Optimal Energy Decay Rate for Rayleigh Beam Equation with Only One Dynamic Boundary Control

In [21], Wehbe considered the Rayleigh beam equation with two dynamical boundary controls and established the optimal polynomial energy decay rate of type 1 t . The proof exploits in an explicit way the presence of two boundary controls, hence the case of the Rayleigh beam damped by only one dynamical boundary control remained open. In this paper, we fill this gap by considering a clamped Rayleigh beam equation subject to only one dynamical boundary feedback. First, we consider the Rayleigh beam equation subject to only one dynamical boundary control moment. In that case, we prove a polynomial decay in 1 t of the energy by using an observability inequality. For that purpose, we give the asymptotic expansion of eigenvalues and eigenfunctions of the undamped underling system. Moreover, using the real part of the asymptotic expansion of eigenvalues of the damped system, we prove that the obtained energy decay rate is optimal. Next, we consider the Rayleigh beam equation subject to only one dynamical boundary control force. Here we use a Riesz basis approach. As before, we start by giving the asymptotic expansion of the eigenvalues and the eigenfunctions of the damped and undamped systems. We next show that the system of eigenvectors of the damped problem form a Riesz basis. Finally, we deduce the optimal energy decay rate of polynomial type in 1


Introduction
In [21], Wehbe considered a Rayleigh beam clamped at one end and subjected to two dynamical boundary controls at the other end, namely y(0, t) = y x (0, t) = 0, t > 0, (1.2) where γ > 0 is the coefficient of moment of inertia, a > 0 and b > 0 are positive constants, η and ξ denote respectively the dynamical boundary control moment and force.The damping of the system is made via the indirect damping mechanism at the right extremity of the beam that involves the following two first order differential equations: ξ t (t) − y t (1, t) + βξ(t) = 0, t > 0, (1.6) where α > 0 and β > 0. The notion of indirect damping mechanisms has been introduced by Russell in [18] and since that time, it retains the attention of many authors.In [21], Wehbe considered the Rayleigh beam equation with two dynamical boundary controls moment and force, i.e., under the conditions a > 0 and b > 0.
The lack of uniform stability was proved by a compact perturbation argument of Gibson and a polynomial energy decay rate of type 1 t is obtained by a multiplier method usually used for nonlinear problems.Finally, using a spectral method, he proved that the obtained energy decay is optimal in the sense that for any ε > 0, we cannot expect a decay rate of type 1 t 1+ε .But in [21] the effect of each control separately on the stability of the Rayleigh beam equation is not investigated.Indeed, the multiplier method exploits in an explicit way the presence of the two boundary controls.Furthermore, the lack of one of this two controls yield this method ineffective.Then, the important and interesting case when the Rayleigh beam equation is damped by only one dynamical boundary control (a = 0 and b > 0 or a > 0 and b = 0) remained open.The aim of this paper is to fill this gap by considering a clamped Rayleigh beam equation subject to only one dynamical boundary feedback.
Optimal Energy Decay Rate of the Rayleigh Beam Equation 133Using an explicit approximation of the characteristic equation, we give the asymptotic behavior of eigenvalues and eigenfunctions of the associated undamped system with the help of Rouché's theorem.Then to prove the polynomial energy decay, we apply a methodology introduced in [2].This requires, on one hand, to establish an observability inequality of solution of the undamped system and on the other hand, to verify the boundedness property of the transfer function.This attend to establish a polynomial energy decay rate of type 1 t for smooth initial data.Finally, using a frequency domain approach, we prove that the obtained energy decay rate is optimal in the sense that for any ε > 0, we cannot expect a decay rate of type 1 t 1+ε .
