A localized nonstandard stabilizer for the Timoshenko beam

Tebou, Louis

doi:10.1016/j.crma.2015.01.004

Cited by 14 publications

(2 citation statements)

References 16 publications

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“…Wehbe and Youssef in [41] proved that the Timoshenko system with one locally distributed viscous feedback is exponentially stable if and only if the wave propagation speeds are equal (i.e., k1 ρ1 = ρ2 k2 ), otherwise, only the polynomial stability holds. Tebou in [38] showed that the Timoshenko beam with same feedback control in both equations is exponentially stable. The stability of the Timoshenko system with thermoelastic dissipation has been studied in [36], [12], [13], and [15].…”

Section: Introductionmentioning

confidence: 99%

A transmission problem for the Timoshenko system with one local Kelvin-Voigt damping and non-smooth coefficient at the interface

Ghader¹,

Wehbe²

2020

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In this paper, we study the indirect stability of Timoshenko system with local or global Kelvin-Voigt damping, under fully Dirichlet or mixed boundary conditions. Unlike [43] and [39], in this paper, we consider the Timoshenko system with only one locally or globally distributed Kelvin-Voigt damping D (see System (1.1)). Indeed, we prove that the energy of the system decays polynomially of type t −1 and that this decay rate is in some sense optimal. The method is based on the frequency domain approach combining with multiplier method.

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Section: Introductionmentioning

confidence: 99%

A transmission problem for the Timoshenko system with one local Kelvin-Voigt damping and non-smooth coefficient at the interface

Ghader¹,

Wehbe²

2020

Preprint

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“…In addition to the previously cited papers. The stability of the Timoshenko system with a different kind of damping has been also studied [15,69,31,33,29,56,30,66,36,21,22,1]. For the stabilization of the Timoshenko beam with nonlinear term, we mention [57,12,55,17,55,27,36].…”

Section: Introductionmentioning

confidence: 99%

Stability and Controllability results for a Timoshenko system

Akil,

Chitour,

Ghader

et al. 2019

Preprint

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In this paper, we study the indirect boundary stability and exact controllability of a one-dimensional Timoshenko system. In the first part of the paper, we consider the Timoshenko system with only one boundary fractional damping. We first show that the system is strongly stable but not uniformly stable. Hence, we look for a polynomial decay rate for smooth initial data. Using frequency domain arguments combined with the multiplier method, we prove that the energy decay rate depends on coefficients appearing in the PDE and on the order of the fractional damping. Moreover, under the equal speed propagation condition, we obtain the optimal polynomial energy decay rate. In the second part of this paper, we study the indirect boundary exact controllability of the Timoshenko system with mixed Dirichlet-Neumann boundary conditions and boundary control. Using non-harmonic analysis, we first establish a weak observability inequality, which depends on the ratio of the waves propagation speeds. Next, using the HUM method, we prove that the system is exactly controllable in appropriate spaces and that the control time can be small.

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