Boundary stabilization of solutions of a nonlinear system of Timoshenko type

EI In fact, they proved that the uniform stability of (9) holds if and only if the wave speeds are equal: K ρ = Iρ ; otherwise only the asymptotic stability has been proved. This result improves earlier ones by Soufyane [19] and Shi and Feng [15], where an exponential decay of the solution energy of (7), together with two locally distributed feedbacks, had been proved. Xu and Yung [22] studied a system of Timoshenko beams with pointwise feedback controls, sought information about the eigenvalues and eigenfunctions of the system, and used this information to
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S.A. Messaoudi, A. Soufyane / Nonlinear Analysis 67 (2007) 2107–2121
where Ω is an open bounded set of Rn with a smooth boundary ∂ Ω = Γ0 ∪ Γ1 . Here Γ0 and Γ1 are closed and disjoint, with meas(Γ0 ) > 0, ν is the unit outer normal, u represents the deflection of the beam from its equilibrium and v is the total rotatory angle of the beam (see [21] for details), and gi (i = 1, 2) are the relaxation functions considered positive non-increasing belonging to W 1,2 (0, +∞). Eqs. (4) and (5) are the nonlocal boundary conditions responsible for the memory effect. Our main interest is in investigating the asymptotic behavior of the solutions of system (1)–(6); that is, whether the dissipation induced by the boundary conditions of the memory effect in (4) and (5) is strong enough to stabilize the system and what type of rate of decay we obtain. Let us mention some known results on the decay rates for the Timoshenko model. In [21], a simple one-dimensional model describing the transverse vibration of a beam is given by the following system of coupled hyperbolic equations: ρ u tt = ( K (u x − v))x , in (0, L ) × (0, +∞) Iρ vtt = ( E I vx )x + K (u x − v), in (0, L ) × (0, +∞), (7)
u (0, L ) = u ( L , t ) = v(0, t ) = v( L , t ) = 0,
and proved that the energy associated with (8) decays exponentially. This result is similar to the one given by Taylor [20] but, as they mentioned, the originality in their work lies in the method they used, which was developed by Liu and Zheng [11]. This method is different from the usual ones such as the classical energy method. It mainly uses the semigroup theory. Soufyane and Wehbe [18] showed that it is possible to stabilize (7) uniformly by using a unique locally distributed feedback. So, they considered ρ u tt = ( K (u x − v))x , in (0, L ) × (0, +∞) in (0, L ) × (0, +∞) ∀t > 0, (9) Iρ vtt = ( E I vx )x + K (u x − v) − bvt ,
Received 15 March 2006; accepted 28 August 2006
Abstract In this paper we are concerned with a multidimensional Timoshenko system subjected to boundary conditions of memory type. We establish general rate decay results. The usual exponential and polynomial decay rates are only special cases. c 2006 Elsevier Ltd. All rights reserved.
Nonlinear Analysis 67 (2007) 2107–2121 /locate/na
Boundary stabilization of solutions of a nonlinear system of Timoshenko type
Salim A. Messaoudi a , Abdelaziz Soufyane b,∗
where the coefficients ρ, Iρ , E , I and K are respectively the density (the mass per unit length), the polar moment of inertia of a cross section, Young’s modulus of elasticity, the moment of inertia of a cross section, and the shear modulus. Problem (7) has been studied by many mathematicians. Kim and Renardy [8] considered (7) together with twoboundary control of the form K v( L , t ) − K ∂u ∂u ( L , t ) = c ( L , t ), ∀t ≥ 0 ∂x ∂t ∂v ∂v E I ( L , t ) = −d ( L , t ), ∀t ≥ 0 ∂x ∂t
and used the multiplier techniques to establish an exponential decay result for the natural energy of (7). They also provided numerical estimates for the eigenvalues of the operator associated with system (7). An analogous result was also established by Feng et al. [6], where the stabilization of vibrations in a Timoshenko system was studied. Raposo et al. [13] studied (7) with homogeneous Dirichlet boundary conditions and two linear frictional dampings. To be precise, they looked into the following system: ρ1 u tt − K (u x − v)x + u t = 0, in (0, L ) × (0, +∞) in (0, L ) × (0, +∞) ∀t > 0 (8) ρ2 vtt − bvx x + K (u x − v) + vt = 0,
a Mathematical Sciences Department, KFUPM, Dhahran 31261, Saudi Arabia b ALHOSN University, Faculty of Engineering and Applied Sciences, PO Box 38772, Abu Dhabi, United Arab Emirates
0 t