Next, we consider the Rayleigh beam equation (1.1)-(1.4)with only one dynamical boundary control force, i.e., when a = 0, b = 1 and ξ solution of (1.6).Here we prefer to use a Riesz basis approach.First, as before we give the asymptotic expansion of the eigenvalues and the eigenfunctions of the damped and undamped systems.Next, we show that the system of eigenvectors of high frequencies of the damped problem is quadratically closed to the system of eigenvectors of high frequencies of the undamped problem.This yields, from [9,Theorem 6.3] and [1, Theorem 1.2.10] that the system of generalized eigenvectors of the damped problem forms a Riesz basis of the energy space.Finally, using [14, Theorem 2.1]) we establish the optimal energy decay rate of polynomial type 1 √ t .
The stabilization of the Rayleigh beam equation retains the attention of many authors.In this regard, different types of dampings have been introduced to the Rayleigh beam equation and several uniform and polynomial stability results have been obtained.Rao [16] studied the stabilization of Rayleigh beam equation subject to a positive internal viscous damping.Using a constructive approximation, he established the optimal exponential energy decay rate.In [12], Lagnese studied the stabilization of system (1.1)- (1.4) with two static boundary controls (the case a > 0, b > 0, η(t) = y xt (1, t) and ξ(t) = y t (1, t)).He proved that the energy decays exponentially to zero for all initial data.Rao in [16] extended the results of [12] to the case of one boundary feedback (the case a > 0, b = 0 and η(t) = y xt (1, t) or a = 0, b > 0 and ξ(t) = y t (1, t)).In the case of one control moment (the case a > 0, b = 0 and η(t) = y xt (1, t)), using a compact perturbation theory due to Gibson [8], he established an exponential stability of system (1.1)- (1.4).Moreover, in the case of one control force (a = 0, b > 0 and ξ(t) = y t (1, t)), he first proved the lack of exponential stability of the system (1.1)- (1.4).Next, he proved that the Rayleigh beam equation can be strongly stabilized by only one control force if and only if the inertia coefficient γ is large enough but he did not studied the decay rate of the energy of the system.In [3], Bassam and al. studied the decay rate of energy of system (1.1)-(1.4)with a = 0, b > 0 and ξ(t) = y t (1, t).First, using an explicit approximation, they gave the asymptotic expansion of eigenvalues and eigenfunctions of the undamped system corresponding to (1.1)-(1.4),then they established the optimal polynomial energy decay rate via an observability inequality of solution of the undamped system and the boundedness of the transfer function associated with the undamped problem.
Let us briefly outline the content of this paper.Section 2 considers the Rayleigh beam equation with only one dynamical boundary control moment and is divided into four subsections.In subsection 2.1, we formulate the system into an evolution equation and we recall the well-posedness property of the problem by the semi-group approach (see [15], [16] and [21]).In subsection 2.2, we propose an explicit approximation of the characteristic equation determining the eigenvalues of the corresponding undamped system.Then, we give an asymptotic expansion of eigenvalues and eigenfunctions of the corresponding operator.In subsection 2.3, we establish a polynomial energy decay rate for smooth initial data.In subsection 2.4, we prove that the obtained energy decay rate is optimal.Section 3 considers the Rayleigh beam equation with only one dynamical boundary control force and is divided into 2 subsections.As before our system can be transformed into an evolution equation and we deduce the well-posedness property of the problem by the semi-group approach.We recall the condition to reach the strong stability of our system (see [16]).In subsection 3.1, we proposed also an explicit approximation of the characteristic equation determining the eigenvalues of the damped and undamped system.Then, we give an asymptotic expansion of eigenvalues and eigenfunctions of the corresponding operators.In subsection 3.2, we show that the system of eigenvectors of the damped problem forms a Riesz basis and we establish the optimal polynomial energy decay rate of type 1 √ t .

Rayleigh beam equation with only one dynamical control moment
In this section, we consider the Rayleigh beam equation with only one dynamical boundary control moment: (2.1) Let y and η be smooth solutions of system (2.1), we define their associated energy by: A direct computation gives Thus the system (2.1) is dissipative in the sense that the energy E(t) is a nonincreasing function of the time variable t.

