Stabilization of Variable Coe ffi cients Euler-Bernoulli Beam with Viscous Damping under a Force Control in Rotation and Velocity Rotation

This paper investigates the problem of exponential stability for a damped Euler-Bernoulli beam with variable coefficients clamped at one end and subjected to a force control in rotation and velocity rotation. We adopt the Riesz basis approach for show that the closed-loop system is a Riesz spectral system. Therefore, the exponential stability and the spectrumdetermined growth condition are obtained.


Introduction
In this paper, we study the exponential stability property of a damped Euler-Bernoulli beams with variable coefficients under a force feedback in rotation and velocity rotation.The equations of motion of the system are described as follows m(x)w tt (x, t) + (EI(x)w xx (x, t)) xx + γ(x)w t (x, t) = 0, 0 < x < 1, t > 0, (1) w (0, t) = w x (0, t) = 0, t > 0, (2) (EI(.)wxx ) x (1, t) = 0, t > 0, (3) where the subscripts t and x denote derivatives with respect to the time t and the position x respectively.w(x, t) stands for the transverse displacement of the beam at the position x and time t.The feedbacks α and β are two given positive constants.We assume that the length of the beam is equal to unity.EI(x) and m(x) are, respectively, the flexural rigidity function and the mass density function of the beam along the spatial variable x satisfying the following condition m(x), EI(x) ∈ C 4 (0, 1), m(x), EI(x) > 0 (5) for all x ∈ [0, 1].Non-homogeneous materials, in particular smart materials used in engineering, are typical examples of the importance of assuming coefficients as variables (Lee & Li, 1998).Moreover, γ(x) is a continuous coefficient function of feedback damping that is assumed to satisfy the condition ) ( m(x) EI( x) Notice that the condition (6) will allow γ to be indefinite in the interval [0, 1].
In the theory of dynamic systems, stability is a matter of great interest to mathematicians and engineers.More particularly, exponential stability is the most desirable stability, especially for damped systems.The study of the case (γ ≡ 0) was done in (Bomisso, Touré & Yoro, 2017) where the authors have obtained a result according to the authors Wang et al (see e.g.Wang, 2004) to show the exponential stability of system (1)-(4).In (Wang, 2004), a question has been raised and is valid for our system (1)-( 4): Due to the nonuniform physical thickness and/or density of the Euler-Bernoulli beam with the variable coefficient damping γ(x) in equation ( 1), what conditions are needed to put onto the damping term to guarantee exponential stability?Here it is very hard to have the exact precise location of the eigenvalues because equation (1) contains variable coefficients and subject to the boundary conditions ( 2)-( 4).This question is treated when (1) is associated to hinged boundary conditions (see Wang, 2004), and when the equation ( 1) is subjected to the force control in position and velocity (see Touré, Coulibaly & Kouassi, 2015).Moreover, in order to study the eigenvalues of systems with variable coefficients, we will used the two steps provided by Birkhoff's works (Birkhoff, 1908) and Naimark's works (Naimark, 1967).This approach was used by many authors for study the Euler-Bernoulli beams equations with variable coefficients (see e.g.Guo, 2002;Guo & Wang, 2006;Wang, 2004;Wang, Xu & Yung, 2005).In our case, we rely on idea of Wang et al (see e.g.Wang, 2004;Wang, Xu & Yung, 2005) in order to study the problem with eigenvalues related to problem (1)-( 4) in the form of an ordinary differential equation L( f ) = λ f with boundary conditions λ-polynomials.We establish conditions on the two feedbacks parameters at the boundary α and β to obtain the property of the Riesz basis and the exponential stability of the system (1)-(4).
Our main contribution is to prove the exponential stability of the perturbed system (1)-( 4).
The rest of the paper is organized as follows.In section 2, the system (1)-( 4) is formulated as an evolution problem and studied in semigroup framework.In section 3, a spectral analysis is made and next, we prove that the system operator has Riesz basis property in the corresponding state space.Consequently, in section 4, we give conditions that ensure the exponential stability of our perturbed system.

