Carleman Estimate for a Singulary Perturbed Degenerated Parabolic Equation

Zeine Sid Elemine; Ibrahima Faye; Alassane Sy; Diaraf Seck

doi:doi:10.11648/j.sjams.20210905.11

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Carleman Estimate for a Singulary Perturbed Degenerated Parabolic Equation

Zeine Sid Elemine, Ibrahima Faye, Alassane Sy, Diaraf Seck

Published in Science Journal of Applied Mathematics and Statistics (Volume 9, Issue 5)

Received: 23 July 2021 Accepted: 18 August 2021 Published: 29 October 2021

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Abstract

In this paper, we are concerned with the the internal control of an elliptic singularly perturbed degenerated parabolic equation. This parabolic equation models sand transport problem near the coast in areas subjected to the tide. We study first the null controllability result of the parabolic equation modeling sand transport equation.The limit problem obtained by homogenization problem is also considered. We use distributed and bounded controls supported on a small open set of the initial domain. We prove the null controllability of the system at any time by using observability inequality for both problem. For this purpose, a specific carleman estimate for the solutions of degenerate adjoint limit problem is also proved.

Published in	Science Journal of Applied Mathematics and Statistics (Volume 9, Issue 5)
DOI	10.11648/j.sjams.20210905.11
Page(s)	113-125
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

PDE Parabolic, Null Controllability, Sand Transport Model, Carlman Estimates, Observability Inequality

References

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[4]	J. M. Coron and S. Guerrero, Singular optimal control: a linear 1D parabolic-hyperbolic: example Ssymptotic Anal 44 237-57.
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[6]	A. Dubova, A. Osses and J. P.Puel, Exact controllability to trajectoire for semi linear heat equations with discontinuous diffusion coefficient ESAIM: Control option calc, 2002 Var 8 621-61.
[7]	A. Dubova, E. Fernandez-Cara, M. Gonzales-Burgos, Controlability results for discontinuous semilinear parabolic partial differenntial equations, C. R. Acad. Sci., t. 326, p. 1391-1395, 1998.
[8]	A. Dubova, E. Fernandez-Cara, M. Gonzales-Burgos and E. Zuazua, On the controlability of parabolic systems withanonlinearterminvolvingthestateandthegradient, SIAM J. Control. Optim, Vol. 41, N. 3. p. 798-819, 2002.
[9]	I. Faye, E. Frénod, D. Seck, Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment, Discrete and Continuous Dynamical Systems, Vol 29 No3 March 2011, pp 1001-1030.
[10]	E. Frénod, Raviart P. A., and E. Sonnendrücker, Asymptotic expansion of the Vlasov equation in a large external magnetic field, J. Math. Pures et Appl. 80 (2001), 815–843.
[11]	A. V. Fursikov, O. Y. Imanuvilov, Controlability of evolution equations, Suhak Kanguirok, Seoul National University, 1996.
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[13]	J. L. Lions, Controlabilé exacte, perturbations et stabilisation de systèmes distribués; Perturbations, Recherche en Mathématiques Appliquées. Masson, 1998.
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[15]	D. L. Russel, Controlability ans stability theory for linear partial differential equations: recent progress and open questions. Siam Review, 20 (4), pp 639-739, 1978.
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APA Style

Zeine Sid Elemine, Ibrahima Faye, Alassane Sy, Diaraf Seck. (2021). Carleman Estimate for a Singulary Perturbed Degenerated Parabolic Equation. Science Journal of Applied Mathematics and Statistics, 9(5), 113-125. https://doi.org/10.11648/j.sjams.20210905.11

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ACS Style

Zeine Sid Elemine; Ibrahima Faye; Alassane Sy; Diaraf Seck. Carleman Estimate for a Singulary Perturbed Degenerated Parabolic Equation. Sci. J. Appl. Math. Stat. 2021, 9(5), 113-125. doi: 10.11648/j.sjams.20210905.11

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AMA Style

Zeine Sid Elemine, Ibrahima Faye, Alassane Sy, Diaraf Seck. Carleman Estimate for a Singulary Perturbed Degenerated Parabolic Equation. Sci J Appl Math Stat. 2021;9(5):113-125. doi: 10.11648/j.sjams.20210905.11

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@article{10.11648/j.sjams.20210905.11,
  author = {Zeine Sid Elemine and Ibrahima Faye and Alassane Sy and Diaraf Seck},
  title = {Carleman Estimate for a Singulary Perturbed Degenerated Parabolic Equation},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {9},
  number = {5},
  pages = {113-125},
  doi = {10.11648/j.sjams.20210905.11},
  url = {https://doi.org/10.11648/j.sjams.20210905.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20210905.11},
  abstract = {In this paper, we are concerned with the the internal control of an elliptic singularly perturbed degenerated parabolic equation. This parabolic equation models sand transport problem near the coast in areas subjected to the tide. We study first the null controllability result of the parabolic equation modeling sand transport equation.The limit problem obtained by homogenization problem is also considered. We use distributed and bounded controls supported on a small open set of the initial domain. We prove the null controllability of the system at any time by using observability inequality for both problem. For this purpose, a specific carleman estimate for the solutions of degenerate adjoint limit problem is also proved.},
 year = {2021}
}

