WELL-POSEDENESS AND ENERGY DECAY OF SOLUTIONS TO A BRESSE SYSTEM WITH A BOUNDARY DISSIPATION OF FRACTIONAL DERIVATIVE TYPE

. We consider the Bresse system with three control boundary conditions of fractional derivative type. We prove the polynomial decay result with an estimation of the decay rates. Our result is established using the semigroup theory of linear operators and a result obtained by Borichev and Tomilov.

Recently, Park and Kang [19] considered the stabilization of the Timoshenko system with weakly nonlinear internal feedbacks.
In [24], Soriano, Wenden Charles, Rodrigo Schulz considered a Bresse system with three internal feedbacks. They proved the exponential decay of the solution.
In [21], Liu and Rao considered a Bresse system coupled with two heat equations. The two wave equations for the longitudinal displacement and the shear angle displacement are effectively globally damped by the dissipation from the two heat equations. The wave equation about the vertical displacement is subject to a weak thermal damping and indirectly damped through the coupling. They establish exponential energy decay rate when the vertical and the longitudinal waves have the same speed of propagation. Otherwise, a polynomial-type decay is established.
The boundary feedback under the consideration here are of fractional type and are described by the fractional derivatives The order of our derivatives is between 0 and 1. These convolutions with locally integrable kernels are not simple to treat: analytically, the singular character of kernel t −α (with 0 < α < 1) problematizes the use of methods and techniques developed for convolution terms with regular and/or integrable kernels. It has been shown (see [16]) that, as ∂ t , the fractional derivative ∂ α t forces the system to become dissipative and the solution to converge to the equilibrium state. Therefore, when applied on the boundary, we can consider them as controllers which help to reduce the vibrations.
Boundary fractional dissipations are not only important from the theoretical point of view but also for applications. They naturally arise in hereditary processes and fractal media to describe memory effects and anomalous phenomena (see [22]). Indeed, it has been observed by experiments that many concepts cannot be described in Newtonian terms. In other words, in many fields, phenomena with strange kinetics cannot be described within the framework of classical theory using integer-order derivatives. It could lead to a more adequate modeling and more robust control performance. For example, in viscoelasticity, due to the nature of the material microstructure, both elastic solid and viscous fluid like response qualities are involved. Using Boltzmann assumptions, we end up with a stress-strain relationship defined by a time convolution. More precisely, the stress at each point and at each instant does not depend only on the present value of the strain but also on the entire temporal prehistory of the motion from 0 up to time t. This is interpreted by a time convolution with a "relaxation function" as kernel. Viscoelastic response occurs in a variety of materials, such as soils, concrete, rubber, cartilage, biological tissue, glasses, and polymers (see [3], [4], [5] and [14]). In our case, the fractional dissipations may simply describe an active boundary viscoelastic damper designed to reduce unwanted vibrations. for the purpose of reducing the vibrations.
Our purpose in this paper is to give a global solvability in Sobolev spaces and energy decay estimates of the solutions to the problem (P ) for linear damping. To obtain global solutions to the problem (P ), we use the argument combining the semigroup theory ( [8]) with the energy estimate method. To prove decay estimates, we use a frequency domain approach and a Theorem of A. Borichev and Y. Tomilov. 2. 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 [15]). Let µ be the function: (1) Then the relationship between the 'input' U and the 'output' O of the system is given by where Proof. Let us set We have Then the function f λ is integrable. Moreover From Theorem 1.16.1 in [25], the function For a real number λ > −η, we have Both holomorphic functions g and λ → (λ + η) α−1 π sin πα coincide on the half line ] − η, +∞[, hence on D following the principle of isolated zeroes.
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: Lemma 2.3. Let (ϕ, φ 1 , ψ, φ 2 , ω, φ 3 ) be a solution of (P ). Then, the energy functional defined by (6) satisfies Proof. Multiplying the first equation in (P ) by ϕ t , the third equation by ψ t , the five equation by ω t , integrating over (0, L) and using integration by parts, we get where ζ i = (π) −1 sin(απ)γ i . Multiplying second, fourth and sixth equations in (P ) by ζ i φ i respectively and integrating over (−∞, +∞), to obtain: (9) From (6), (8) and (9) we get This completes the proof of the lemma.
3. Global existence. In this section we will give well-posedness results for problem (P ) using semigroup theory. Let us introduce the semigroup representation of the Bresse system (P ). Let U = (ϕ, ϕ t , φ 1 , ψ, ψ t , φ 2 , ω, ω t , φ 3 ) T and rewrite (P ) as where the operator A is defined by with domain where, the energy space H is defined as We show that the operator A generates a C 0 -semigroup in H. In this step, we prove that the operator A is dissipative. Let U = (ϕ, u, φ 1 , ψ, v, φ 2 , ω,ω, φ 3 ) T . Using (10), (7) and the fact that we get Consequently, the operator A is dissipative. Now, we will prove that the operator λI −A is surjective for λ > 0. For this purpose, let ( Suppose that we have found ϕ, ψ and ω. Therefore, the first, the fourth and the seventh equation in (15) give It is clear that u ∈ H 1 L (0, L), v ∈ H 1 L (0, L) andω ∈ H 1 L (0, L). Furthermore, by (15) we can find φ i (i = 1, 2, 3) as