Well-posedness and strong stability of the problem
In this subsection, we will study the existence, uniqueness and the asymptotic behavior of the solution of system (2.1).We start our study by formulating the problem in an appropriate Hilbert space.We first introduce the following spaces: and the energy space endowed with the usual inner product Identify L 2 (0, 1) with its dual so that we have the following continuous embedding Let y and η be smooth solutions of system (2.1).Then, multiplying the first equation of the system (2.1) by Φ ∈ W and integrating by parts yields Now we define the following linear operators Then, by means of Lax-Milgram theorem (see [6]), we see that A (resp C) is the canonical isomorphism from W into W ′ (resp from V into V ′ ).On the other hand, using the usual trace theorems and Poincaré inequality, we easily check that the operator B is continuous for the corresponding topology.Therefore, using the operators A, B and C and the continuous embedding (2.7), we formulate the variational equation (2.8) as: Assume that Ay + Bη ∈ V ′ , then we obtain: (2.12) Next we introduce the linear unbounded operator A 0 by and the linear bounded operator B as follows Then, denoting u = (y, y t , η) the state of system (2.1) and define A α = A 0 + αB with D(A α ) = D(A 0 ), we can formulate the system (2.1) into a first-order evolution equation It is easy to show that −A 0 is m-dissipative and −B is dissipative in the energy space H. Therefore the operator −A α generates a C 0 -semigroup (e −tAα ) t≥0 of contractions in the energy space H following Hille-Yosida's theorem (see [15]).Hence, we have the following results concerning the existence and uniqueness of the solution of the problem (2.16): Theorem 2.1.For any initial data u 0 ∈ H, the problem (2.16) has a unique weak solution Moreover, we characterize the space D(A 0 ) by the following proposition.
Proposition 2.2.Let u = (y, z, η) ∈ H. Then u ∈ D(A 0 ) if and only if the following conditions hold: (2.17) In particular, the resolvent (I + A 0 ) −1 of −A 0 is compact on the energy space H and the solution of the system (2.1) satisfies (2.18)

✷
Optimal Energy Decay Rate of the Rayleigh Beam Equation 137The proof is same as in Rao [16,Proposition 2.3] (see also Wehbe [21]) so we omit the details here.Now we investigate the strong stability of the problem (2.16) by the following theorem: Theorem 2.3.For any γ > 0, the semigroup of contractions e −tAα is strongly asymptotically stable on the energy space H, i.e. for any u 0 ∈ H, we have

19)
Proof: The proof is same as in Rao [16,Theorem 3.1], it is based on the spectral decomposition theory of Sz-Nagy-Foias [19], Foguel [7] and Benchimol [4].In order to prove (2.19), it is sufficient to show that there is no spectrum in imaginary axis.We omit the details here.✷ Further, since A 0 is skew adjoint operator and B is compact, then using a compact perturbation method of Russel [17], we deduce that the system (2.16) is not uniformly stable (see Rao [16], and Wehbe [21]).

Polynomial Stability for smooth initial data
Our main result in this subsection is the following polynomial-type decay estimate: Theorem 2.4.(Polynomial energy decay rate) Let γ > 0. For all initial U 0 ∈ D(A 0 ), there exists a constant c > 0 independent of U 0 , such that the solution of the problem (2.16) satisfies the following estimate:

