TAHRI Mohamed,BENKHEDDA Hanane and BENAISSA Abbes

Laboratory of Analysis and Control of PDEs,Djillali Liabes University,P.O.Box 89,Sidi Bel Abbes 22000,Algeria.

Abstract. In this paper,we study the energy decay rate for a one-dimensional nondegenerate wave equation under a fractional control applied at the boundary.We proved the polynomial decay result with an estimation of the decay rates. Our result is established using the frequency-domain method and Borichev-Tomilov theorem.

Key Words: Nondegenerate wave equation; fractional boundary control; Frequency domain method;Optimal polynomial stability.

1 Introduction

In this paper, we are concerned with the boundary stabilization of convolution type for nondegenerate wave equation of the form

where the coefficientais a positive function on[0,1].

Up to now, there are many works concerning the stabilization and controllability of nondegenerate wave equation with different types of dampings (see e.g. [1–4] and the references therein). In [4], fora(x)=a1x+a0: the authors have established asymptotics stabilization under boundary conditions of the form

It has been shown in[1],fora∈H1(0,1),a(x)≥a0>0,that the feedback law

exponentially stabilizes equation (1.1) under appropriate assumptions on the functionF. Another stabilization result for equation(1.1) has also been established in [3] via the action of the following feedback:

In [2], the authors considered the following modelization of a flexible torque arm controlled by two feedbacks depending only on the boundary velocities:

where

They proved the exponential decay of the solutions.

Motivated by the work of[2],a feedback control depending only on the velocity has been proposed in[5]for the system(1.1)and an asymptotic convergence result has been established(see also[6–8]).

In this article,we are concerned with the system

where Γ is the usual Euler gamma function and(0<α<1).

Although there is quite a bit of work on damping mechanisms for beam models of this kind, there does not seem to be much about damping involving fractional derivatives. In[11], Mbodje studies the energy decay of the wave equation(witha≡1) with a boundary fractional derivative control. He used a diffusive representation and the semigroup theory to establish the strong asymptotic stability under the conditionη=0 and a polynomial type decay rateE(t)≤C/tifη/=0.

The main result of this paper is to show that system (P) is not exponentially stable for a general nondegenerate functiona. Furthermore,we prove that the solution decays to zero polynomially whentgoes to infinity for general initial data taken in the domain of A and for a general nondegenerate functionafor both casesη>0 andη=0.

Fractional Boundary dissipations can be encountered in many physical,chemical,biological, and economical phenomena(see [12–14]). In recent years, the control of PDEs with boundary control of convolution type has become an active area of research because it improves the performance of the systems.

This work is divided into five sections. In Section 2,we give preliminary results and we reformulate the system(P)into an augmented system by coupling the nondegenerate wave equation with a suitable diffusion equation.In Section 3,we convert the system into an evolution equation in an appropriate Hilbert space,and then prove the well-posedness of our problem by semigroup theory. In Section 4,we prove lack of exponential stability by spectral analysis. In Section 5, we study asymptotic stability of above model and we establish an optimal polynomial energy decay depending with the parameterαfor smooth solution.

2 Preliminary results

Leta∈C([0,1])∩C1(]0,1])be a function satisfying the following assumptions:

2.1 Augmented model

This section is concerned with the reformulation of the model (P) into an augmented system.For that,we need the following claims.

Theorem 2.1(see[11]).Let κ be the function:

We are now in a position to reformulate system(P). Indeed,by using Theorem 2.1,system(P)may be recast into the augmented model:

We define the energy associated to the solution of the problem (P′) by the following formula:

DifferentiatingEin a formal way, using(P′) and integrating by parts,we obtain after a straightforward computation the following Lemma.

Lemma 2.2.Let(u,θ)be a regular solution of the problem(P′). Then, the energy functional defined by(2.7)satisfies

Remark 2.1. For an initial datum inD(A)(see Theorem 3.1 below),we know that(w,θ)is of classC1in time,thus we can defferentiate the energyE(t).

3 Well-posedness

The energy space associated to system(P′)is

We have the following existence and uniqueness result.Theorem 3.1(Existence and uniqueness).

