

φ in C 3 [0, π].
First we deal with the right-hand side of (41). We obtain by using ( 27) and ( 28)

(g n , φw n,x ) H ≤ C∥g n ∥ H ∥w n,x ∥ H ≤ C∥g n ∥ H ∥w n ∥ V → 0. (42)
Notice that

(f n , φw n,x ) H = π 0 [f n φw n,x + αf n,x (φw n,x ) x ]dx = - π 0 [(f n φ) x w n + αf n,xx φw n,x ]dx + α(φ(π)f n,x (π)w n,x (π) -φ(0)f n,x (0)w n,x (0)). (43)
By ( 26) and ( 28) and H 2 (0, π) included in C 1 (0, π), we deduce that

β n π 0 [(f n φ) x w n + αf n,xx φw n,x ]dx ≤ C∥f n ∥ V ∥β n w n ∥ H → 0 (44)
and that

β n φ(π)f n,x (π)w n,x (π) = φ(π)o(β n w n,x (π)), β n φ(0)f n,x (0)w n,x (0) = φ(0)o(β n w n,x (0)). ( 45
)
It follows from (43), ( 44) and (45) that

(iβ n f n , φw n,x ) H = φ(π)o(β n w n,x (π)) + φ(0)o(β n w n,x (0)) + o(1). ( 46
)
Now we turn to the left-hand side of (41). Using integrating by parts we have

(-β 2 n w n , φw n,x ) H = -β 2 n π 0 [w n φw n,x + αw n,x (φw n,x ) x ]dx =β 2 n π 0 [φ x |w n | 2 + φw n,x w n + αw n,xx φw n,x ]dx -β 2 n α(φ(π)|w n,x (π)| 2 -φ(0)|w n,x (0)| 2 ).
Hence we have 2Re(-

β 2 n w n , φw n,x ) H =β 2 n π 0 φ x [|w n | 2 -α|w n,x | 2 ]dx -β 2 n α(φ(π)|w n,x (π)| 2 -φ(0)|w n,x (0)| 2 ). ( 47
)
Denote I := (R{w n,xxx } x , φw n,x ) H . Since Rf xx = (Rf -f )/α and R is an isomorphism from L 2 (0, π) to V , then we obtain by integrating by parts

I = - π 0 {w n,xxx }(φw n,x ) x dx = - ξ 0 + η ξ + π η [w n,xxx φ x w n,x + φw n,xxx w n,xx ]dx = π 0 [-φ x |w n,xx | 2 + 3φ x |w n,xx | 2 + w n,xx φ xx w n,x ]dx + ξ 0 + η ξ + π η φw n,xx w n,xxx dx + φ x (η)w n,x (η)[w n,xx ] η + φ x (ξ)w n,x (ξ)[w n,xx ] ξ + φ(η)[|w n,xx | 2 ] η + φ(ξ)[|w n,xx | 2 ] ξ = π 0 3φ x |w n,xx | 2 dx + ξ 0 + η ξ + π η φw n,xx w n,xxx dx + φ x (η)w n,x (η)[w n,xx ] η + φ x (ξ)w n,x (ξ)[w n,xx ] ξ + φ(η)[|w n,xx | 2 ] η + φ(ξ)[|w n,xx | 2 ] ξ + π 0 [w n,x φ xx w n,xx -φ xxx |w n,x | 2 -w n,x φ xx w n,xx ]dx + ξ 0 + η ξ + π η w n,x φ x w n,xxx dx + φ x (η)w n,x (η)[w n,xx ] η + φ x (ξ)w n,x (ξ)[w n,xx ] ξ + φ xx (π)|w n,x (π)| 2 -φ xx (0)|w n,x (0)| 2 = ξ 0 + η ξ + π η [w n,x φ x w n,xxx + φw n,xx w n,xxx ]dx + π 0 [3φ x |w n,xx | 2 -φ xxx |w n,x | 2 ]dx + φ(η)[|w n,xx | 2 ] η + φ(ξ)[|w n,xx | 2 ] ξ + 2Re(φ x (η)w n,x (η)[w n,xx ] η + φ x (ξ)w n,x (ξ)[w n,xx ] ξ ) + φ xx (π)|w n,x (π)| 2 -φ xx (0)|w n,x (0)| 2 , and hence 2ReI = π 0 [3φ x |w n,xx | 2 -φ xxx |w n,x | 2 ]dx + φ(η)[|w n,xx | 2 ] η + φ(ξ)[|w n,xx | 2 ] ξ + 2Re(φ x (η)w n,x (η)[w n,xx ] η + φ x (ξ)w n,x (ξ)[w n,xx ] ξ ) + φ xx (π)|w n,x (π)| 2 -φ xx (0)|w n,x (0)| 2 .
Using a classical interpolation inequality (see [START_REF] Lions | Non-homogeneous boundary value problems and applications[END_REF]), for all s in [0, 2), ∥u∥ 2 H s (0,π) ≤ C∥u∥ H 2 (0,π) ∥u∥ L 2 (0,π) , and (28) and β n → ∞, we obtain that w n → 0 in H s (0, π) for all s in [0, 2). Hence w n → 0 in C 1 (0, π) since embedding H 2 (0, π) → C 1 (0, π) is continuous. And we have [w n,xx ] η → 0 and [w n,xx ] η → 0 because of ( 14) and (36). Therefore,

2ReI = π 0 3φ x |w n,xx | 2 dx + φ(η)[|w n,xx | 2 ] η + φ(ξ)[|w n,xx | 2 ] ξ + o(1). Notice that [|w n,xx | 2 ] η =w n,xx (η + )w n,xx (η + ) -w n,xx (η -)w n,xx (η -) =[w n,xx ] η w n,xx (η + ) + w n,xx (η -)[w n,xx ] η =2Re([w n,xx ] η w n,xx (η + )) -|[w n,xx ] η | 2 =2κRe((v n,x (η) -v n,x (ξ))w n,xx (η + )) + o(1).
Similarly,