✷
For this aim, we need first to analyze the spectrum of the operator A 0 .Next, We will apply a method introduced by Ammari and Tucsnak in [2], where the polynomial stability for the damped problem is reduced to an observability inequality of the corresponding undamped problem (via the spectral analysis), combined with the boundedness property of the transfer function of the associated undamped system.
2.2.1.Spectral analysis of the operator A 0 .Since A 0 is closed with a compact resolvent, its spectrum σ(A 0 ) consists entirely of isolated eigenvalues with finite multiplicities (see [11]).Moreover, as the coefficients of A 0 are real then the eigenvalues appear by conjugate pairs.Further, the eigenvalues of A 0 are on the imaginary axis.
Proof: First, a straightforward computation shows that 0 ∈ σ(A 0 ) and is simple.
Later, assume that there exists λ ∈ σ(A 0 ) such that λ is not simple.As A 0 is a skewadjoint operator we deduce that there correspond at least two independent eigenvectors U 1 = (y 1 , z 1 , η 1 ) and U 2 = (y 2 , z 2 , η 2 ).Then, is also an eigenvector associated with λ with η 3 = 0, hence the contradiction with the first part of the proof.✷ Now, in order to get a better knowledge of the spectrum we compute the characteristic equation.Thus let λ = iµ, µ ∈ R * , be an eigenvalue of A 0 and U = (y, z, η) ∈ D(A 0 ) be an associated eigenfunction.Then we have (2.22) Then, using (2.9)-(2.11)we interpret (2.22) as the following variational equation Equivalently, the function y is determined by the following system: (2.23) We have found that λ = iµ = 0 is an eigenvalue of A 0 if and only if there is a non trivial solution of (2.23).The general solution of the first equation of (2.23) is given by where Here and below, for simplicity we denote r i (µ) by r i .Thus the boundary conditions in (2.23) may be written as the following system: where g i (µ) = r i r 2 i + γµ 2 e ri and h i (µ) = r i (r i + 1) e ri .Consequently (2.23) admits a non-trivial solution if and only if f (µ) := det M (µ) = 0. Finally, we have found that λ = iµ is an eigenvalue of A 0 if and only if µ satisfies the characteristic equation f (µ) = 0. Proposition 2.6.(Spectrum of A 0 ) There exists k 0 ∈ N * , sufficiently large, such that the spectrum σ(A 0 ) of A 0 is given by: where J 0 is a finite set and κ 0 j , µ k ∈ R.Moreover, µ k satisfies the following asymptotic behavior: where

31)
Proof: The proof is decomposed into two steps.
Step 1. First, we start by the expansion of r 1 and r 3 when |µ| → ∞.After some computations we find and This gives and (2.37) Next, using (2.32)-(2.37),we find the asymptotic behavior of and ). (2.41) Similarly, we get

