Dynamics of Timoshenko system with time-varying weight and time-varying delay

Abstract

This paper is concerned with the well-posedness of global solution and exponential stability to the Timoshenko system subject with time-varying weights and time-varying delay. We consider two problems: full and partially damped systems. We prove existence of global solution for both problems combining semigroup theory with the Kato's variable norm technique. To prove exponential stability, we apply the Energy Method. For partially damped system the exponential stability is proved under assumption of equal-speed wave propagation in the transversal and angular directions. For full damped system the exponential stability is obtained without the hypothesis of equal-speed wave propagation.

Stabilization of swelling porous elastic soils with fluid saturation, time varying-delay and time-varying weights

“…There are also several studies considering nonlinear models with a delay where the existence of attractors is investigated, among them, Timoshenko systems [7,9,20,25], poroelastic systems [6] and suspension bridge [18,42]. Based on the work mentioned above about the problem of swelling of the one-dimensional porous elastic soils and in references [40,41], we design and propose to study the exponential stability for the following system…”

Equivalence between exponential stabilization and observability inequality for magnetic effected piezoelectric beams with time-varying delay and time-dependent weights

“…Besides, Nonato et al [27,28] considered exponential stablility for Timoshenko and thermoelastic laminated beam system with nonlinear weights and time-varying delay. Unlike previous works, the dampings δ 1 and δ 2 depend on time t. Under appropriate assumptions about the weights of the damping δ 1 and δ 2 , the authors obtained the exponential decay of system.…”

“…It is worth mentioning that although (1)-( 2) is formed by two coupled wave equations, one of them with a delay mechanism, it is not immediate that for systems of this nature, the exponential decay of the solution happens, once that there are systems formed by coupled wave equations with delay terms whose solution does not decay exponentially [29]. In some cases, such as the Timoshenko model, which is also a system formed by two coupled wave equations, when subjected to damping of the delay type in one of the equations, exponential decay is obtained, provided the equality of speeds occurs [16,27,37].…”

“…Lemma 2.3. [2,27] Let (v, p, z) be a solution to system (17)- (25). Then the energy functional defined by (26…”