Semigroup Formulation
We define the following functional spaces: and with the inner product where w = ( f 1 , f 2 ) T ∈ H, v = (g 1 , g 2 ) T ∈ H and ∥.∥ H denotes the corresponding norm.We recall also the definitions of the following spaces : ⊂ H→H an unbounded linear operator with the domain defined as So, we can written (1)−(4) as a first order evolution problem where z (t) = (w, w t ) T , z (0) = (w 0 , v 0 ) T .Furthermore, notice that ) is a linear operator and bounded on H with A 0 denotes the operator in the undamped case γ (x) ≡ 0.
Two results are immediate and the first one is a consequence of the perturbation theory of semigroups (see e.g.Pazy, 1983).
Theorem 1 Let operators A γ and A 0 be defined as before.Thus A 0 is a dissipative operator and generates a C 0 -semigroup of contractions on H denoted by {S (t)} t≥0 and therefore A γ is a generator of the contraction semi-group e A γ t on H denoted by {T (t)} t≥0 .
Proof.The first assertion has been proved in (Bomisso, Touré & Yoro, 2017).The authors have used the Lumer-Phillips Theorem (see, e.g., Pazy, 1983, pp.14) and Hille-Yosida-Phillips Theorem in order to show that the dissipative operator A 0 generates a C 0 -semigroup S (t) = e A 0 t on H, such that Hence, the perturbation theory (see Theorem 1.1 of Pazy, 1983) allows us to deduce that Theorem 2 The operator A γ has compact resolvents and 0 ∈ ρ ) .
Proof.The first assertion is trivial.It remains only to show that 0 ∈ ρ The solution of above system is obtained straightforward after computation: . Moreover, using Sobolev's Embedding Theorem, we deduce that A −1 γ is a compact operator on the Hilbert space H.

