Stabilization for partially dissipative laminated beams with non-constant coefficients
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Stabilization for partially dissipative laminated beams with non-constant coefficients M. S. Alves and R. N. Monteiro
Abstract. We address the stability of solutions corresponding to weakly damped two-layered Timoshenko beams with space variable coefficients. The mathematical model is composed of a system of coupled wave equations and describes the slip effect produced by a thin adhesive layer joining the beams. For the considered PDE system, combined with different boundary conditions, the main result is that the decay rates for the energy depend on local equality of the wave propagation velocity. The latter is achieved by introducing new observability estimates for the present model. We also study the problem with constant coefficients. Here, two main results are obtained: (i) for mixed boundary condition type, we characterize the exponential decay rate by assuming the equality of the wave propagation velocity and (ii) without imposing this assumption, we establish optimal rational decay rates. Mathematics Subject Classification. 35 L 70, 35 B 40. Keywords. Laminated Timoshenko systems, Dissipative systems, Exponential stability, Rational decay rate.
1. Introduction The PDE model: This discussion is devoted to the layered Timoshenko system resulted from the composition of two identical beams joined by a thin layer of adhesive in the interface. The resulting model is written for variables (w, ψ, s) = (w(x, t), ψ(x, t), s(x, t)) denoting the transverse displacement, the rotation angle produced by the beam deflection and the slip along the interface at time t and longitudinal spatial variable x, respectively. With these variables, as derived in [11,12], the laminated beam—in terms of equations—is written as follows ρwtt + G(ψ − wx )x = 0 in (0, ) × (0, ∞), Iρ (3s − ψ)tt − G(ψ − wx ) − D(3s − ψ)xx = 0 in (0, ) × (0, ∞),
(1)
3Iρ stt + 3G(ψ − wx ) + 4γs + 4βst − 3Dsxx = 0 in (0, ) × (0, ∞). The positive physical parameters ρ, G, Iρ , D, γ, β describe the density of the beam, the shear stiffness, the mass moment of inertia, the flexural rigidity, the adhesive stiffness of the beams and the adhesive damping parameter, respectively. To establish the model of the present manuscript, we first resort to results on the system (1). It has been shown in [31] that system (1), without assuming equal speeds of propagation of waves, does not reach the exponential stability. This last, motivated us to consider the weakly damped model with variable coefficients, that is, we study the problem with an extra mechanical dissipation acting only on the equation for the transverse displacement w and with physical parameters depending on the spatial 0123456789().: V,-vol
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variable x ∈ (0, ). In this direction, we are interested in studying the problem given by ρwtt + [G(3s − ξ − wx )]x + αwt = 0 in (0, ) × (0, ∞), Iρ ξtt − G(3s − ξ − wx ) − [Dξx ]x = 0 in (0, ) × (0, ∞),
(2)
3Iρ stt + 3G(3s − ξ − wx ) + 4γs + 4βst − 3[Dsx ]x = 0 in (0, ) × (0,
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