Exponential Stabilty for a Timoshenko System with Nonlocal Delay

The purpose of this paper is to study the Timoshenko system with the nonlocal time-delayed condition. The well-posedness is proved by Hille-Yosida theorem. Exploring the dissipative properties of the linear operator associated with the full damped model, we obtain the exponential stability by using Gearhart-Huang-Prüss theorem.


INTRODUCTION
The history of nonlocal problems with integral conditions for partial differential equations is recent and goes back to [4]. In particular, a review of the progress related to the nonlocal models with integral type was given in [3] with many discussions about physical justifications, advantages, and numerical applications. For a nonlocal hyperbolic equation with integral conditions of the 1st kind, we cite [20]. Dissipative properties associated with the Timoshenko system have been studied by several authors by considering the dissipative mechanism of frictional or viscoelastic type. An interesting problem was brought out when the dissipation acts in different ways on the domain. For the case of terms of time-varying delay in the internal feedbacks, the stability result of the Timoshenko system can be found in [10]. On the other hand, for the case of delay and boundary feedback, we can see in [24]. The Timoshenko beam system with delay in the boundary control was studied in [25] where the exponential stabilization result is proved via a test of exact observability of the system. Distributed delay in the boundary control was considered in [13]. Distributive delay in a Timoshenko-type system of thermoelasticity of type III was considered in [8] and then, in [9], the same problem was dealt with constant delay. The Timoshenko system with second sound and the internal distributed delay was investigated in [2].
The transmission problem with delay in porous-elasticity was considered in [21]. For a nonlinear Timoshenko system with delay, we cite [5] and reference therein. As far as we know, there is no result for the Timoshenko system with nonlocal delay.
Let Ω = (0, L) be an interval on R. In this paper, ϕ = ϕ(x,t) describe the small transverse displacement of the beam and ψ = ψ(x,t) the rotation angle of a filament of the beam in Ω, respectively at the time t. For a constant c > 0 and F, G : (0, c) → R bounded functions, we define the nonlocal time delayed integral of the 1st kind condition by (1.1) These conditions (1.1) are called nonlocal because the integral is not a pointwise relation, so it provides a problem with distributed delay. Well-posedness and exponential stability for this kind of nonlocal time-delayed for a wave equation were studied in [17,22] by different techniques.
Let b, k, α, β be positive constants. The Timoshenko system with frictional damping and nonlocal time-delayed condition is given by We consider the Dirichlet boundary conditions as follows Here the initial data We use the Sobolev spaces with its properties as in Adams [1] and the semigroup theory ( see Pazy [18]). In this paper, we apply the semigroup technique for dissipative systems (see Liu and NONATO, RAPOSO and NGUYEN 397 Zheng [14]), that is different from some others in the literature, for example, like as the energy method (see Rivera [23]), the direct method (see Kormonik [11,12]) and the Nakao's method (see [15]). This manuscript is organized as follows. In Section 2, we deal with the semigroup setting where we prove the well-posedness of the system. In section 3, we show the exponential stability by using the Gearhart-Huang-Prüss theorem, [6,7,19].

SEMIGROUP SETUP
As in Nicaise and Pignotti [17] we introduce the new variables The new variables z, y satisfy Moreover, using the approach as in [16], the equations respectively. The problem (1.2)-(1.7) is equivalent to with the Dirichlet boundary condition (1.8) and z(x, ρ,t, s) = y(x, ρ,t, s) = 0 on the boundary x = 0, L.
Defining U = (ϕ, ψ, u, v, z, y) T , u = ϕ t and v = ψ t , we formally get that U satisfies the Cauchy problem where the operator A is defined by We introduce the energy space The domain of A is defined by Clearly, D(A ) is dense in H and independent of time t > 0. Next, we will prove that the operator A is dissipative. Proof.
The well-posedness of (2.5)-(2.12) is ensured by the following theorem. Proof. We will use the Hille-Yosida theorem. Since A is dissipative and D(A ) is dense in H , it is sufficient to show that A is maximal; that is, I − A is surjective. Given H = (h 1 , h 2 , . . . , h 6 ) ∈ H , we must show that there exists U = (ϕ, ψ, u, v, z, y) ∈ D(A ) satisfying (I − A )U = H which is equivalent to sy(x, ρ,t, s) + y ρ (x, ρ,t, s) = sh 6 . (2.26) Suppose that we have found ϕ and ψ with the appropriated regularity. Therefore, (2.21) and (2.22) give It is clear that u, v ∈ H 1 0 (Ω).
Applying the classical elliptic regularity, it follows from (2.30) that Therefore, the operator I − A is surjective. As consequence of the Hille-Yosida theorem [14, Theorem 1.2.2, page 3], we have that A generates a C 0 -semigroup of contractions S(t) = e tA on H . From semigroup theory, U(t) = e tA U 0 is the unique solution of (2.13) satisfying (2.19) and (2.20). The proof is complete.

EXPONENTIAL STABILITY
The necessary and sufficient conditions for the exponential stability of the C 0 -semigroup of contractions on a Hilbert space were obtained by Gearhart [6] and Huang [7] independently, see also Pruss [19]. We will use the following result due to Gearhart.
To prove (3.2) we use contradiction argument again. If (3.2) is not true, there exists a real sequence ζ n , with ζ n → ∞ and a sequence of vector functions V n ∈ H that satisfies (λ n I − A ) −1 V n H V n H ≥ n, where λ n = iζ n .
Hence (λ n I − A ) −1 V n H ≥ n V n H . (3.10) Since λ n ∈ ρ(A ) it follows that there exists a unique sequence U n = (ϕ n , ψ n , u n , v n , z n , y n ) T ∈ D(A ), with unit norm in H such that (λ n I − A ) −1 V n = U n .
Denoting ξ n = λ n U n − A U n we have from (3.10) that ξ n H ≤ 1 n and then ξ n → 0 strongly in H as n → ∞.