Exponential Stability for Damped Shear Beam Model and New Facts Related to the Classical Timoshenko System With a Distributed Delay Term

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In the case of the wave equations, Nicaise and Pignotti [8] investigated exponential stability results with delay concentrated at τ for the system under the condition µ 2 < µ 1 , by combining inequalities due to Carleman estimates and compactness-uniqueness arguments. Later, they also obtain in [13] the exponential stability with distributed delay of the system under the assumption (2).
Recently in the case of the wave equation with dynamical control, Silga and Bayili [2] studied a wave equation set in a bounded domain with a dynamical control and prove that if the delay term is small enough, then the system with delay Vol. 15, No. 3;2023 has the same (polynomial) decay rate than the one without delay.
Later in [14] they also investigate the case of the wave equation set in a bounded domain with a distributed delay on dynamical control To do this they use the assumption (2).
Motivated by the above results we will establish under the hypothesis (2) the well-posedness and the exponnential stability of the system (1).
This paper is organized as follows. In Section 2, we give the decrease for the energy of the system and the well-posedness of problems (1) using the Faedo-Galerkin method. And finally in Section 3, we will give the exponential stability of the problem using a Lyapounov fuction.

Well-Posedness of the Problem
In this section we will give well-posedness results for problem (1) using Faedo-Galerkin method. To this aim, we introducing the following auxiliary change of variable The problem (1) is now equivalent to Let us consider the Hilbert spaces and H 2 * (0, L) = H 2 (0, L) ∩ H 1 * (0, L). We equipped H with the norm Define the energy of a solution (θ, θ t , ψ, z) of (9) as We can now prove that the energy is decreasing. More precisely, we have the following result.
Proposition 2.1. For any regular solution of problem (9), the energy is decreasing and there exists a positive constant C such that d dt Démonstration. Multiplying , in (9) , the first equation by θ t , the second equation by ψ t and the third equation by α(s)η, and then integrating the first and the second over [0; L] and the third over (0, L) × (0, 1) × (τ 0 , τ 1 ). By integrating by parts and using the boundary conditions, we obtain and 1 2 Now summing relations (10), (11) and (12) we obtain Then from the definition of the energy, it follows that We also know that Then using the relations (14) in (13) , its follows that This proves that We therefore deduce that the energy of the system (9) is dissipative.
To state the main result in this section, we start by defining what we mean by a weak solution of problem (9) as following The main result of this section is the following : Moreover, if U 0 = (θ 0 , θ 1 , ψ 0 , η 0 ) ∈ H 1 , then problem (9) admits a unique stronger weak solution U = (θ, θ t , ψ, η) which satisfies In both cases, the solution (θ, θ t , ψ, η) depends continuously on the initial data in H. Particulary, the system (9) admit a unique weak solution.
Démonstration will be the key to prove the existence of a global solution. The proof will be done in six steps Step 1. : Faedo-Galerkin approximations.
We consider the finite-dimensional subspaces H n , V n and W n defined by H n = span u 1 , u 2 , ..., u n , V n = span v 1 , v 2 , ..., v n and W n = span w 1 , w 2 , ..., w n . Now we will find an approximate solution in the form for all u ∈ H n , v ∈ V n , w ∈ W n , with initial conditions such that strongly in H and where a jn , b jn and c jn are time-dependent coefficients. Applying standard theory of ordinary differential equations, the finite dimensional problem (19) − (20) has a solution (a jn , b jn , c jn ), 1 ≤ j ≤ n defined on [0, t n ) for every n ∈ N . Then the a priori estimates that follow imply that in fact t n = T, ∀T > 0.

Exponential Stability
In this section, our aim is to study the exponential stability of the systelm (1) . More precisely we prove the following result.
Theorem 3.1. Let the assumption (2) be satisfied. Then, there exist positive constants M and K such that, for any solution of (9) Démonstration. Let where c is a constant whose conditions we will specify later.
We return to the proof of theorem 3.1.
This completes the proof.