ABBES BENAISSA AND ABDERRAHMANE KASMI
By using (15) and (16) the functions ϕ, ψ and ω satisfying the following system Solving system (18) is equivalent to finding (ϕ, ψ, ω) ∈ (H 2 (0, L) ∩ H 1 L (0, L)) 3 such that . By using (19) and (17) the functions ϕ, ψ and ω satisfying the following system Inserting (21) into (20), we get where the bilinear form a : and L(w, χ, ζ) It is easy to verify that a is continuous and coercive, and L is continuous. So applying the Lax-Milgram theorem, we deduce that for all (w, Applying the classical elliptic regularity, it follows from (22) . Therefore, the operator λI − A is surjective for any λ > 0. Consequently, using Hille-Yosida theorem, we have the following results. (1) If U 0 ∈ D(A), then system (10) has a unique strong solution (1) If U 0 ∈ H, then system (10) has a unique weak solution U ∈ C 0 (IR + , H).

4.
Lack of exponential stability. We first state three well-known theorems.
Our main result is the following Theorem 4.4. The semigroup generated by the operator A is not exponentially stable.
We state and prove a proposition that will be needed later. We consider the case when l → 0 i.e, when (P ) takes the following form System (P 0 ) can be reduced to the Timoshenko system and an independent wave equation: The abstract formulation of (P 0 ) is: with domain Proposition 1. The semigroup generated by operator A 0 is not exponentially stable.
Proof. This result is due to the fact that a subsequence of eigenvalues of A 0 is close to the imaginary axis.
Let H 1 , be the subspaces of H defined by Observe that the generator A 0 becomes the operator A 1 defined by We first compute the characteristic equation that gives the eigenvalues of A 1 . Let λ be an eigenvalue of A 1 with associated eigenvector U = (0, 0, 0, 0, 0, 0, ω,ω, φ 3 ) T .
with the following conditions The matrix of the system determining is not singular. Set The characteristic polynomial ofB is We find the roots Here and below, for simplicity we denote t i (λ) by t i . The solution ω is given by Thus the boundary conditions may be written as the following system: Hence a non-trivial solution ω exists if and only if the determinant ofM (λ) vanishes. Set f (λ) = detM (λ), thus the characteristic equation is f (λ) = 0. Our purpose in the sequel is to prove, thanks to Rouché's Theorem, that there is a subsequence of eigenvalues for which their real part tends to 0.
In the sequel, since A 1 is dissipative, we study the asymptotic behavior of the large eigenvalues λ of A 1 in the strip −α 0 ≤ R(λ) ≤ 0, for some α 0 > 0 large enough and for such λ, we remark that e ti , i = 1, 2 remains bounded. where Moreover for all |k| ≥ N , the eigenvalues λ k are simple.
Proof.The proof is decomposed in three steps: Step 1.
Step 2. We look at the roots of f 0 . From (37), f 0 has one familie of roots that we Now with the help of Rouché's Theorem, we will show that the roots off are close to those of f 0 . Let us start with the first family. Changing in (36) the unknown λ by u = 2 ρ1 K λL then (36) becomes The roots of f 0 are u k = i(k+ 1 2 ) rL π, k ∈ Z, and setting u = u k + re it , t ∈ [0, 2π], we can easily check that there exists a constant C > 0 independent of k such that |e u + 1| ≥ Cr for r small enough. This allows to apply Rouché's Theorem. Consequently, there exists a subsequence of roots off which tends to the roots u k of f 0 . Equivalently, it means that there exists N ∈ IN and a subsequence {λ k } |k|≥N of roots of f (λ), such that λ k = λ 0 k + o(1) which tends to the roots Step 2, we can write Using (40), we get Substituting (41) into (36), using thatf (λ k ) = 0, we get: and hence The operator A 1 has a non exponential decaying branche of eigenvalues. Thus the proof of Proposition 1 is complete.