[|w n,xx | 2 ] ξ = -2κRe((v n,x (η) -v n,x (ξ))w n,xx (ξ + )) + o(1) = -2κRe((v n,x (η) -v n,x (ξ))([w n,xx ] ξ + w n,xx (ξ -))) + o(1) = -2κRe((v n,x (η) -v n,x (ξ))w n,xx (ξ -)) + o(1). Consequently, we have 2ReI = π 0 3φ x |w n,xx | 2 dx + 2κφ(η)Re((v n,x (η) -v n,x (ξ))w n,xx (η + )) -2κφ(ξ)Re((v n,x (η) -v n,x (ξ))w n,xx (ξ -)) + o(1). (48)
Gathering together (41), ( 42), ( 46), ( 47) and (48) with φ(x) = x, we obtain

3 π 0 |w n,xx | 2 dx + β 2 n π 0 [|w n | 2 -α|w n,x | 2 ]dx = (απβ n w n,x (π) + o(1))β n w n,x (π) -2κRe((v n,x (η) -v n,x (ξ))(ηw n,xx (η + ) -ξw n,xx (ξ -))) + o(1). ( 49
)
Now we estimate w n,xx (η + ) and w n,xx (ξ -) as n → ∞. We have the following result.

-β 2 n w n,x + R{w n,xxx } xx = g n,x + iβ n f n,x in L 2 (0, π). ( 50
)
Multiplying each term in (40) by φw n,x where φ in C 3 [0, π] is a real function, and integrating over (0, ξ) and (η, π), we obtain

-β 2 n ξ 0 w n φw n,x dx + ξ 0 R{w n,xxx } x φw n,x dx = ξ 0 (g n + iβ n f n )φw n,x dx, (51) -β 2 n π η w n φw n,x dx + π η R{w n,xxx } x φw n,x dx = π η (g n + iβ n f n )φw n,x dx. ( 52
)
Multiplying each term in (50) by (φw n,x ) x and integrating over (0, ξ) and (η, π), we obtain

-β 2 n ξ 0 w n,x (φw n,x ) x dx + ξ 0 R{w n,xxx } xx (φw n,x ) x dx = ξ 0 (g n + iβ n f n ) x (φw n,x ) x dx, (53)
and

-β 2 n π η w n,x (φw n,x ) x dx + π η R{w n,xxx } xx (φw n,x ) x dx = π η (g n + iβ n f n ) x (φw n,x ) x dx. (54)
Adding (51) by α•(53) and adding (52) by α•(54), we obtain

-β 2 n ξ 0 [w n φw n,x + αw n,x (φw n,x ) x ]dx + ξ 0 [R{w n,xxx } x φw n,x + αR{w n,xxx } xx (φw n,x ) x ]dx = ξ 0 [g n φw n,x + αg n,x (φw n,x ) x ]dx + iβ n ξ 0 [f n φw n,x + αf n,x (φw n,x ) x ]dx, (55)
and

-β 2 n π η [w n φw n,x + αw n,x (φw n,x ) x ]dx + π η [R{w n,xxx } x φw n,x + αR{w n,xxx } xx (φw n,x ) x ]dx = π η [g n φw n,x + αg n,x (φw n,x ) x ]dx + iβ n π η [f n φw n,x + αf n,x (φw n,x ) x ]dx. (56)
Denote ( 55) and (56) by I ξ 1 + I ξ 2 = I ξ 3 + I ξ 4 and I η 1 + I η 2 = I η 3 + I η 4 respectively. Now we estimate each term of (55) and (56). Obviously

|I ν 3 | ≤ C∥g n ∥ H ∥w n ∥ V = o(1), ν = ξ, η. (57)
Using integrating by parts, we have

I ξ 4 =iβ n ξ 0 [f n φw n,x -αf n,xx φw n,x ]dx + iβ n [φ(ξ)f n,x (ξ)w n,x (ξ) -φ(0)f n,x (0)w n,x (0)].
Notice that

iβ n ξ 0 [f n φw n,x -αf n,xx φw n,x ]dx ≤ C∥f n ∥ V ∥β n w n ∥ H = o(1),
then we obtain

I ξ 4 = iβ n [φ(ξ)f n,x (ξ)w n,x (ξ) -φ(0)f n,x (0)w n,x (0)] + o(1), (58)
and similarly obtain

I η 4 = iβ n [φ(π)f n,x (π)w n,x (π) -φ(η)f n,x (η)w n,x (η)] + o(1). ( 59
)
Again using integrating by parts, we deduce

I ξ 1 =β 2 n ξ 0 [φ x |w n | 2 + w n,x φw n + αw n,xx φw n,x ]dx -β 2 n φ(ξ)|w n (ξ)| 2 -β 2 n α(φ(ξ)|w n,x (ξ)| 2 -φ(0)|w n,x (0)| 2 ),
and therefore,

2ReI ξ 1 =β 2 n ξ 0 φ x [|w n | 2 -α|w n,x | 2 ]dx -β 2 n φ(ξ)|w n (ξ)| 2 -β 2 n α(φ(ξ)|w n,x (ξ)| 2 -φ(0)|w n,x (0)| 2 ). ( 60
)
Similarly we have

2ReI η 1 =β 2 n π η φ x [|w n | 2 -α|w n,x | 2 ]dx + β 2 n φ(η)|w n (η)| 2 -β 2 n α(φ(π)|w n,x (π)| 2 -φ(η)|w n,x (η)| 2 ). ( 61
)
Finally we deal with I ξ 2 and I η 2 . We have

I ξ 2 = ξ 0 [-R{w n,xxx }(φw n,x ) x + (R{w n,xxx } -{w n,xxx })(φw n,x ) x ]dx + R{w n,xxx }(ξ)φ(ξ)w n,x (ξ) = - ξ 0 w n,xxx (φw n,x ) x dx + R{w n,xxx }(ξ)φ(ξ)w n,x (ξ) = : I ξ 21 + I ξ 22 . (62)
First notice that

