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Carleman Estimate for a Singulary Perturbed Degenerated Parabolic Equation

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), 2021. Published by Science Publishing Group

Keywords

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

References
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[2] P. Cannarsa, P.Martinez and J. Vancostenoble, Global Carleman Estimates for Degenerate Parabolic Operators with Applications, American Mathematical Society,Vol 329, Num. 1133, 2016.
[3] J. M. Coron and Guerrero S, Null Controllability of the N-Dimensional Stokes system with N-1 scolar controls, J. of Diff. Equ., 2009 246 2908-21.
[4] J. M. Coron and S. Guerrero, Singular optimal control: a linear 1D parabolic-hyperbolic: example Ssymptotic Anal 44 237-57.
[5] J. M. Coron and S. Guerrero, Null Controllability of the N-Dimensional Stokes system with N-1 scolar controls, J. Diff. Equ. 2009, 246 2908-21.
[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.
[12] G. Lebeau, L. Robbiano, Contrˆ ole exact de l’équation de la chaleur, Séminiare Equations aux dérivées partielles (dit”Goulaouic- Schwarttz), ppp 1-11, 94-95.
[13] J. L. Lions, Controlabilé exacte, perturbations et stabilisation de systèmes distribués; Perturbations, Recherche en Mathématiques Appliquées. Masson, 1998.
[14] K. Mauffrey, Contrˆ olabilité de systèmes gouvernés par des ´ quations aux dérivées partielles, Analysis of PDEs [math.AP]. Université de Franche-Comté, 2012. French.
[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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    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
  • Department of Mathematics, Faculty of Sciences and Technologies, Cheikh Anta Diop University, Dakar, Senegal

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

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

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

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