Spectral Analysis of Operator A γ
Spectral analysis is one of the methods used today to determine the behavior of eigenvalues of operators of dynamic systems.In what follows, we will rely on idea of Wang et al ( Wang, Xu & Yung, 2005) in order to study the eigenvalues problem associated to system (1)-( 4).
But first, we recall the following definitions and notations useful in the sequel.
Let L ( f ) be an ordinary differential operator of order n = 2m ∈ N defined as under the following boundary conditions where 15) are sufficiently smooth in (0, 1) , and that the boundary conditions are normalized in the sense that K = n ∑ j=1 k j is minimal with respect to all equivalent boundary conditions (see Naimark, 1967).
Definition 3 The boundary problem (17) with ( 16) is said to be regular if the coefficients F k k 0 in (18) are nonzero.Furthermore, the regular boundary problem (17) with ( 16) is said to be strongly regular if the zeros of ∆ (ρ) are asymptotically simple and isolated one from another.
Let W m 2 (0, 1) be the usual Sobolev space of order m and let Let H be a Hilbert space defined as which denotes its corresponding norm and let A be a operator in H defined by The Theorem 4 used in (Wang, Xu & Yung, 2005) was presented in (Wang, 2003).The reader may also refer to chapter 3 of (Wang, 2004).
Theorem 4 If the ordinary differential system with parameter has strongly regular boundary conditions, then the generalized eigenfunction system of A form a Riesz basis in the Hilbert space H.
According to Theorem 2, A γ has a compact resolvent.Thus, the spectrum of A denoted by σ(A γ ), consists only of isolated eigenvalues, which distribute in conjugate pairs on the complex plane.Now, the eigenvalue problem of operator A γ can be investigated.Let λ be an eigenvalue of the spectrum σ(A γ ) and Φ = (ϕ, Ψ) denoting its corresponding eigenfunction.Thus, Ψ = λϕ with ϕ satisfies the following equations: In order to solve (21), spatial transformations as introduced in (Guo, 2002) are performed, which convert the first equation of ( 21) into a more convenient form.For this purpose, for 0 < x < 1, the system ( 21) is firstly rewritten as : Moreover, introduce the following space transformation in order to transform the coefficient function appearing with ϕ in the first expression of ( 22) into a constant.Let Thus, using again its boundary conditions, the system ( 22) can be transformed as and Next, we use the idea of Naimark presented in Chapter 2 of (Naimark, 1967) for solve (25).Then, in order to cancel the third derivative term a (z) f ′′′ (z) in ( 25), we introduce a new invertible space transformation (25) can be written as follows, for any 0 < z < 1: x (1) Since the above transformations are invertible, the obtained system ( 32) is equivalent to the original problem (21).
To further solve the eigenvalue problem (32), the complex plane is divided into eight distinct sectors: Moreover, we denote the roots of equation θ 4 + 1 = 0 by ω 1 , ω 2 , ω 3 , ω 4 such that inequalities holds Re (ρω 1 ) ≤ Re (ρω 2 ) ≤ Re (ρω 3 ) ≤ Re (ρω 4 ) , ∀ρ ∈ S n . (42) Clearly, the choices in the sector S 1 satisfying (42) are given as follows Notice that, similary, the choices can be obtained for other sectors.In the following, we study the asymptotic behavior of the eigenvalues only for the sectors S 1 and S 2 because the work done in these two sectors is valid in the other sectors.
In the following, we will use the notation: We also need the following Lemma: 2 , then the inequalities holds Re (ρω 1 ) ≤ − |ρ| δ, Re (ρω 4 ) ≥ |ρ| δ and e . Using Lemma 5, the asymptotic expressions for the boundary conditions for large enough |ρ| , are obtained for s = 1, 2, 3, 4, Notice that λ 0 is the eigenvalue of (32) if and only if ρ satisfies the characteristic equation By substitution, the following expression is obtained Moreover, expanding the exponential function according to its Taylor series, we get Therefore, we have and ω 2 2 = i.So we have where n = N, N + 1, . . .with N large enough.
We make the same work for the sector S 2 because the eigenvalues of system (32) can be got by a similar computation with the following choices such that the inequality ( 42) is satisfied Re (ρω 1 ) ≤ Re (ρω 2 ) ≤ Re (ρω 3 ) ≤ Re (ρω 4 ) , ∀ρ ∈ S 2 .
Hence, in sector S 2 , the characteristic determinant ∆ (ρ) of ( 21) : After computation, we have with N large enough.Again, using Furthermore, λ n (n = N, N + 1, . ..) with sufficiently large modulus are simple and distinct except for finitely many of them, and satisfy Remark 1 This Theorem is a bit of surprise because although the beam is nonuniform, we obtain an asymptotic uniform rate of decay in terms of the viscous damping function and thus the important question asked in introduction is answered.
Moreover, with reference to (Naimark, 1967), we can say that the eigenvalues generated by the other sectors S n coincide with those determined in the sectors S 1 and S 2 .

Riesz Basis Property of the Generalized Eigenfunctions of A γ
In what follows, we follow an idea of Wang in (Wang, Xu & Yung, 2005) in order to discuss the Riesz basis property of the eigenfunctions of operator A γ of system (14).To begin, we prove that the generalized eigenfunctions of Notice that L is invertible and is a bounded operator on H. Also, we define the following ordinary differential operator: z 2 x (1) where the coefficients are defined by ( 25)-( 29).Let A be defined as in ( 19), η ∈ σ (A) be an eigenvalue of A and ( f, g) be an eigenfunction corresponding to η, then we obtain g = η f and f satisfies the following equation: (61) A form an unconditional basis in the Hilbert space H. Thus let us choose a transformation L such thatL ( f, g) = (ϕ, ψ) with ϕ (x) = f (z) , ψ (x) = g (z) ,