∥R{w n,xxx }∥ C 0 (0,π) ≤C∥R{w n,xxx }∥ H ≤ C∥{w n,xxx }∥ H -1 (0,π) =C∥w n,xxx -[w n,xx ] ξ δ ξ -[w n,xx ] η δ η ∥ H -1 (0,π) ≤C(∥w n,xxx ∥ H -1 (0,π) + [w n,xx ] ξ + [w n,xx ] η ) ≤C∥w n ∥ V + o(1) = O(1),
and therefore,

I ξ 22 ≤ ∥R{w n,xxx }φw n,x ∥ C 0 (0,π) = o(1). ( 63
)
Then we estimate I ξ 21 . We have

I ξ 21 = - ξ 0 [w n,xxx φ x w n,x + w n,xxx φw n,xx ]dx = ξ 0 [-φ x |w n,xx | 2 + 3φ x |w n,xx | 2 + w n,xx φ xx w n,x + w n,xx φw n,xxx ]dx -φ x (ξ)w n,x (ξ)w n,xx (ξ -) -φ(ξ)|w n,xx (ξ -)| 2 = ξ 0 [3φ x |w n,xx | 2 + w n,xx φw n,xxx + w n,x φ x w n,xxx -φ xxx |w n,x | 2 ]dx -φ x (ξ)w n,x (ξ)w n,xx (ξ -) -φ(ξ)|w n,xx (ξ -)| 2 -φ x (ξ)w n,x (ξ)w n,xx (ξ -) + φ xx (ξ)|w n,x (ξ)| 2 -φ xx (0)|w n,x (0)| 2 . So 2ReI ξ 21 = ξ 0 [3φ x |w n,xx | 2 -φ xxx |w n,x | 2 ]dx -2φ x (ξ)Re(w n,x (ξ)w n,xx (ξ -)) -φ(ξ)|w n,xx (ξ -)| 2 + φ xx (ξ)|w n,x (ξ)| 2 -φ xx (0)|w n,x (0)| 2 =3 ξ 0 φ x |w n,xx | 2 dx -φ(ξ)|w n,xx (ξ -)| 2 -2φ x (ξ)Re(w n,x (ξ)w n,xx (ξ -)) + o(1). (64)
Consequently, gathering together (62), ( 63) and ( 64), we obtain

2ReI ξ 2 =3 ξ 0 φ x |w n,xx | 2 dx -φ(ξ)|w n,xx (ξ -)| 2 -2φ x (ξ)Re(w n,x (ξ)w n,xx (ξ -)) + o(1), (65)
and similarly we have

2ReI η 2 =3 π η φ x |w n,xx | 2 dx + φ(η)|w n,xx (η + )| 2 + 2φ x (η)Re(w n,x (η)w n,xx (η + )) + o(1). (66)
Taking φ(x) = x and gathering together (55), ( 57), ( 58), ( 60) and (65), we obtain

3 ξ 0 |w n,xx | 2 dx + β 2 n ξ 0 [|w n | 2 -α|w n,x | 2 ]dx -2Re(w n,x (ξ)w n,xx (ξ -)) -ξ(|w n,xx (ξ -)| 2 + |β n w n (ξ)| 2 + α|β n w n,x (ξ)| 2 ) =β n w n,x (ξ)o(1) + o(1), (67)
and similarly, gathering together (56), ( 57), ( 59), ( 61) and (66), we obtain

3 π η |w n,xx | 2 dx + β 2 n π η [|w n | 2 -α|w n,x | 2 ]dx + 2Re(w n,x (η)w n,xx (η + )) + η(|w n,xx (η + )| 2 + |β n w n (η)| 2 + α|β n w n,x (η)| 2 ) =πα|β n w n,x (π)| 2 + β n (w n,x (π) + w n,x (η))o(1) + o(1), (68)
Focus on (68), since w n,x (η) = o(1), we have

2Re(w n,x (η)w n,xx (η + )) = w n,xx (η + )o(1).
Using Cauchy inequality, we deduce

|w n,xx (η + )| 2 = O(|β n w n,x (π)| 2 + 1).
Taking difference of ( 68) and (67), we obtain

3 π η - ξ 0 |w n,xx | 2 dx + β 2 n π η - ξ 0 [|w n | 2 -α|w n,x | 2 ]dx + η(|w n,xx (η + )| 2 + |β n w n (η)| 2 + α|β n w n,x (η)| 2 ) + ξ(|w n,xx (ξ -)| 2 + |β n w n (ξ)| 2 + α|β n w n,x (ξ)| 2 ) =πα|β n w n,x (π)| 2 -2Re[w n,x (η)w n,xx (η + ) + w n,x (ξ)w n,xx (ξ -)] + β n (w n,x (π) + w n,x (η))o(1) + β n w n,x (ξ)o(1) + o(1).
Using w n,x (ξ) = o(1), w n,x (η) = o(1) and Cauchy inequality, we obtain

|w n,xx (ξ -)| 2 = O(|β n w n,x (π)| 2 + 1).
This concludes the proof of Proposition 4.4.

Thanks to Proposition 4.4 and ( 49), finally we arrive to

3 π 0 |w n,xx | 2 dx + β 2 n π 0 [|w n | 2 -α|w n,x | 2 ]dx = O(|β n w n,x (π)| 2 ) + o(1),
which is equivalent to (29). This concludes the proof of Lemma 4.1.

Proof of Lemma 4.3.

To simplify the notations we drop the subscript n in the proof of Lemma 4.3. In order to help the reading of the proof of (30) we split it in three steps as follows.

Step 1. We compute the expression (80) of w x (π) by solving a 4 × 4 linear system and denote w x (π) := H 1 /H 2 ; Step 2. We estimate the upper bound of |H 1 |;

Step 3. We estimate the lower bound of |H 2 |.

Let us now proceed with the formal proof of (30).

