Controllability of the Kirchhoff System for Beams as a Limit of the Mindlin–Timoshenko System

Araruna, F. D.; Zuazua, Enrique

doi:10.1137/060659934

Cited by 26 publications

(15 citation statements)

References 11 publications

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“…Controllability properties of these linear systems have also been studied. More precisely, using Fourier series decompositions and filtering techniques, it was shown in [9] that the exact controllability of the Kirchhoff system may be obtained as the limit, when k → ∞, of a partial controllability property for the Mindlin-Timoshenko system. Note however that these techiques, based in the Fourier analysis, cannot be applied in the present nonlinear context.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic limits and stabilization for the 1D nonlinear Mindlin-Timoshenko system

2010

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This paper shows how the so called von Kármán model can be obtained as a singular limit of a modified Mindlin-Timoshenko system when the modulus of elasticity in shear k tends to infinity, provided a regularizing term through a fourth order dispersive operator is added. Introducing damping mechanisms, the authors also show that the energy of solutions for this modified Mindlin-Timoshenko system decays exponentially, uniformly with respect to the parameter k. As k → ∞, the authors obtain the damped von Kármán model with associated energy exponentially decaying to zero as well.

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Section: Introductionmentioning

confidence: 99%

“…For the linear case, that is, in the absence of the term ψ x η x + 1 2 ψ 2 x x , and without the equation for η, it was proved in [3] (see also [9]) that the linear Mindlin-Timoshenko system…”

Section: Introductionmentioning

confidence: 99%

Asymptotic limits and stabilization for the 1D nonlinear Mindlin-Timoshenko system

2010

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“…For the linear case, that is, in the absence of the term ψ x η x + 1 2 ψ 2 x x , and without the equation for η, it was proved in [8] (see also [1]) that the linear Mindlin-Timoshenko system…”

mentioning

confidence: 97%

“…Controllability properties of these systems have also been studied in [1]. The problem of singular perturbations and stabilization related to the nonlinear von Kármán model has also been intensively investigated.…”

mentioning

confidence: 99%

Asymptotics and stabilization for dynamic models of nonlinear beams

Araruna¹,

Silva²,

Zuazua³

2010

Proc. Estonian Acad. Sci.

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We prove that the von Kármán model for vibrating beams can be obtained as a singular limit of a modified Mindlin-Timoshenko system when the modulus of elasticity in shear k tends to infinity, provided a regularizing term through a fourth-order dispersive operator is added. We also show that the energy of solutions for this modified Mindlin-Timoshenko system decays exponentially, uniformly with respect to the parameter k, when suitable damping terms are added. As k → ∞, one deduces the uniform exponential decay of the energy of the von Kármán model.

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“…The observability is obtained by a so-called spectral compensation argument, which states that the bad behavior of the part of the spectrum which accumulates to σ ess (A) is somehow compensated by the suitable gap of the discrete part. In a different context, we also mention [3,17,19] for the controllability of systems with spectral accumulation point.…”

Section: Introduction and Problem Statement Let ω = (0 1)mentioning

confidence: 99%

Exact Boundary Controllability of a System of Mixed Order with Essential Spectrum

Khodja¹,

Mauffrey²,

Münch³

2011

SIAM J. Control Optim.

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Abstract. We address in this work the exact boundary controllability of a linear hyperbolic system of the form u ′′ + Au =0withu =( u 1 ,u 2 )T posed in (0,T) × (0, 1) 2 . A denotes a self-adjoint operator of mixed order that usually appears in the modelization of a linear elastic membrane shell. The operator A possesses an essential spectrum which prevents the exact controllability from holding uniformly with respect to the initial data u 0 ,u 1 . We show that the exact controllability holds by one Dirichlet control acting on the first variable u 1 for any initial data u 0 ,u 1 generated by the eigenfunctions corresponding to the discrete part of the spectrum of A. The proof relies on a suitable observability inequality obtained by way of a full spectral analysis and the adaptation of an Inghamtype inequality for the Laplacian in two spatial dimensions. This work provides a nontrivial example of a system controlled by a number of controls strictly lower than the number of components. Some numerical experiments illustrate our study.

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Controllability of the Kirchhoff System for Beams as a Limit of the Mindlin–Timoshenko System

Cited by 26 publications

References 11 publications

Asymptotic limits and stabilization for the 1D nonlinear Mindlin-Timoshenko system

Asymptotic limits and stabilization for the 1D nonlinear Mindlin-Timoshenko system

Asymptotics and stabilization for dynamic models of nonlinear beams

Exact Boundary Controllability of a System of Mixed Order with Essential Spectrum

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