(1) If U0∈D(A),then system(3.1)has a unique strong solution

(2) If U0∈H,then system(3.1)has a unique weak solution

Proof.We use the semigroup approach. First,we prove that A is dissipative. Indeed,forU∈D(A)and using(2.8),(3.1)and the fact that

we have

Hence, A is dissipative. Next, we show thatγI−A is surjective forγ>0. That is, forG=(g1,g2,g3)T∈H,we have to findU=(w,v,θ)T∈D(A)such that

i.e

Supposewis found with the appropriate regularity. Then,(3.7)1and(3.7)3yield

Also,substituting the equation(3.8)into the equation(3.7)2,we get

Inserting(3.13)into(3.12),we get

4 Lack of exponential stability

In this section we prove the lack of exponential decay of the solutions of system(3.1). In order to state and prove our stability results,we need the following Theorem.

Theorem 4.1([16]).Let S(t)be a C0-semigroup of contractions on Hilbert space with generatorA. Then S(t)is exponentially stable if and only if

Our main result is stated as follows:

Theorem 4.2.The semigroup generated by the operatorAis not exponentially stable.

Proof.We will examine two cases.

• Caseη=0:

•Caseη/=0:

We aim to show that an infinite number of eigenvalues of A approach the imaginary axis which prevents the system(P)from being exponentially stable. Indeed we first compute the characteristic equation that gives the eigenvalues of A. Letγbe an eigenvalue of A with associated eigenvectorU=(w,v,θ)T. Then AU=γUis equivalent to

From(4.1)1−(4.1)2for suchγ,we find

Using the boundary conditions and(4.1)3,we deduce that

Our purpose is to prove, thanks to Rouch´e’s Theorem, that there is a subsequence of eigenvalues for which their real part tends to 0.

In the sequel, since A is dissipative, we study the asymptotic behavior of the large eigenvaluesγof A in the strip −α0≤ℜ(γ)≤0,for someα0>0 small enough.

Lemma 4.1.There exists N∈INsuch that

Moreover for all|k|≥N,the eigenvalues γk are simple.

Proof.The proof is decomposed in three steps:

Writing(4.3)in the standard form of a linear differential operator with homogeneous boundary conditions,we obtain

In order to simplify the computations,we introduce a spatial-scale transformation inx

Then Eq.(4.5)has the form

Equation (4.7) can be further simplified by applying another invertible transformation(see[17]):

To asymptotically estimate the solutions to the eigenvalue problem(4.9), we proceed as in[17].

Lemma 4.2.The equation

where

For simplicity,we introduce the following notation:[a]i:=a+O(γ−i)fori=1,2. From Lemma 4.2,one can write the asymptotic solution of(4.9)as follows:

Note thatf0andf1remain bounded in the strip −α0≤R(γ)≤0.

Now with the help of Rouch´e’s Theorem, we will show that the roots of ˜fare close to those off0. Let us start with the first family. Changing in (4.13) the unknownγbyu=2hγthen(4.13)becomes

From(4.19)we have in that case|k|1−αRγk～β,with

The operator A has a non exponential decaying branche of eigenvalues. Thus the proof is complete.

5 Asymptotic behavior

5.1 Strong stability of the system

To prove that the semigroup(etA)t≥0is strongly asymptotically stable, we shall apply a version of the Arendt-Batty and Lyubich-Vu for Hilbert spaces[18,19].

Theorem 5.1([18,19]).LetAbe the generator of a uniformly bounded C0-semigroup{S(t)}t≥0on a Hilbert spaceH. If:

(i)Adoes not have eigenvalues on iIR.

(ii) The intersection of the spectrum σ(A)with iIRis at most a countable set,

then the semigroup{S(t)}t≥0is asymptotically stable,i.e,‖S(t)z‖H→0as t→∞for any z∈H.

Our next main result in this part is the following theorem.

Theorem 5.2.The C0-semigroup etAis strongly stable inH; i.e, for all U0∈H, the solution of(3.1)satisfies

For the proof of Theorem 5.2,we need the following two lemmas.

Lemma 5.1. Adoes not have eigenvalues on iIR.

Proof.We will argue by contraction. Suppose that there isγ∈IR such thatiγis an eigenvalue for A and hence one can find a corresponding eigenfunctionU=(w,v,θ)∈D(A).Consequently,we have

Our immediate aim is to prove that this equation has onlyU=0 as a solution, which contradicts the definition of an eigenfunction. Firstly, the equation(5.1) is equivalent to consider the following system

Secondly,we will consider two cases:

• Caseγ/=0: Taking theL2(0,1)-inner product withUof both sides of (5.1) and using(3.5),we immediately obtain

We deduce thatwsatisfies the boundary value problem:

U∈D(A), then the regularity is sufficiently for applying an integration on the second integral in the left hand side in Eq. (5.8). Then we obtain

Using Green formula and the boundary conditions,we get

We deduce that

Using Eq.(5.2)1,we obtain

Consequently, using equations (5.11), (5.12) and (5.4), we deduce that the only solution of(5.1)is the null one.