Step 1. (Computation of the expression (80) of w x (π)): We compute w x (π) by solving system (40). Setting F = g + iβf , we solve the system

           -β 2 w + R{w xxx } x = F in (0, π), w(0) = w(π) = w xx (0) = w xx (π) = 0, [w] ξ = [w] η = [w x ] ξ = [w x ] η = 0, [w xx ] η = -[w xx ] ξ = κ(iβ(w x (η) -w x (ξ)) -(f x (η) -f x (ξ))).
We denote F = (I -α∂ xx ) F . Here we firstly assume (w, v) belongs to D(A) ∩ {(w, v)|w ∈ H 4 (0, ξ) ∩ H 4 (ξ, η) ∩ H 4 (η, π)} and then obtain the upper bound (30) of w x (π). Then the density argument shows that the same bound holds for (w, v) in D(A). Since w belongs to [START_REF] Ammari | Decay rates for a beam with pointwise force and moment feedback[END_REF]). Then we need to solve

V ∩ H 4 (0, ξ) ∩ H 4 (ξ, η) ∩ H 4 (η, π), we have [w xxx ] ξ = [w xxx ] η = 0 (see
           {w xxx } x + αβ 2 w xx -β 2 w = F in (0, π), w(0) = w(π) = w xx (0) = w xx (π) = 0, [w] ξ = [w] η = [w x ] ξ = [w x ] η = [w xxx ] ξ = [w xxx ] η = 0, [w xx ] η = -[w xx ] ξ = κ(iβ(w x (η) -w x (ξ)) -(f x (η) -f x (ξ))).
Let λ, µ, σ and γ be given numbers. We first solve the following Cauchy problem in (0, ξ)

w xxxx + αβ 2 w xx -β 2 w = F in (0, ξ), w(0) = 0, w x (0) = λ, w xx (0) = 0, w xxx (0) = µ, (69)
and then solve the following (backwards) Cauchy problem in (η, π)

w xxxx + αβ 2 w xx -β 2 w = F in (η, π), w(π) = 0, w x (π) = σ, w xx (π) = 0, w xxx (π) = γ, (70)
and finally solve the following Cauchy problem in (ξ, η)

     w xxxx + αβ 2 w xx -β 2 w = F in (ξ, η), w(ξ + ) = w(ξ -), w x (ξ + ) = w x (ξ -), w xxx (ξ + ) = w xxx (ξ -), w xx (ξ + ) = w(ξ -) -κ(iβ(w x (η + ) -w x (ξ -)) -(f x (η) -f x (ξ))). (71)
Then λ, µ, σ and γ have to be chosen such that

w(η + ) = w(η -), w x (η + ) = w x (η -), w xxx (η + ) = w xxx (η -), w xx (η + ) = w(η -) + κ(iβ(w x (η + ) -w x (ξ -)) -(f x (η) -f x (ξ))). (72)
The ODE w xxxx + αβ 2 w xx -β 2 w = F can be written as

d dx     w w x w xx w xxx     = M     w w x w xx w xxx     +     0 0 0 F     , with M =     0 1 0 0 0 0 1 0 0 0 0 1 β 2 0 -αβ 2 0     ,
where β > 0 is large enough. Solving the eigenfunction y 4 + αβ 2 y 2 -β 2 = 0, we have y 1,2 = ±a and y 3,4 = ±ib, where

a 2 := -αβ 2 + β α 2 β 2 + 4 2 , -b 2 := - αβ 2 + β α 2 β 2 + 4 2 , (73)
with a, b > 0. With the notation of a and b, we obtain that the solution matrix e xM corresponding to M has the form

e xM = 1 a 2 + b 2     u ′ 1 (x) u 1 (x) u ′ 2 (x) u 2 (x) u ′′ 1 (x) u ′ 1 (x) u ′′ 2 (x) u ′ 2 (x) u ′′′ 1 (x) u ′′ 1 (x) u ′′′ 2 (x) u ′′ 2 (x) u ′′′′ 1 (x) u ′′′ 1 (x) u ′′′′ 2 (x) u ′′′ 2 (x)     , (74)
where

u 1 (x) = b 2 a sinh(ax) + a 2 b sin(bx), u 2 (x) = 1 a sinh(ax) - 1 b sin(bx).
Therefore, solution of (69) is

    w w x w xx w xxx     (x) = e xM     0 λ 0 µ     + x 0 e (x-y)M F (y)dy     0 0 0 1     , x ∈ (0, ξ),
and we obtain

    w w x w xx w xxx     (ξ -) = e ξM     0 λ 0 µ     + ξ 0 e (ξ-y)M F (y)dy     0 0 0 1     . (75)
Similarly, solving (70) backward, we have

    w w x w xx w xxx     (η + ) = e (η-π)M     0 σ 0 γ     + η π e (η-y)M F (y)dy     0 0 0 1     . (76)
Denoting

G := κ(iβ(w x (η + ) -w x (ξ -)) -(f x (η) -f x (ξ))
) and solving (71) forward, we have

    w w x w xx w xxx     (η -) =e (η-ξ)M     w(ξ -) w x (ξ -) w xx (ξ -) -G w(ξ -)     + η ξ e (η-y)M F (y)dy     0 0 0 1     =e ηM     0 λ 0 µ     -Ge (η-ξ)M     0 0 1 0     + η 0 e (η-y)M F (y)dy     0 0 0 1     . (77)
Then thanks to (75), ( 76) and ( 77), ( 72) is equivalent to

e (η-π)M     0 σ 0 γ     + η π e (η-y)M F (y)dy     0 0 0 1     =e ηM     0 λ 0 µ     + G(I -e (η-ξ)M )     0 0 1 0     + η 0 e (η-y)M F (y)dy     0 0 0 1     , namely     0 σ 0 γ     = e πM     0 λ 0 µ     + G(e (π-η)M -e (π-ξ)M )    0 0 1 0     + π 0 e (π-y)M F (y)dy     0 0 0 1     . ( 78
)
Next we reorganize the linear system (78). First we introduce some notions

