Exponential stabilization of laminated beams with history memories

Abstract

In this paper we consider laminated beams modelled from the well established Timoshenko system, which is a structure given by two identical layers uniform on top of each other, taking into account that an adhesive of small thickness is bonding the two surfaces and produces an interfacial slip. By using semi-group approach, we prove the global well-posedness of the system when a viscoelastic dissipation acts on the three equations. In addition, we prove that the dissipation through memory terms is strong enough to get exponential decay of the system without the condition of equal wave speeds.

Laminated Timoshenko beams with interfacial slip and infinite memories

“…Namely, under some restrictions on the parameters and with one or two kernels converging exponentially to zero at infinity, the exponential stability was proved in Lo and Tatar 1,8,22 . Similar stability results with finite or infinite memory terms were obtained in previous studies 23–27 …”

“…1,8,22 Similar stability results with finite or infinite memory terms were obtained in previous studies. [23][24][25][26][27] For the stability of Bresse systems 28 with infinite memories, we refer the readers to Guesmia 29,30 and Guesmia and Kirane, 31,32 and the references therein.…”

Exponential and polynomial stabilization of laminated beams with two history memories

“…The laminated beam with history memory was firstly studied in [12], in which the author considered three laminated system by a viscoelastic damping generated by an infinite memory and acting only on one equation and proved that if the infinite memory is effective on transverse displacement, the model is not exponentially stable independently of the values of the parameters, but the exponential stability is equivalent to the equality between the three speeds of wave propagations if the infinite memory is effective on the second or the third equation. Feng et al [7] showed that when the system is influenced by three histories, one for each equation, it is exponentially stable without any restrictions on the coefficients. Guesmia et al [13] considered a laminated beam when two infinite memories are effective on the first and the second equations in (1) and proved a general stability of the system without any restrictions on the parameters by assuming a wider class of memory effect.…”

Nonuniform laminated beam of Lord–Shulman type

“…As can be seen from the existing literature, in the theory of laminated beams, several Dirichlet-Neumann conditions can be considered 7,11 , as well as some boundary feedback control 6,29 . In this paper, we adopt the boundary conditions described in (3) due to their physical nature and also for technical reasons.…”

“…In this paper, we study the qualitative properties of solutions of system ( 5)- (11). More precisely, we prove that the system is globally well-posed by using the semigroup theory of linear operators together with the Lumer-Phillips theorem.…”