.45)
Optimal Energy Decay Rate of the Rayleigh Beam Equation 141Now, using (2.26) and (2.32)-( 2.45), we can write M (µ) as follows where Again after some computations, we find the following asymptotic development of where (2.47) For convenience we set that has the same root as f , except 0.
Step 2. We look at the roots of S. Is is easy to see that the roots of f 0 are given by: Then, with the help of Rouché's theorem, there exists k 0 ∈ N * large enough, such that for all |k| ≥ k 0 the large roots of S (denoted by µ k ) are close to α k .More precisely, there exists k 0 ∈ N * large enough, such that the splitting of σ(A 0 ) given in (2.27)-(2.28)holds and we have Equivalently we can write which implies Inserting the previous identity in (2.50) we directly get (2.29).✷ Eigenvectors of A 0 .According the decomposition of the spectrum σ(A 0 ) of A 0 , a set of eigenvectors associated with σ(A 0 ) is given as follows: where (2.55) Now, for |k| ≥ k 0 and µ = µ k , we give a solution up to a factor of problem (2.23) and some appropriated asymptotic behavior.
Proposition 2.7.Let |k| ≥ k 0 .Then, a solution y k of the undamped initial value problem (2.23) with µ = µ k satisfies the following estimations: (2.57) 23) amounts to find a solution C(µ k ) = 0 of the system (2.26) of rank three.For clarity, we divide the proof into two steps.
Step 1. Estimate of y k,x (1).For simplicity of notation we write C(µ k ) = (c 1 , c 2 , c 3 , c 4 ).Since we search C(µ k ) up to a factor we choose c 3 = 1, the possibility of this choice will be justify later.Therefore (2.26) becomes Next, using Cramer's rule, we obtain where and First we study the behavior of α 1 .Inserting (2.32) and (2.33) (with µ = µ k ) in (2.59) we find after some computations Now inserting (2.53) in (2.51) and (2.52) we have Inserting (2.29) and (2.64) in (2.63) we find again after some computations Similarly long computations left to the reader yields and (2.68) Remark that α 3 = 0 provided we have chosen k 0 large enough; for this reason our choice c 3 = 1 is valid.Substituting (2.65)-(2.68)into (2.58),we obtain (2.69) Finally we have found that a solution (2.26) has the form: where Note that the corresponding solution y k of (2.23) is given by (2.24).From equation (2.24), we have where we recall that for i = 1, ..., 4, r i = r i (µ k ) are given by (2.25) and c i , i = 1, ..., 4, satisfy (2.69).Therefore using the series expansion (2.29), (2.32), (2.33) and (2.69) we easily find Step 2. Estimates of y k W and y k V .We start with where and First, since r 2 = −r 1 ∈ R (for |k| large enough) and r 3 = −r 4 ∈ iR, we directly find In addition using the identity Optimal Energy Decay Rate of the Rayleigh Beam Equation 145where Similarly, we easily prove that Therefore, we deduce that We will establish an observability inequality for the undamped problem corresponding to (2.80) in following Lemma: Lemma 2.8.Let γ > 0. There exist T > 0 and C T > 0 such that the solution U of the problem satisfies the following observability inequality where (D(A 0 )) ′ is the dual of D(A 0 ) with respect to the scalar product in H.
Proof: Let U 0 ∈ D(A 0 ), then we can write where denotes the set of normalized eigenvectors of A 0 such that and From (2.83) we obtain Consequently, we have , ∀t > 0.
The spectral gap is satisfied by the eigenvalues of A 0 because they are simple and for k large enough, we have µ k+1 − µ k ≥ π 4 √ γ , in other words, there exists d > 0, such that min Thus, using Ingham's inequality (see [10]), we deduce that there exist T > 0 and c T > 0 such that On the other hand, using (2.56)-(2.57)and (2.85) we get (2.88) Optimal Energy Decay Rate of the Rayleigh Beam Equation 147Therefore, we deduce from (2.87), Proposition 2.5 and (2.29) that: The proof of lemma is completed. ✷ Next, we introduce the transfer function H: The transfer function H defined in (2.89) is bounded on C ω .
Proof: First, since A 0 generate a C 0 -semigroup of contractions, we deduce (see Corollary I.3.6 of [15]): Next, combining this estimate with the boundedness of the operators B and B * , we deduce the boundedness of the function H on C ω .✷ Proof of the Theorem 2.4.The polynomial energy estimate (2.20) is obtained by application of Theorem 2.4 in [2] on the first order problem with Y 1 = D(A 0 ),