a := -     u 1 (π) u ′ 1 (π) u ′′ 1 (π) u ′′′ 1 (π)     , b :=     u 2 (π) u ′ 2 (π) u ′′ 2 (π) u ′′′ 2 (π)     , c :=     0 a 2 + b 2 0 0     , d :=     0 0 0 a 2 + b 2     e :=     u ′ 2 (π -η) -u ′ 2 (π -ξ) u ′′ 2 (π -η) -u ′′ 2 (π -ξ) u ′′′ 2 (π -η) -u ′′′ 2 (π -ξ) u ′′′′ 2 (π -η) -u ′′′′ 2 (π -ξ)     , j := π 0     u 2 (π -y) u ′ 2 (π -y) u ′′ 2 (π -y) u ′′′ 2 (π -y)     F (y)dy, ∆ = iβ a 2 + b 2 ξ 0 u ′ 2 (ξ -y)F (y)dy + π η u ′ 2 (η -y)F (y)dy + f x (η) -f x (ξ),
and constant S = iκβ(a 2 + b 2 ) -1 . Then substituting solution matrix (74) and G into (78), we have

a + Su ′ 1 (ξ)e b + Su ′ 2 (ξ)e c -Su ′ 1 (π -η)e d -Su ′ 2 (π -η)e     λ µ σ γ     = j -κ∆e. (79)
According to the solution of linear system and recalling w x (π) = σ, we have

w x (π) = det a + Su ′ 1 (ξ)e b + Su ′ 2 (ξ)e j -κ∆e d -Su ′ 2 (π -η)e det a + Su ′ 1 (ξ)e b + Su ′ 2 (ξ)e c -Su ′ 1 (π -η)e d -Su ′ 2 (π -η)e , (
a 2 + b 2 = β α 2 β 2 + 4 ∼ αβ 2 , b 2 ∼ αβ 2 , a 2 = αβ 2 2 1 + 4 α 2 β 2 -1 ∼ αβ 2 2 1 2 4 α 2 β 2 = 1 α . (81)
Step

|H 15 | ≤ C(a 2 + b 2 )|f x (η) -f x (ξ)| ≤ C(a 2 + b 2 )∥f ∥ V ≤ Cβ 2 ∥f ∥ V . (83)
Since F = (I -α∂ xx )(g + iβf ), we deduce by integrating by parts that + iκβα(P 4 (π)g ′ (π) -P 4 (η)g ′ (η)). We first deal with trace terms. Expressions of P 2 , P 3 and P 4 show that P 4 (π) -P 2 (π) = 0, P 2 (0) -P 3 (0) = 0, P 3 (ξ) = 0, P 4 (η) = 0, therefore, trace terms in H 12 + H 13 + H 14 vanish. Then we obtain

|H 12 + H 13 + H 14 | ≤Cβ(∥P 2 ∥ H + ∥P 3 ∥ H + ∥P 4 ∥ H )∥g∥ H + Cβ 2 (∥P 2 ∥ L 2 (0,π) + ∥P 3 ∥ L 2 (0,π) + ∥P 4 ∥ L 2 (0,π) )∥f ∥ V .
Noticing that

∥P j ∥ L 2 (0,π) ≤ C, ∥P j ∥ H ≤ Cb ≤ Cβ, j = 2,
H 2 = det a b c d + det Su ′ 1 (ξ)e b c d + det a Su ′ 2 (ξ)e c d + det a b -Su ′ 1 (π -η)e d + det a b c -Su ′ 2 (π -η)e = -(a 2 + b 2 ) 2 (a 2 + b 2 ) 2 ab sinh(aπ) sin(bπ) + iκβ(a 2 + b 2 ) sin(bπ) 2b (2 cosh(aπ) + cosh a(π -2ξ) + cosh a(π -2η) -2 cosh a(π -η + ξ) -2 cosh a(π -η -ξ)) + sinh(aπ) 2a (2 cos b(π -η + ξ) + 2 cos b(π -η -ξ) -2 cos(bπ) -cos b(π -2ξ) -cos b(π -2η))]} = : -(a 2 + b 2 ) 2 H ′ 2 . ( 87
)
Then we estimate the lower bound of |H

′ 2 |. Denote h(b) =2 cos b(π -η + ξ) + 2 cos b(π -η -ξ) -2 cos(bπ) -cos b(π -2ξ) -cos b(π -2η) and l(a) =2 cosh a(π -η + ξ) + 2 cosh a(π -η -ξ) -2 cosh(aπ) -cosh a(π -2ξ) -cosh a(π -2η).
With the asymptotic property of a and b, namely (81), simple calculation shows that there exist C 1 , C 2 > 0 such that for any fixed 0 < ξ < η < π, we have C 1 ≤ -l(a) ≤ C 2 , for β large enough. Then we obtain

|H ′ 2 | 2 =(a 2 + b 2 ) 2 a 2 + b 2 ab 2 sinh(aπ) 2 sin(bπ) 2 +κ 2 β 2 sinh(aπ) 2a h(b) - sin(bπ) 2b l(a) 2 ≥C ′ β 4 (β 2 sin(bπ) 2 + β 2 (h(b)) 2 -sin(bπ) 2 ) ≥Cβ 6 (sin(bπ) 2 + (h(b)) 2 )
for β large enough. It follows that

|H ′ 2 | ≥ Cβ 3 sin(bπ) 2 + (h(b)) 2 . (88)
Therefore, gathering together (80), (82), (86), (87) and (88), we finally obtain that

|βw x (π)| ≤ C ∥g∥ H + ∥f ∥ V sin(bπ) 2 + (h(b)) 2 . ( 89
)
This concludes the proof of Lemma 4.3.

5.

