EXPONENTIAL STABILITY FOR A NONLINEAR TIMOSHENKO SYSTEM WITH DISTRIBUTED DELAY

. This paper is concerned with a nonlinear Timoshenko system modeling clamped thin elastic beams with distributed delay time. The distributed delay is deﬁned on feedback term associated to the equation for rotation angle. Under suitable assumptions on the data, we establish the exponential stability of the system under the usual equal wave speeds assumption.


Introduction
In this work, we consider the following non linear Timoshenko system with distributed delay, where t denotes the time variable and x the space variable along a beam of length 1 in its equilibrium configuration. Here, ϕ = ϕ(x, t) and ψ = ψ(x, t) denotes the transverse displacement of the beam and the rotation angle of its filament, respectively. The term µ 1 ψ t represents a frictional damping and f (ψ) is a forcing term. The coefficients, ρ 1 , ρ 2 , k are positive constants represent the density, the polar momentum of inertia of a cross section, shear modulus respectively, and b = EI where E is the young's modulus of elasticity, I is the moment of inertia cross-section.
This type of problems (without delay), has been considered, first in [15] where µ 1 = µ 2 = f = 0. The stability of this problems has received much attention in last years, we can find in the literature many results about different stability of Timoshenko systems depending, in particular, on the weights µ 1 and µ 2 (see [14]) Recently also a great consideration ha been addressed to time delay effects. On such problems, it was showed that a small delay acted on a boundary control, or internal can destabilize a system which is uniformly asymptotically stable in the absence of delays. See for instance ( [5]) .
In [13] S. Nicaise and C. Pignotti examined a system of wave equation with initial feedback where a ∈ L 2 (Ω) is a function chosen with some assumptions. They proved that the above system is exponentially stable under the condition Similarly result was obtained by the authors when the distributed delay acted on the part of boundary.
In [11] Mustapha considered a Timoshenko system of thermoelasticity of type III with distributed delay and establish the stability for the case of equal and non equal speeds of wave propagation .Appalara [1] investigated a thermo-elastic system of Timoshenko type with second sound and distributed delay in (0, 1)(0, α), this system is exponentially stable regardless the speeds of wave propagation.
and obtained an exponential stability under equal wave speeds.
Recently S. A. Messaoudi, B. Said-Houari [10] established the stability of a thermoelastic Timoshenko system of type III with past history and distributed delay for the cases of equal and non equal speeds of wave propagation respectively.
In the present work, we extend the result of Feng and Pelier, [6] where constant delay is replaced by distributed delay.
Therefore, the problem (1.1) is equivalent to with the following initial and boundary conditions Concerning the weight of the delay, we only assume that In addition, we give some hypothesis on the forcing term f (ψ(x, t)). We assume that f : IR → IR satisfies the following condition We introduce the Hilbert space, We introduce two new dependent variables ϕ t = u and ψ t = v, then the system (2.1)-(2.2) can be written as Clearly, D(A) is dense in H, we have the following existence and uniqueness result (see [6]).

Stability result
In this section, we use the energy method to show that the solution of problem (2.1)-(2.2) decays exponentially, below we shall give the stability result.
Theorem 3.1. Assume that(2.4)-(2.5) and µ 2 < µ 1 hold. Assume that ρ1 ρ2 = k b also holds. Then, with respect to mild solutions, there exist 1 > 0 and 2 > 0 such that To achieve our goal we state and prove the following lemmas.

9)
Proof. Differentiating I 1 (t), we obtain and using (2.1) 1 , (2.1) 2 , we get We have the following inequalities. Proof. We multiply equation (3.13) by w, integrate by parts and use the Cauchy-Schwarz inequality to obtain (3.14) Next, we differentiate (3.13) with respect to t and by the same procedure as above, we obtain This completes the proof of Lemma (3.3).
Proof. By differentiation I 3 (t) and using (2.1) 1 , (2.1) 2 , we obtain By using Young's inequality, we have Using Young's and Cauchy Schwarz inequalities, we get Young's, Cauchy Schwarz and Poincaré inequalities lead to Next, in order to handle the boundary terms, appearing in (3.23) , we define the function x ∈ (0, 1) So, we have the following result.
Lemma 3.8. Let (ϕ, ψ, z) be the solution of (2.1)-(2.2). Then, there exists two positive constants β 1 and β 2 such that the Lyapunov functional L(t) satisfies Proof. Let Exploiting Young's, Poincaré and Cauchy-Schwarz inequalities, we obtain , we get First, we choose ε 1 small enough such that After, we take N 1 large so that Then, we select N 2 large to satisfies By finally choose N large enough (even larger so that 1.1 remains valid) such that we obtain (3.35). The proof is complete.