Optimal polynomial decay rate
The aim of this subsection is to prove the following optimality result.
Theorem 2.10.(Optimal decay rate) The energy decay rate (2.20) is optimal in the sense that for any ǫ > 0, we can not expect the decay rate 1 t 1+ǫ for all initial data U 0 ∈ D(A 0 ).✷ To prove this theorem, we need the asymptotic behavior of the eigenvalues of the operator A α .Let λ = α be an eigenvalue of A α and U = (y, z, η) be an associated eigenfunction, then we obtain A α U = λU .Equivalently, we have the following system: = 0, The general solution of the first equation of (2.90) is given by where Here and below, for simplicity we denote R i (λ) by R i .Thus the boundary conditions in (2.90) may be written as the following system: where we have set Since A α is closed with a compact resolvent, its spectrum consists entirely of isolated eigenvalues with finite multiplicities.Further as the coefficients of A α are real, the eigenvalues appear by conjugate pairs.Proposition 2.11.There exists a positive constant c such that any eigenvalue λ of A α satisfies 0 < ℜ(λ) ≤ c.
Proof: Obviously, we already know that the real part of any eigenvalue of A α is positive, so we only have to prove that it is upper bounded.Let λ = α be an eigenvalue of A α and U = (y, −λy, y x (1)) an associated eigenvector such that U H = 1.Multiplying the first equation of the system (2.90) by y and integrating by parts yields (2.95) then the imaginary part of the equation (2.94) gives (2.96) Assume that v = 0 then H = 1, it follows from the previous identity that for u large Consequently (2.94) implies which is not possible.Therefore, for u large enough, we deduce from (2.96) that ℑ(λ) = v = 0. Finally, taking the real part of the equation (2.92) with v = 0, we obtain Hence the contradiction with U 2 H = 1 if u is large enough.✷ In the following proposition we study the spectrum of A α : Proposition 2.12.(Spectrum of A α ) There exists k 1 ∈ N * sufficiently large such that the spectrum σ(A α ) of A α is given by: where and J is a finite set.Moreover, λ k is simple and satisfies the following asymptotic behavior where (2.101) Proof: The proof is divided into three steps.
Step 1 furnishes an asymptotic development of the characteristic equation for large λ.
Step 2 uses Rouché's theorem to localize high frequency eigenvalues.In step 3, we perform a limited development stopped when a non zero real part appear.
Step 1. First, We start by the expansion of R 1 and R 3 when |λ| → ∞ and Next, using (2.102) and (2.103), we find the asymptotic behavior of and ). (2.107) Similarly, we get Optimal Energy Decay Rate of the Rayleigh Beam Equation 151where N (λ) is given by , where , and Then, after some computations, we find the following asymptotic development of where ) where and where (2.117) As the real part of λ is bounded, then the functions f i are bounded for i ∈ {0, 1, 2}.
For convenience we set (2.118) Step 2. We look at the roots of S. It is easy to see that the roots of f 0 are simple and given by: where α k is defined in (2.30).Then, with the help of Rouché's theorem there exists k 1 large enough such that for all |k| ≥ k 1 the large eigenvalues of σ(A α ) (denoted by λ k ) are simple and close to z k .More precisely, there exists k 1 ∈ N * large enough, such that the splitting of σ(A α ) given in (2.97)-(2.98)holds and we have (2.120) Equivalently, we can write (2.121) Step 3. Determination of ǫ k .First, using (2.118) and the identities (2.113)-(2.116)we have On the other hand, using (2.121) we find and (2.126) Similarly, we get (2.127) Now, using (2.121), (2.126) and (2.127) we get Therefore  The table below confirms this behavior.
Optimal Energy Decay Rate of the Rayleigh Beam Equation 155Thus, we deduce lim k→+∞ Finally, thanks to Theorem 2.4 in [5], we deduce that the trajectory e tAα U0 decays slower that 1 on the time t → +∞.Then we cannot expect the energy decay rate 1 t 1+ǫ .✷

Rayleigh beam equation with only one dynamical boundary control force
In this section, we consider the Rayleigh beam equation with only one dynamical boundary control force: First, let y and ξ be smooth solutions of system (3.1).We define its associated energy by: Then the system (3.1) is dissipative in the sense that its energy E(t) is a nonincreasing function of the time variable t.Let Φ ∈ W . Integrating by parts, we transform (3.1) into a variational equation: According, we define the continuous operator B as follows: Assume that Ay ∈ V ′ , then we can formulate the variational equation (3.4) as: where the operators A and C are defined in (2.9) and (2.11).Now define the energy space H = W × V × C endowed with the usual inner product and where W and V are given in (2.4) and (2.5).Next, we introduce the linear unbounded operator A0 and the linear bounded operator B as follows: and Then, denoting by U = (y, yt, ξ) the state of the system (3.1) and define , we can formulate the system into an evolution equation It is easy to prove that − A β is a maximal dissipative operator in the energy space H, therefore it generates a C0-semigroup (e −t A β ) t≥0 of contractions in the energy space H using Hille-Yosida's theorem (see Pazy [15]).In addition, it is easy to show that an element U = (y, z, ξ) ∈ D(A β ) if and only if y ∈ H 3 (0, 1) ∩ W, z ∈ W and yxx(1) = 0.In particular, the resolvent Consequently, the spectrum of A β (respectively A0) consists entirely of isolated eigenvalues with finite multiplicities.Moreover, since the coefficients of A β (respectively A0 ) are real, their eigenvalues appear by conjugate pairs.Theorem 4.2 of [16] shows that the semi-group of contractions (e −tA ) t≥0 is strongly asymptotically stable in the energy space H, i.e. for any u0 ∈ H, we have lim Using a numerical program we find Moreover, from Theorem 4.3 of [16] there exists a infinite numbers of 0 < γ < γ 0 such that the operator A β has eigenvalues on the imaginary axis and therefore for which problem (3.1) is not stable.Further, we know that the Rayleigh beam is not uniformly exponentially stable neither with one boundary direct control force (see [16]) nor with two dynamical boundary control (see [21]).Then, we look for a optimal polynomial energy decay rate for smooth initial data.

