Uniform boundary stabilization for the finite difference discretization of the 1-D wave equation

Boujaoui, H. El; Maniar, Lahcen; Bouslous, H.

doi:10.1007/s13370-016-0406-3

Cited by 6 publications

(7 citation statements)

References 22 publications

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“…Proof: The first limit is proved in [5]. The second limit follows from the same argument together with Theorem 3.…”

Section: Semi-discretizations Of (1) In the X−variablementioning

confidence: 78%

“…The proof outlined in [5] is based off of the similar decomposition technique as in [23] yet the "direct Fourier filtering" is implemented to the control-free model, ξ = 0 in (1). The major drawback of the exponential stability results in both [5] and [23] is that the proof of the exponential stability result solely relies on an observability result of the control-free model. This together with the decomposition argument make the maximal decay rate analysis, and therefore the analysis of finding the optimal feedback gain to achieve the maximal decay rate, more complicated.…”

Section: Introductionmentioning

confidence: 99%

“…The techniques in the proof prevents the rigorous investigation of the maximal decay rate in terms of the numerical filtering parameter. Later on, with the direct Fourier filtering technique, as in [9], the exponential stability for the Finite Difference-based model is shown to be retained [5] as the discretization parameter approaches zero. The proof outlined in [5] is based off of the similar decomposition technique as in [23] yet the "direct Fourier filtering" is implemented to the control-free model, ξ = 0 in (1).…”

Section: Introductionmentioning

confidence: 99%

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Revisiting the Direct Fourier Filtering Technique for the Maximal Decay Rate of Boundary-damped Wave Equation by Finite Differences and Finite Elements

Ozer,

Emran

2023

Preprint

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The one-dimensional PDE model of the wave equation with a state feedback controller at its boundary, which describes wave dynamics of a wide-range of controlled mechanical systems, has exponentially stable solutions. However, it is known that the reduced models of the wave equation by the standard Finite Differences and Finite Elements suffer from the lack of exponential stability (and exact observability without a state feedback controller) uniformly as the discretization parameter tends to zero. This is due to the loss of uniform gap among the high-frequency eigenvalues as the discretization parameter tends to zero. One common remedy to overcome this discrepancy is the direct Fourier filtering of the reduced models, where the high-frequency spurious eigenvalues are filtered out. After filtering, besides from the strong convergency, the exponential decay rate, mimicking the one for the partial differential equation counterpart, can be retained uniformly. However, the existing results in the literature are solely based on an observability inequality of the control-free model, to which the filtering is implemented. Moreover, the decay rate as a function of the filtering parameter is implicit. In this paper, exponential stability results for both filtered Finite Difference and Finite Element reduced models are established directly by a Lyapunov-based approach and a thorough eigenvalue estimation.The maximal decay rate is explicitly provided as a function of the feedback gain and filtering parameter. Our results, expectedly, mimic the ones of the PDE counterpart uniformly as the discretization parameter tends to zero. Several numerical tests are provided to support our results. Mathematics Subject Classification (2010). 35Q60, 35Q93; 74F15; 35Q74, 93B52.

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“…Proof: The first limit is proved in [5]. The second limit follows from the same argument together with Theorem 3.…”

Section: Semi-discretizations Of (1) In the X−variablementioning

confidence: 78%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Revisiting the Direct Fourier Filtering Technique for the Maximal Decay Rate of Boundary-damped Wave Equation by Finite Differences and Finite Elements

Ozer,

Emran

2023

Preprint

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“…This phenomenon is due to the fact that the semi-discrete dynamics lead to high-frequency spurious solutions which propagate with arbitrary small velocity and make the discrete controls diverge when the mesh-size h goes to zero. We notice that the uniform controllability question is closely related to the uniform observability [3,8,16] and the uniform stabilization problems for the discrete systems [5,9,25].…”

Section: Abdeladim El Akri and Lahcen Maniarmentioning

confidence: 99%

“…. , ψ N ) and the matrix −∂ 2 h is defined by (9). Now, we expand the initial data ϕ 0 h , ϕ 1 h , ψ 0 h , ψ 1 h on the basis (e n (h)) 1≤n≤N given by (11)…”

Section: Abdeladim El Akri and Lahcen Maniarmentioning

confidence: 99%

Uniform indirect boundary controllability of semi-discrete <inline-formula><tex-math id="M1">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M2">\begin{document}$ d $\end{document}</tex-math></inline-formula> coupled wave equations

Akri¹,

Maniar²

2020

Mathematical Control &Amp; Related Fields

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In this paper, we treat the problem of uniform exact boundary controllability for the finite-difference space semi-discretization of the 1-d coupled wave equations with a control acting only in one equation. First, we show how, after filtering the high frequencies of the discrete initial data in an appropriate way, we can construct a sequence of uniformly (with respect to the mesh size) bounded controls. Thus, we prove that the weak limit of the aforementioned sequence is a control for the continuous system. The proof of our results is based on the moment method and on the construction of an explicit biorthogonal sequence.

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Uniform interior stabilization for the finite difference fully-discretization of the 1-D wave equation

Boujaoui

2018

Afr. Mat.

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Uniform boundary stabilization for the finite difference discretization of the 1-D wave equation

Cited by 6 publications

References 22 publications

Revisiting the Direct Fourier Filtering Technique for the Maximal Decay Rate of Boundary-damped Wave Equation by Finite Differences and Finite Elements

Revisiting the Direct Fourier Filtering Technique for the Maximal Decay Rate of Boundary-damped Wave Equation by Finite Differences and Finite Elements

Uniform indirect boundary controllability of semi-discrete <inline-formula><tex-math id="M1">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M2">\begin{document}$ d $\end{document}</tex-math></inline-formula> coupled wave equations

Uniform interior stabilization for the finite difference fully-discretization of the 1-D wave equation

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