•Caseγ=0: In this case,by(5.2)1,we havev=0 which gives thatθ=0 by(5.2)3.

Multiplying equation(5.2)2byw,using Green formula and the boundary conditions,we get

Hencewis constant in(0,1). Asw(1)=0,then

HenceUmust be the trivial solution of(5.1), which is the desired result. The proof has been completed.

Lemma 5.2.We have

where IR∗=IR−{0}.

Proof.•Caseγ/=0:

We will prove that the operatoriγI−A is surjective forγ/=0. For this purpose, letG=(g1,g2,g3)T∈H,we seekX=(w,v,θ)T∈D(A)solution of the following equation

and Using the compactness embedding fromL2(0,1) intoH−1L(0,1) and fromH1L(0,1) intoL2(0,1) we deduce that the operatorLγis compact fromL2(0,1) intoL2(0,1). Consequently,by Fredholm alternative,proving the existence ofwsolution of(5.20)reduces to proving that 1 is not an eigenvalue ofLγ. Indeed if 1 is an eigenvalue, then there existsw/=0,such that

We deduce thatU=0.

•Caseγ=0 andη/=0: Using Lax-milgram theorem,we obtain the result.

5.2 Polynomial Stability(for η/=0)

In order to establish the polynomial energy decay rate,we need the following theorem.

Theorem 5.3([20]).Let S(t)be a bounded C0-semigroup on a Hilbert spaceHwith generatorA. If

for some l>0,then there exist a positive constant c such that

Our main result is the following.

Theorem 5.4.The semigroup SA(t)t≥0is polynomially stable and

Moreover,the rate of energy decay t2/(1−α)is optimal for general initial data in D(A).

Proof.GivenG=(g1,g2,g3)T∈H,letU=(w,v,θ)T∈D(A)be the solution of the resolvent equation(iγI−A)U=G,forγ∈IR,i.e.,

• Step 1 Taking the real part of the inner product of(iγI−A)UwithUin H and using(3.5),we get

We conclude that

From the boundary condition

By multiplying(5.30)by(iγ+ξ2+η)−2|ξ|,we get

Hence, by taking absolute values of both sides of (5.31), integrating over the interval]−∞,+∞[ with respect to the variableξand applying Cauchy-Schwartz inequality, we obtain

for a positive constant C.

Proof.To get(5.35),let us multiply the equation(5.25)2by 2(c0+c1)ψwx+c0w.Integrating on(0,1)we obtain

The conclusion then follows by applying Theorem 5.3.

5.3 Polynomial Stability(for η=0)

By theorem 4.2 (see case 1) 0 is a spectral point. Therefore it is convenient to have the following generalization of theorem 5.3 at hand:

Theorem 5.5([21]).Let S(t)be a bounded C0-semigroup on a Hilbert spaceHwith generatorA. Assume that σ(A)∩iIR={0}and that there exist ϑ>1and υ>0such that

Then there exist constants C,t0>0such that for all t≥t0and U0∈D(A)∩R(A)we have

where ς=max{ϑ,υ}.

Our main result is the following.

Theorem 5.6.The semigroup SA(t)t≥0is polynomially stable and

Proof.First forγlarge enough,from the estimation in the proof of Theorem 5.4,we have

Forγnear 0,we have from(5.33)

Now,from the boundary conditions,we have

and

Substitution of inequalities(5.42)into(5.43)and(5.44),we obtain that

Substitution of inequalities(5.42)and(5.45)into(5.40)and using(5.46),we obtain that

This completes the proof of the theorem.

6 Conclusions and future works

We have studied the boundary stabilization of the non-degenerate wave system with dissipation law of fractional derivative type. Using a spectral analysis we have proved a non-uniform stability. Using Arendt-Batty Theorem,we have proved the strong asymptotic stability. Using a frequency domain approach, we prove some polynomial energy decay rate depending on parameterα.

It is interesting to extend the results of this paper to the following problem

Acknowledgement

We would like to thank very much the referees for suggesting references [1,5–8] and for their important suggestions which allow us to correct and improve this paper. The authors thank D.G.R.S.D.T for supporting this work.