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TY  - JOUR
T1  - Carleman Estimate for a Singulary Perturbed Degenerated Parabolic Equation
AU  - Zeine Sid Elemine
AU  - Ibrahima Faye
AU  - Alassane Sy
AU  - Diaraf Seck
Y1  - 2021/10/29
PY  - 2021
N1  - https://doi.org/10.11648/j.sjams.20210905.11
DO  - 10.11648/j.sjams.20210905.11
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 113
EP  - 125
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20210905.11
AB  - In this paper, we are concerned with the the internal control of an elliptic singularly perturbed degenerated parabolic equation. This parabolic equation models sand transport problem near the coast in areas subjected to the tide. We study first the null controllability result of the parabolic equation modeling sand transport equation.The limit problem obtained by homogenization problem is also considered. We use distributed and bounded controls supported on a small open set of the initial domain. We prove the null controllability of the system at any time by using observability inequality for both problem. For this purpose, a specific carleman estimate for the solutions of degenerate adjoint limit problem is also proved.
VL  - 9
IS  - 5
ER  -

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Author Information

Zeine Sid Elemine

Department of Mathematics, Faculty of Sciences and Technologies, Cheikh Anta Diop University, Dakar, Senegal
Ibrahima Faye

Department of Mathematics, Faculty of Applied Sciences and Communications Technologies, Alioune Diop University, Bambey, Senegal
Alassane Sy

Department of Mathematics, Faculty of Applied Sciences and Communications Technologies, Alioune Diop University, Bambey, Senegal
Diaraf Seck

Department of Mathematics of Decision, Faculty of Economics and Management, Cheikh Anta Diop University, Dakar, Senegal

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Plain Text BibTeX RIS

APA Style

Zeine Sid Elemine, Ibrahima Faye, Alassane Sy, Diaraf Seck. (2021). Carleman Estimate for a Singulary Perturbed Degenerated Parabolic Equation. Science Journal of Applied Mathematics and Statistics, 9(5), 113-125. https://doi.org/10.11648/j.sjams.20210905.11

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ACS Style

Zeine Sid Elemine; Ibrahima Faye; Alassane Sy; Diaraf Seck. Carleman Estimate for a Singulary Perturbed Degenerated Parabolic Equation. Sci. J. Appl. Math. Stat. 2021, 9(5), 113-125. doi: 10.11648/j.sjams.20210905.11

Copy | Download

AMA Style

Zeine Sid Elemine, Ibrahima Faye, Alassane Sy, Diaraf Seck. Carleman Estimate for a Singulary Perturbed Degenerated Parabolic Equation. Sci J Appl Math Stat. 2021;9(5):113-125. doi: 10.11648/j.sjams.20210905.11

Copy | Download

@article{10.11648/j.sjams.20210905.11,
  author = {Zeine Sid Elemine and Ibrahima Faye and Alassane Sy and Diaraf Seck},
  title = {Carleman Estimate for a Singulary Perturbed Degenerated Parabolic Equation},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {9},
  number = {5},
  pages = {113-125},
  doi = {10.11648/j.sjams.20210905.11},
  url = {https://doi.org/10.11648/j.sjams.20210905.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20210905.11},
  abstract = {In this paper, we are concerned with the the internal control of an elliptic singularly perturbed degenerated parabolic equation. This parabolic equation models sand transport problem near the coast in areas subjected to the tide. We study first the null controllability result of the parabolic equation modeling sand transport equation.The limit problem obtained by homogenization problem is also considered. We use distributed and bounded controls supported on a small open set of the initial domain. We prove the null controllability of the system at any time by using observability inequality for both problem. For this purpose, a specific carleman estimate for the solutions of degenerate adjoint limit problem is also proved.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Carleman Estimate for a Singulary Perturbed Degenerated Parabolic Equation
AU  - Zeine Sid Elemine
AU  - Ibrahima Faye
AU  - Alassane Sy
AU  - Diaraf Seck
Y1  - 2021/10/29
PY  - 2021
N1  - https://doi.org/10.11648/j.sjams.20210905.11
DO  - 10.11648/j.sjams.20210905.11
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 113
EP  - 125
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20210905.11
AB  - In this paper, we are concerned with the the internal control of an elliptic singularly perturbed degenerated parabolic equation. This parabolic equation models sand transport problem near the coast in areas subjected to the tide. We study first the null controllability result of the parabolic equation modeling sand transport equation.The limit problem obtained by homogenization problem is also considered. We use distributed and bounded controls supported on a small open set of the initial domain. We prove the null controllability of the system at any time by using observability inequality for both problem. For this purpose, a specific carleman estimate for the solutions of degenerate adjoint limit problem is also proved.
VL  - 9
IS  - 5
ER  -

Copy | Download