The lack of exponential stability and the lack of polynomial stability with high decay rate (Proof of Theorem 1.6). In this section, we prove two results in Theorem 1.6 together by providing a counter example. The exponential stability of ( 5) is equivalent to iR is included in ρ(A) and the resolvent estimate

sup β≥1 ∥(iβ -A) -1 ∥ < ∞ (90)
holds due to a well-known result (see [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF][START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF]). Combined with the discussion in the previous section, our aim is to prove

sup β≥1 |β| -l ∥(iβ -A) -1 ∥ = ∞ (91)
for 0 ≤ l < 1 2 . First we give the inspiration of Theorem 1.6. Recalling Remark 4.2, we notice that actually we can deduce (28) and (29) from ( 24) and ( 25) with l = 0. Therefore, if we try to prove the exponential stability in the same way as in the proof of Theorem 1.5, we are necessary to claim

inf b≥1 sin(bπ) 2 + (h(b)) 2 ≥ C > 0.
However, this claim is false. In fact

inf b≥1 sin(bπ) 2 + (h(b)) 2 ≤ inf b∈N * sin(bπ) 2 + (h(b)) 2 = inf b∈N * |h(b)| =8 inf b∈N * sin b η + ξ 2 sin b η -ξ 2 2 . Since η±ξ 2π are irrational numbers, inf b≥1 sin(bπ) 2 + (h(b)) 2 ≤ 8 inf b∈N * sin b η + ξ 2 sin b η -ξ 2 2 = 0.
These facts inspire us to prove the exponential stability never holds no matter where ξ and η are located, as precisely stated in Theorem 1.6. Now let us turn to prove Theorem 1.6. Our aim is to find

β n in R, (w n , v n ) in D(A) for n = 1, 2, . . . such that ∥(w n , v n )∥ H = 1, β n → ∞ (92) and |β| l (f n , g n ) → (0, 0) in H, (93)
for 0 ≤ l < 1 2 where (f n , g n ) := (iβ n -A)(w n , v n ). If such β n and (w n , v n ) can be found, then (91) holds which completes the proof of Theorem 1.6.

To find such β n and (w n , v n ), we need to use Proposition 2.3 on simultaneous approximation. If η±ξ 2π are rational numbers, then the system is not strongly stable. Now assume η±ξ 2π are irrational numbers. Applying Proposition 2.3 we obtain the existence of a strictly increasing sequence of positive integers {q n } n≥1 such that

sin q n η + ξ 2 ≤ π √ q n , sin q n η -ξ 2 ≤ π √ q n ∀n ≥ 1. ( 94
)
As we mentioned in Remark 4.2, β n , w n and v n should satisfy some necessary conditions of (92) and (93),

|v n,x (η) -v n,x (ξ)| 2 → 0 as n → ∞, and lim n→∞ ∥w n ∥ 2 V = lim n→∞ ∥v n ∥ 2 H = lim n→∞ ∥β n w n ∥ 2 H = 1 2 ,
i.e., (36) and (28). We set

β n = q 2 n 1 + αq 2 n → ∞ as n → ∞, (95)
and

v n (x) = 1 √ πd n sin(q n x), (96)
where {d n } n≥1 is a sequence that will be chosen later. Since q n is integer, we have v n belongs to V and

∥v n ∥ 2 H = 1 + αq 2 n 2d 2 n .
Therefore, it is necessary that d n must satisfy

lim n→∞ d n = +∞, lim n→∞ 1 + αq 2 n d 2 n = 1. (97)
Using (94), ( 96) and (97), we obtain that

|v n,x (η) -v n,x (ξ)| = q n √ πd n sin q n η + ξ 2 sin q n η -ξ 2 → 0. ( 98
)
Then we need to define w n . Notice that v n belongs to C ∞ [0, π] and

[w n,xx ] η = -[w n,xx ] ξ =κ(v n,x (η) -v n,x (ξ)) = - κq n √ πd n sin q n η + ξ 2 sin q n η -ξ 2 ,
therefore, from the definition of f n we have

[f n,xx ] η = -[f n,xx ] ξ = iL n ,
where

L n = κβ n (v n,x (η) -v n,x (ξ)) = - κq n β n √ πd n sin q n η + ξ 2 sin q n η -ξ 2 . (99)
Using ( 94), ( 95) and (97), we have

|L n | ≤ κπ 3 2 β n d n ≤ L ∀n ≥ 1, (100)
where L > 0 is a constant. Assume

c n = C q m n , (103)
where 2l < m < 2 -2l and C > 0 is a constant such that c 1 < η-ξ 2 . Note that 0 ≤ l < 1 2 implies the existence of m. And define

ϕ n =                      0, x ∈ (0, ξ), L n c n (x -ξ), x ∈ (ξ, ξ + c n ), L n , x ∈ (ξ + c n , η -c n ), - L n c n (x -η), x ∈ (η -c n , η), 0, x ∈ (η, π), (104)
Then we have Next we calculate ∥w n ∥ V . Since (f n , g n ) := (iβ n -A)(w n , v n ) and the definition of A, i.e. [START_REF] Guo | Dynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observation[END_REF], we obtain that

w n = 1 iβ n (f n + v n ) = 1 β n (F n -iv n ),
which implies that

w n,xx = 1 β n (F n,xx -iv n,xx ) = 1 β n (ϕ n -L n I (ξ,η) -iv n,xx ).
Then we have

∥w n ∥ 2 V = ∥w n,xx ∥ 2 L 2 (0,π) = 1 β 2 n (∥ϕ n -L n I (ξ,η) ∥ 2 L 2 (0,π) + ∥v n,xx ∥ 2 L 2 (0,π) ) = 1 β 2 n 2 3 L 2 n c n + q 4 n 2d 2 n = 1 + αq 2 n 2d 2 n + 2L 2 n c n 3β 2 n → 1 2 . ( 106
)
According to (92), we also need ∥w n ∥ 2 V + ∥v n ∥ 2 H = 1, which implies that

d 2 n = 1 + αq 2 n 1 -(2L 2 n c n )/(3β 2 n ) . (107)
The We know from [START_REF] Guo | On the dynamic behavior and stability of controlled connected Rayleigh beams under pointwise output feedback[END_REF] that for any ψ in L 2 (0, π),

Rψ =c sinh x √ α - 1 √ α x 0 sinh x -s √ α ψ(s)ds, c = √ α sinh π √ α -1 π 0 sinh π -s √ α ψ(s)ds.
Then we can calculate Rt n and Rs n as

t n = - 1 β n v n,xxx = q 3 n √ πβ n d n cos(q n x), Rt n = q n d n π(1 + αq 2 n ) cos(q n x) -cosh x √ α + sinh π √ α -1 cosh π √ α -cos(q n π) sinh x √ α , s n = 1 β n ϕ n,x =
L n β n c n (I (ξ,ξ+cn) -I (η-cn,η) ),

Rs n =                            0, x ∈ (0, ξ), L n β n c n 1 -cosh x -ξ √ α , x ∈ (ξ, ξ + c n ), L n β n c n 1 -cosh c n √ α , x ∈ (ξ + c n , η -c n ), L n β n c n cosh x -η + c n √ α -cosh c n √ α ,
x ∈ (η -c n , η),

0,

x ∈ (η, π).

