Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks

Kirane, Mokhtar; Said‐Houari, Belkacem; Anwar, M. Naim

doi:10.3934/cpaa.2011.10.667

Cited by 67 publications

(43 citation statements)

References 16 publications

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“…For a similar result, we refer to Datko et al (1986), Datko (1988), Datko (1993) and Xu et al (2006). If α j = 0 and β j = 0 but α j > β j > 0, we can assert from known results that the system can be stabilised exponentially by the velocity and angle velocity feedback for any τ > 0, here we refer to Said-Houari and Laskri (2010), Kirane et al (2011) and Said-Houari and Soufyane (2012) for Timoshenko beam with interior and boundary feedback controls. But when α j < β j , the closed loop systems are unstable.…”

Section: Dα(s)u(t + S)mentioning

confidence: 89%

Stabilisation of Timoshenko beam system with delay in the boundary control

Wang

2013

International Journal of Control

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In this paper, we consider the exponential stabilisation problem of a Timoshenko beam with delay in boundary control. Suppose that the controller outputs of the forms α 1 u 1 (t) + β 1 u 1 (t − τ ) and α 2 u 2 (t) + β 2 u 2 (t − τ ) where u 1 (t) and u 2 (t) are the inputs of boundary controllers. In the past, most of the stabilisation results are required α j > β j > 0, j = 1, 2. In the present paper, we shall give a new dynamic feedback control law that makes the system exponential stabilisation ∀τ > 0 provided that |α j | = |β j |(j = 1, 2). The exponential stabilisation result is proved via test of exact observability of the system.

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Section: Dα(s)u(t + S)mentioning

confidence: 89%

Stabilisation of Timoshenko beam system with delay in the boundary control

Wang

2013

International Journal of Control

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“…The work [28] has been extended to the case of time-varying delay of the form ( , − ( )) by Kirane, Said-Houari and Anwar [12]. First, by using the variable norm technique of Kato, and under some restrictions on the parameters 1 , 2 and on the delay function ( ), the system has been shown to be well-posed.…”

Section: Introductionmentioning

confidence: 98%

A stability result of a Timoshenko system in thermoelasticity of second sound with a delay term in the internal feedback

Ouchenane

2014

Georgian Mathematical Journal

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In this paper, we consider a one-dimensional linear thermoelastic system of Timoshenko type with a delay term in the feedback. The heat conduction is given by Cattaneo's law. Under an appropriate assumption between the weight of the delay and the weight of the damping, we prove the well-posedness of the problem. Furthermore, an exponential stability result is shown without the usual assumption on the wave speeds. To achieve our goals, we make use of the semigroup method and the energy method.

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“…The feedback control laws were designed to stabilise system under different conditions, and many excellent results were obtained, such as the exponential stability (Kim & Renardy, 1987;Kirane, Said-Houari, & Anwar, 2011;Liu, Zhang, Han, & Xu, 2015;Morgül, 1992;Shi, Hou, & Feng, 1998;Zhang, Dawson, De Queiroz, & Vedagarbha, 1997), the spectral asymptotic expressions (Shubov, 1999;Vu, Wang, Xu, & Yung, 2005) and the Riesz basis property (Xu, 2005;Xu & Feng, 2002;Xu & Wang, 2013). These works mentioned above are based on the ideal situation without uncertainties coming from either the internal or the external disturbances.…”

Section: Introductionmentioning

confidence: 99%

Stabilisation of Timoshenko beam system with a tip payload under the unknown boundary external disturbances

Liu

Zhang

2015

International Journal of Control

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This paper is concerned with the boundary stabilisation of a Timoshenko beam with a tip payload under the boundary external disturbances. Nonlinear feedback control laws are designed to reduce the effects of the external disturbances. Since the controlled system is nonlinear, the well posedness of the nonlinear closed-loop system is investigated using the theory of nonlinear maximal monotone operators and the variational principle. In addition, we show the exponential stability of the nonlinear controlled system using the Lyapunov direct method. In the end, the numerical simulations are displayed to illustrate the effectiveness of the proposed boundary controls.

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Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks

Cited by 67 publications

References 16 publications

Stabilisation of Timoshenko beam system with delay in the boundary control

Stabilisation of Timoshenko beam system with delay in the boundary control

A stability result of a Timoshenko system in thermoelasticity of second sound with a delay term in the internal feedback

Stabilisation of Timoshenko beam system with a tip payload under the unknown boundary external disturbances

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