The general solution of (52) must be of the form where t i (λ) (i = 1, . . . , 6) are the roots of (53) such that t 2 (λ) = −t 1 (λ), t 4 (λ) = −t 3 (λ), t 6 (λ) = −t 5 (λ). t 3 (λ) 2 and t 5 (λ) 2 are complex conjugate. Then, we write (45) uniquely in function of ψ. From (46) 1 we have From (47) 1 , after derivation, we get Thus we find We deduce that From (46) 3 and (47) 1 we have Then from (46) 2 and (47) 2 and (56), we get Thus the boundary conditions (45) may be written as the following system: Thus the boundary conditions may be written as the following system: where C(λ) = (c 1 , c 2 , c 3 , c 4 , c 5 , c 6 ) T and . We remark that f l (λ) = detM l is a smooth function with the parameter l. In the expansion of t i (λ)(i = 1, . . . , 6) and h i (t j )(i = 1, . . . 5, j = 1, . . . , 6), the parameter l appears only in lower terms. Hence, in the development of detM l in power series following λ, we obtain same development as detM 0 modulo lower terms depending on l. Hence A l (if we note A by A l ) and A 0 have same branches of eigenvalues modulo lower terms depending on l.
From Proposition 1 and the fact that f 0 (λ) = detM 0 = 0 give the eigenvalues of Bresse system when l = 0 we conclude our result.

Remark 1.
We can also show the lack of exponential stability by proving that the second condition in Theorem 4.1 does not hold. In particular, it can be shown that there is a sequence λ n ∈ IR diverging to ∞, and a bounded sequence F n ∈ H such that (iλ n − A) −1 F n → ∞ for all n large enough .
We give an idea of the proof in the Appendix.
5. Asymptotic stability. In this section, we use a general criteria of Arendt-Batty in [2] to show the strong stability of the C 0 -semigroup e tA associated to the system (P ) in the absence of the compactness of the resolvent of A. Our main result is the following theorem: Let us first prove Lemma 5.3.

BRESSE SYSTEM WITH BOUNDARY FRACTIONAL DERIVATIVE CONTROLS 4389
Thus, by using again the inequality 2P Q ≤ P 2 + Q 2 , P ≥ 0, Q ≥ 0, we get We deduce that It follows that Let us introduce the following notation Lemma 5.8. Let q ∈ H 1 (0, L). We have that for a positive constant C.
Proof. To get (97), let us multiply the equation (88) 2 by qϕ x Integrating on (0, L) we obtain Since iλϕ x = u x + f 1x taking the real part in the above equality results in Performing an integration by parts we get Similarly, multiplying equation (88) 5 by qϕ x , integrating on (0, L) and taking the real part we obtain Since iλψ x = v x + f 4x taking the real part in the above equality results in Performing an integration by parts we get Finally, multiplying equation (88) 8 by qω x , integrating on (0, L) and taking the real part, after some algebric manipulations we obtain (99) for Since iλω x =ω x + f 7x taking the real part in the above equality results in Performing an integration by parts we get for λ = 0. Since that ϕ = u+f1 iλ , ψ = v+f4 iλ and ω =ω +f7 iλ we obtain E ϕ (L) + E ψ (L) + E ω (L) Appendix. We will show the lack of exponential stability by frequency domain method.