Then we calculate the single term in G n . We have Consequently, we obtain

G n = L 2 n αβ 2 n c n 1 -cosh c n √ α + 2L 2 n √ αβ 2 n c 2 n sinh c n √ α + 2q 2 n π √ αd 2 n (1 + αq 2 n ) sinh π √ α -1 cos(q n π) -cosh π √ α . (109)
On the one hand, l < 1 2 and (97) imply that

2q 2 n β 2l n π √ αd 2 n (1 + αq 2 n ) sinh π √ α -1 cos q n π -cosh π √ α → 0 as n → ∞.
On the other hand, since c n → 0 as n → ∞, using Taylor expansion, we obtain

1 -cosh c n √ α = - c 2 n 2α + O(c 4 n ), sinh c n √ α = c n √ α + O(c 3 n ).

Lemma 4 . 1 .

 41 Let β n > 0 and (w n , v n ) in D(A) for n ≥ 1 satisfy (24) and[START_REF] Weiss | Exponential stabilization of a Rayleigh beam using collocated control[END_REF]

Lemma 4 . 3 .

 43 Let β n and (w n , v n ) be the same as in Lemma 4.1. Then there exist a constant C > 0 such that

Proposition 4 . 4 .

 44 Let β n and (w n , v n ) be the same as in Lemma 4.1, then we have|w n,xx (η + )| 2 = O(|β n w n,x (π)| 2 + 1), |w n,xx (ξ -)| 2 = O(|β n w n,x (π)| 2 + 1). Since we do not know the asymptotic behavior of |β n w n,x (π)| 2 right now, we add 1 into O to specify that |w n,xx (η + )| 2 and |w n,xx (ξ -)| 2 are only bounded terms if β n w n,x (π) → 0. Proof of Proposition 4.4. Since (40) holds in H, we obtain by differentiating both sides of (40)

1 ,H 1 ==(a 2 + b 2 ) 2 π 0 [P 1 0 P 3 (P 4

 112201034 2. (Estimation of the upper bound of |H 1 |): We calculate the expression of H det a b j d + det Su ′ 1 (ξ)e b j d + det a Su ′ 2 (ξ)e j d + det a b -κ∆e d + det a b j -Su ′ 2 (π -η)e (y) + iκβP 2 (y)]F (y)dy -iκβ ξ (y)F (y)dy + H 15 = : (a 2 + b 2 )2 (H 11 + H 12 + H 13 + H 14 + H 15 ),(82)whereP 1 (y) = (a 2 + b 2 )ab (sinh(aπ) sin(by) -sin(bπ) sinh(ay)),P 2 (y) = sin(by) 2b [-2 cosh a(π -η + ξ) -2 cosh a(π -η -ξ) + 2 cosh(aπ) + cosh a(π -2ξ) + cosh a(π -2η)] + cosh(aη) -cosh(aξ) 2b [sin b(π -η -y) -sin b(π -η + y) -sin b(π -ξ -y) -sin b(π + ξ -y)] + sin(bπ) 2b [cosh a(η + ξ -y) + cosh a(η -ξ -y) -cosh a(2ξ -y) -cosh(ay)] + sinh(ay) 2a [-2 cos b(π -η + ξ) -2 cos b(π -η -ξ) + 2 cos(bπ) + cos b(π -2ξ) + cos b(π -2η)] + cos(bη) -cos(bξ) 2a [sinh a(π -η -y) -sinh a(π -η + y) -sinh a(π -ξ -y) -sinh a(π + ξ -y)] + sinh(aπ) 2a [cos b(η + ξ -y) + cos b(η -ξ -y) -cos b(2ξ -y) -cos(by)], P 3 (y) = -cosh(aη) -cosh(aξ) 2b [sin b(π + ξ -y) + sin b(π -ξ + y)] + sin(bπ) 2b [cosh a(η + ξ -y) + cosh a(η -ξ + y) -cosh a(2ξ -y) -cosh(ay)] -cos(bη) -cos(bξ) 2a [sinh a(π + ξ -y) + sinh a(π -ξ + y)] + sinh(aπ) 2a [cos b(η + ξ -y) + cos b(η -ξ + y) -cos b(2ξ -y) -cos(by)], P 4 (y) = -cosh(aη) -cosh(aξ) 2b [sin b(π + η -y) + sin b(π -η + y)] + sin(bπ) 2b[cosh a(2η -y) + cosh(ay)-cosh a(η + ξ -y) -cosh a(η -ξ -y)] -cos(bη) -cos(bξ) 2a [sinh a(π + η -y) + sinh a(π -η + y)] + sinh(aπ) 2a [cos b(2η -y) + cos(by) -cos b(η + ξ -y) -cos b(η -ξ -y)],andH15 =κ(a 2 + b 2 ) 1 a (cos(bη) -cos(bξ)) sinh(aπ) -1 b (cosh(aη) -cosh(aξ)) sin(bπ) (f x (η) -f x (ξ)).