Analysis of eigenvalues and eigenvectors of the operator A β for β ≥ 0
In this subsection, we study the eigenvalues and the eigenvectors of the operator A β for β ≥ 0. First, let λ = β be an eigenvalue of the operator A β and U = (y, z, ξ) be an associated eigenfunction, then we have A β U = λU .Equivalently, λ and y verify the following system: The general solution of the system (3.10) is where Ri(λ), i = 1, .., 4 are given in (2.92).Next, using the boundary conditions, we may write the system (3.10) as follows: Optimal Energy Decay Rate of the Rayleigh Beam Equation 157where and where Remark 3.1.First, like we did in Proposition 2.11, we find that the real part of any eigenvalue λ of A β is bounded, i.e.
Next, let λ 0 be an eigenvalue of A0 and U 0 = (y 0 , z 0 , ξ 0 ) ∈ D( A0) an associated eigenvector.Then, like we did in Proposition 2.5, we can easily prove that λ 0 is simple and ξ 0 = 0. ✷ Next, we study the asymptotic behavior of the eigenvalues of the operators A β in the following proposition: Then there exists k β ∈ N * sufficiently large such that the spectrum σ(A β ) of A β is given by where where J β is a finite set and λ β,k is simple and satisfies the following asymptotic behavior: ) .

Proof:
The proof uses the same strategy than the one from Proposition 2.12.For the sake of completeness, we give the details.For simplicity, we denote Ri(λ) by Ri.
For convenience we set Step 2. We look at the roots of S β .It is easy to see that the roots of f 0 are simple and given by: Then, with the help of Rouché's theorem, there exists k β ∈ N * large enough, such that ∀|k| ≥ k β the large eigenvalues of ).
On the other hand, using (3.39) we obtain and Similarly we get ) and Next, using (3.47) we find the first development of ζ k,β given by ).Then, inserting (3.48) in (3.47) we obtain where and where ).Then, substituting (3.51) into (3.50)yields Optimal Energy Decay Rate of the Rayleigh Beam Equation 163where and where E and F are given in (3.18) is the eigenvector associated with the eigenvalue λ β,k of high frequency, and by {Φ β,j,l } m β,j l=1 the Jordan chain of root vectors associated with the eigenvalue λ β,j of low frequency (Φ 0,j,l are in fact eigenvectors of A0) .Thus we obtain a system of root vectors of β : and we deduce that Step 1. Determination of y k .Since we search C(λ k ) up to factor we choose c4(λ k ) = 1, the possibility of this choice will be justify later.Therefore (3.12) becomes Next, using Cramer's rule, we obtain Optimal Energy Decay Rate of the Rayleigh Beam Equation First, we study the behavior of b1.Inserting (2.102) and (2.103) (with λ = λ k ) in (3.63) we find after some computations b1 = −2γ Now, using the asymptotic behavior (3.16) we find ). (3.68) Then, inserting (3.68) in (3.67) we find again after some computations Similarly long computations left to the reader yield b2 Remark that b4 = 0 provided we have chosen k β large enough, for this reason our choice c4(λ k ) = 1 is valid.Substituting (3.69)-(3.72)into (3.62),we deduce Finally we have found that a solution of (3.12) has the form: ).
Moreover, using the asymptotic behavior (3.79)-(3.80)we find that G k is given as follows: This completes the proof.✷

Riesz basis and polynomial stability with optimal decay rate
Our main result is the following optimal polynomial-type decay estimation.(3.103) , S. Nicaise, M. A. Sammoury and A. Wehbe

1 0 e rx dx = e r r − 1 r
for r = 0 and the asymptotic behavior (2.32)-(2.33)we find that