On the exponential stabilization of a flexible structure with dynamic delayed boundary conditions via one boundary control only

Chentouf, Boumediène; Mansouri, Sabeur

doi:10.1016/j.jfranklin.2020.10.027

Cited by 8 publications

(11 citation statements)

References 34 publications

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“…Exponential stability. The main objective of this section is to prove the exponential stability of the unique solution to the closed-loop system (10). Moreover, a decay rate will be provided by proceeding as in [30,31].…”

Section: Well-posedness Of the Problem Let Us First Introduce A New State Variablementioning

confidence: 99%

“…Note that γ and C 0 are appropriate positive constants that will be fixed later on, while ε is an arbitrary positive constant. Moreover, F is a negative function defined by (see [34] and [2,3,9,10,12] for similar situations)…”

Section: Well-posedness Of the Problem Let Us First Introduce A New State Variablementioning

confidence: 99%

“…Then, the exponential convergence outcome is proved to the detriment of the presence of two extra controls, namely, one interior damping of type Kelvin-Voigt (y xt ) x (x, t) and a second boundary control (applied at the other end of the cable) that contains two terms y xt (1, t) and y t (1, t) [8]. Motivated by this last study, the authors in [10] managed to considerably improve the stability result of [8] by showing that the Kelvin-Voigt damping (y xt ) x (x, t) and the boundary control at x = 1 are both superfluous.…”

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confidence: 96%

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Boundary stabilization of a flexible structure with dynamic boundary conditions via one time-dependent delayed boundary control

Chentouf¹,

Mansouri²

2022

DCDS-S

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<p style='text-indent:20px;'>This article deals with the dynamic stability of a flexible cable attached at its top end to a cart and a load mass at its bottom end. The model is governed by a system of one partial differential equation coupled with two ordinary differential equations. Assuming that a time-dependent delay occurs in one boundary, the main concern of this paper is to stabilize the dynamics of the cable as well as the dynamical terms related to the cart and the load mass. To do so, we first prove that the problem is well-posed in the sense of semigroups theory provided that some conditions on the delay are satisfied. Thereafter, an appropriate Lyapunov function is put forward, which leads to the exponential decay of the energy as well as an estimate of the decay rate.</p>

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Section: Well-posedness Of the Problem Let Us First Introduce A New State Variablementioning

confidence: 99%

Section: Well-posedness Of the Problem Let Us First Introduce A New State Variablementioning

confidence: 99%

mentioning

confidence: 96%

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Boundary stabilization of a flexible structure with dynamic boundary conditions via one time-dependent delayed boundary control

Chentouf¹,

Mansouri²

2022

DCDS-S

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“…2) The reader is also referred to [6,8,10,19,38] for the modeling of other cases of crane systems. As a matter of fact, there exists in literature a considerable number of research papers that studied the system (1.1) or its variants [36].…”

mentioning

confidence: 99%

“…As a matter of fact, there exists in literature a considerable number of research papers that studied the system (1.1) or its variants [36]. However, instead of discussing a rich literature on the subject and going over dozens of articles, we refer the reader to the recent works [6,8,19], where considerable efforts have been made in this regard. In turn, it is worth mentioning that, to the authors' best knowledge, no attempt has been made concerning the stabilization of the crane system (1.1) when the control U (t) is nonlinear and at the same time subject to a nonlinear delay term.…”

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confidence: 99%

On the stabilization of a flexible structure via a nonlinear delayed boundary control

Chentouf

Feng

2022

DCDS-B

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<p style='text-indent:20px;'>The main concern of this article is to investigate the stabilization problem of a flexible structure with one dynamical boundary condition subject to two nonlinearities in the proposed boundary control. Specifically, the first nonlinearity is related to the velocity term, while the second one arises from the delayed term. Despite such a situation, it is shown that the system has a global solution by means of semigroups theory of nonlinear operators. More importantly, we prove that the system energy decays. An estimate of the decay rate of the energy is explicitly provided depending on the type of the nonlinearities.</p>

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Well‐posedness and stability of a nonlinear time‐delayed dispersive equation via the fixed point technique: A case study of no interior damping

Ammari

Chentouf

Smaoui

2022

Math Methods in App Sciences

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The primary concern of this article is to establish the well‐posedness as well as the exponential stability of the zero solution to a nonlinear time‐delayed dispersive equation of order four in a bounded interval. The main ingredient of the proof is the exploitation of Schauder Fixed Point Theorem. This outcome considerably improves an earlier result of Ammari et al. (2021) in the sense that we prove that no interior damping control is required. Furthermore, our results remain valid regardless of the sign of the antidiffusion parameter. Numerical simulations are also given as an illustration of our theoretical result.

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On the exponential stabilization of a flexible structure with dynamic delayed boundary conditions via one boundary control only

Cited by 8 publications

References 34 publications

Boundary stabilization of a flexible structure with dynamic boundary conditions via one time-dependent delayed boundary control

Boundary stabilization of a flexible structure with dynamic boundary conditions via one time-dependent delayed boundary control

On the stabilization of a flexible structure via a nonlinear delayed boundary control

Well‐posedness and stability of a nonlinear time‐delayed dispersive equation via the fixed point technique: A case study of no interior damping

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