H 11 = π 0 P 1 2 + b 2 ab π 0 [ 0 ( 0 [P 2 0 P 2 H 13 = -iκβ ξ 0 P 3 -iκβ ξ 0 [P 3 0 P 3

 01200020213030303 (y)F (y)dy = a 2 + b 2 ab π 0 (sinh(aπ) sin(by) -sin(bπ) sinh(ay))F (y)dy = a (sinh(aπ) sin(by) -sin(bπ) sinh(ay))g(y) + α(b sinh(aπ) cos(by) -a sin(bπ) cosh(ay))g ′ (y)]dy +iβ π sinh(aπ) sin(by) -sin(bπ) sinh(ay))(f (y) -αf ′′ (y))dy , hence |H 11 | ≤Cβ∥ sinh(aπ) sin(b•)-sin(bπ) sinh(a•)∥ H ∥g∥ H + Cβ 2 ∥ sinh(aπ) sin(b•) -sin(bπ) sinh(a•)∥ L 2 (0,π) ∥f ∥ V .Noticing that∥ sinh(aπ) sin(b•) -sin(bπ) sinh(a•)∥ L 2 (0,π) ≤C, ∥ sinh(aπ) sin(b•) -sin(bπ) sinh(a•)∥ H ≤Cb ≤ Cβ, we have |H 11 | ≤ Cβ 2 (∥g∥ H + ∥f ∥ V ).(84)Next, we estimate H 12 + H 13 + H 14 . (y)g(y) + αP ′ 2 (y)g ′ (y)]dy -κβ 2 π (y)(f (y) -αf ′′ (y))dy-iκβα(P 2 (π)g ′ (π) -P 2 (0)g ′ (0)),(y)F (y)dy = (y)g(y) + αP ′ 3 (y)g ′ (y)]dy + κβ 2 ξ (y)(f (y) -αf ′′ (y))dy + iκβα(P 3 (ξ)g ′ (ξ) -P 3 (0)g ′ (0)), H 14 = -iκβ π η P 4 (y)F (y)dy = -iκβ π η [P 4 (y)g(y) + αP ′ 4 (y)g ′ (y)]dy + κβ 2 π η P 4 (y)(f (y) -αf ′′ (y))dy

  f n = iF n , F n is a real function, therefore, we have [F n,xx ] η = -[F n,xx ] ξ = L n . As required, w n belongs to V ∩ H 3 (0, ξ) ∩ H 3 (ξ, η) ∩ H 3 (η, π) and w n,xx (0) = w n,xx (π) = 0, therefore, F n belongs to V ∩ H 3 (0, ξ) ∩ H 3 (ξ, η) ∩ H 3 (η, π) and F n,xx (0) = F n,xx (π) = 0. Then we have F n,xxx = {F n,xxx } + [F n,xx ] η δ η + [F n,xx ] ξ δ ξ = {F n,xxx } + L n (δ η -δ ξ ).(101)Therefore, we denoteF n,xx := ϕ n -L n I (ξ,η) ,(102)where ϕ n belongs to H, ϕ n,x = {F n,xxx } and I (ξ,η) is the characteristic function of (ξ, η). Since (93), we require|β n | l ∥F n ∥ V → 0, i.e., |β n | l ∥F n,xx ∥ L 2 (0,π) → 0, namely |β n | l ∥ϕ n -L n I (ξ,η) ∥ L 2 (0,π) → 0.Here we use piecewise linear function to approximate L n I (ξ,η) . Let

  ∥ϕ n -L n I (ξ,η) ∥ 2 L 2 (0,π) = 2 ξ+cn ξ L n c n (x -ξ) -L n 2 dx = 2 3 L 2 n c n .(105)Since |L n | ≤ L, q n → ∞, m > 2l and (103), we have |β n | l ∥ϕ n -L n I (ξ,η) ∥ L 2 (0,π) → 0.

β 2 n ∥v n ∥ 2 H = β 2 n 1

 221

  80) and denote by H 1 the numerator of w x (π) and by H 2 the denominator of w x (π), i.e., w x (π) = H 1 /H 2 . Our aim is to estimate the upper bound of |βw x (π)| as β → ∞. Therefore, we need to estimate the upper bound of |H 1 | and the lower bound of |H 2 |. Before we estimate |H 1 | and |H 2 |, we claim some property of a and b. Recalling (73), as β → ∞, we have

  Then we estimate H 1 , namely H 11 , H 12 , H 13 , H 14 and H 15 , as β → ∞. Using (81), we have

  3, 4, we have |H 12 + H 13 + H 14 | ≤ Cβ 2 (∥g∥ H + ∥f ∥ V ). |H 11 + H 12 + H 13 + H 14 + H 15 | ≤ Cβ 2 (∥g∥ H + ∥f ∥ V ). (86) In the estimate of H 11 +H 12 +H 13 +H 14 +H 15 , we involve the value of g ′ at a point. In fact, we assume g is smooth enough at first and then use the density argument to claim the estimate (86) also holds for g in H. Step 3. (Estimation of the lower bound of |H 2 |): We calculate the expression of H 2 ,

	(85)
	Gathering together (83), (84) and (85), we obtain

  last condition we need to check is|β n | l (iβ n v n + R{w n,xxx } x ) → 0 in H. We denote {w n,xxx } as {w n,xxx } = 1 β n ({F n,xxx } -iv n,xxx ) = 1 β n (ϕ n,x -iv n,xxx ) =: s n + it n . (108) Denote G n := ∥iβ n v n + R{w n,xxx } x ∥ 2H , then by integrating by parts, we have Rtn ) x + β n v n ) 2 + (Rs n,x ) 2 + α((Rt n ) xx + β n v n,x ) 2 + α(Rs n,xx ) 2 ]dx Rt n,x ) 2 + 2β n v n (Rt n ) x + (β n v n ) 2 + (Rs n,x ) 2 + (Rt n -t n )Rt n,xx + 2(Rt n -t n )β n v n,x + α(β n v n,x ) 2 + (Rs n -s n )Rs n,xx ]dx [(β n v n ) 2 -t n Rt n,xx -2β n t n v n,x + α(β n v n,x ) 2 -s n Rs n,xx ]dx Rt n -s n Rs n ) -2β n t n v n,x dx.

			π
	G n = [((= 0 π 0 [(= π
		0
	=	0	π	β 2 n (v 2 n + αv 2 n,x ) +	1 α	(t 2 n + s 2 n -t n

Acknowledgments. Yubo Bai would like to acknowledge the funding from the China Scholarship Council (No. 202206100101). Zhiqiang Wang is partially supported by the Natural Sciences Foundation of China, under grant 11971119.

Then we have

Now we have already proven β n → ∞ and (w n , v n ) in D(A) such that (92) and (93) hold. Then we have proved that for any 0 < ξ < η < π

for 0 ≤ l < 1 2 . This concludes the proof of Theorem 1.6.