In this paper we study the control of an ill-posed system relating to the Cauchy problem for an elliptical operator. The control of Cauchy systems for an elliptical operator has already been studied by many authors. But it still seems to be globally an open problem. Of all the studies that have been done on this problem, it is assumed that the set of admissible couple-state must be nonempty to make sense of the problem. This is the case of J. L. Lions in [6] who gave various examples of the admissible set to make a sense of the problem. O. Nakoulima in [9] uses the regularization-penalization method to approach the problem by a sequence of well-posed control problems, he obtains the convergence of the processus in a particular case of the admissible set. G. Mophou and O. Nakoulima in [10] do the same study and obtain the convergence of the processus when the interior of the admissible set is non empty. In this work, we give an approximate solution without an additional condition on the set of admissible couple-state.We propose a method which consists in associating with the singular control problem a "family" of controls of well posed problems. We propose as an alternative the stackelberg control which is a multiple-objective optimization approach proposed by H. Von Stackelberg in [12].
| Published in |
American Journal of Applied Mathematics (Volume 8, Issue 5)
This article belongs to the Special Issue Numerical Analysis and Control Theory |
| DOI | 10.11648/j.ajam.20200805.15 |
| Page(s) | 271-277 |
| 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 |
Systems Governed by PDEs, Stackelberg Control, Cauchy Problem, Cost Function
| [1] | M. Barry, O. Nakoulima, G. B. Ndiaye, Cauchy System for Parabolic Operator, International Journal of Evolution Equations 8, no. 4 (2013) pp. 277-290. |
| [2] | M. Barry, G. B. Ndiaye, Cauchy system for an hyperbolic operator, Journal of Nonlinear Equations and Applications. Volume 2014, Number 4 pp. 37-52. |
| [3] | P. M. Fall, O. Nakoulima, A Sène: On a non-classical boundary values problem for the heat equation. African Diaspora Journal of mathematics (ADJM), Volume 16, Number 2, pp. 59-71 (2014). |
| [4] | J. P. Kernevez, Enzyme mathematics, North-Holland Publishing Company, 1980. |
| [5] | J. L. Lions. Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Dunod, Gauthier-Villars, Paris 1968. |
| [6] | J. L. Lions, Contrôle des systèmes distribués singuliers. BORDAS, Gauthier-Villars, Paris, 1983 ISBN 2-04-015539-2. |
| [7] | J. L. Lions and E. MAGENES, Problèmes aux limites non homogènes et applications, DUNOD, vol1 Paris, 1968. |
| [8] | J. L. Lions. Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal. Springer-Verlag. Berlin. Heidelberg. New York 1973. |
| [9] | O. Nakoulima, Contrôle de systèmes mal posés de type elliptique, J. Math. Pures et Appl., 1994, 73, 441-453. |
| [10] | O. Nakoulima, G. M. Mophou, Control of Cauchy System for an Elliptic Operator, Acta Mathematica Sinica (English Series) 25, no. 11 (2009) pp. 1819-1834. |
| [11] | J. Smoler, Shock waves, and reaction-diffusion equations, Springer Ver-lag, Berlin-Heidelberg, New York, 1983. |
| [12] | H. V. Stackelberg. Markform undGleichgewicht. Springer, Berlin, Ger-many 1934. |
APA Style
Sadou Tao. (2020). Control of Cauchy Problem for a Laplacian Operator. American Journal of Applied Mathematics, 8(5), 271-277. https://doi.org/10.11648/j.ajam.20200805.15
ACS Style
Sadou Tao. Control of Cauchy Problem for a Laplacian Operator. Am. J. Appl. Math. 2020, 8(5), 271-277. doi: 10.11648/j.ajam.20200805.15
AMA Style
Sadou Tao. Control of Cauchy Problem for a Laplacian Operator. Am J Appl Math. 2020;8(5):271-277. doi: 10.11648/j.ajam.20200805.15
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Download@article{10.11648/j.ajam.20200805.15,
author = {Sadou Tao},
title = {Control of Cauchy Problem for a Laplacian Operator},
journal = {American Journal of Applied Mathematics},
volume = {8},
number = {5},
pages = {271-277},
doi = {10.11648/j.ajam.20200805.15},
url = {https://doi.org/10.11648/j.ajam.20200805.15},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200805.15},
abstract = {In this paper we study the control of an ill-posed system relating to the Cauchy problem for an elliptical operator. The control of Cauchy systems for an elliptical operator has already been studied by many authors. But it still seems to be globally an open problem. Of all the studies that have been done on this problem, it is assumed that the set of admissible couple-state must be nonempty to make sense of the problem. This is the case of J. L. Lions in [6] who gave various examples of the admissible set to make a sense of the problem. O. Nakoulima in [9] uses the regularization-penalization method to approach the problem by a sequence of well-posed control problems, he obtains the convergence of the processus in a particular case of the admissible set. G. Mophou and O. Nakoulima in [10] do the same study and obtain the convergence of the processus when the interior of the admissible set is non empty. In this work, we give an approximate solution without an additional condition on the set of admissible couple-state.We propose a method which consists in associating with the singular control problem a "family" of controls of well posed problems. We propose as an alternative the stackelberg control which is a multiple-objective optimization approach proposed by H. Von Stackelberg in [12].},
year = {2020}
}
TY - JOUR T1 - Control of Cauchy Problem for a Laplacian Operator AU - Sadou Tao Y1 - 2020/09/21 PY - 2020 N1 - https://doi.org/10.11648/j.ajam.20200805.15 DO - 10.11648/j.ajam.20200805.15 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 271 EP - 277 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20200805.15 AB - In this paper we study the control of an ill-posed system relating to the Cauchy problem for an elliptical operator. The control of Cauchy systems for an elliptical operator has already been studied by many authors. But it still seems to be globally an open problem. Of all the studies that have been done on this problem, it is assumed that the set of admissible couple-state must be nonempty to make sense of the problem. This is the case of J. L. Lions in [6] who gave various examples of the admissible set to make a sense of the problem. O. Nakoulima in [9] uses the regularization-penalization method to approach the problem by a sequence of well-posed control problems, he obtains the convergence of the processus in a particular case of the admissible set. G. Mophou and O. Nakoulima in [10] do the same study and obtain the convergence of the processus when the interior of the admissible set is non empty. In this work, we give an approximate solution without an additional condition on the set of admissible couple-state.We propose a method which consists in associating with the singular control problem a "family" of controls of well posed problems. We propose as an alternative the stackelberg control which is a multiple-objective optimization approach proposed by H. Von Stackelberg in [12]. VL - 8 IS - 5 ER -
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