Research ArticleOpen AccessGoogle Scholar indexed
Boundary Control for Cooperative Elliptic Systems under Conjugation Conditions
Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City, Cairo, Egypt
Department of Mathematics, Faculty of Science, Al-Azhar University [for Girls], Nasr City, Cairo, Egypt
Department of Mathematics, Faculty of Science, Ibb University, Ibb, Yemen
- 1 Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City, Cairo, Egypt
- 2 Department of Mathematics, Faculty of Science, Al-Azhar University [for Girls], Nasr City, Cairo, Egypt
- 3 Department of Mathematics, Faculty of Science, Ibb University, Ibb, Yemen
Advances in Pure Mathematics·Volume 11 (2021)·Pages 457–471·Published 17 May 2021·DOI10.4236/apm.2021.115032
Copy link · social · email
Abstract
The existence and uniqueness of the state for 2 × 2 Dirichlet cooperative elliptic systems under conjugation conditions are proved using Lax-Milgram lemma, then the boundary control for these systems is discussed. The set of equations and inequalities that characterizes this boundary control is found by theory of Lions, Sergienko and Deineka. The problem for cooperative Neumann elliptic systems under conjugation conditions is also considered. Finally, the problem for n × n cooperative elliptic systems under conjugation conditions is established.
KeywordsCooperative SystemsConjugation ConditionsDirichlet and Neumann ConditionsExistence and Uniqueness of SolutionsBoundary Control
- Lions, J.L. (1971) Optimal Control of a System Governed by Partial Differential Equations. Springer-Verlag, New York, 170. https://doi.org/10.1007/978-3-642-65024-6
- Kotarski, W. and El-Saify, H.A. (2006) Optimality of the Boundary Control Problem for n × n Parabolic Lag System. Journal of Mathematical Analysis and Applications, 319, 61-73.
- Serag, H.M. and Qamlo, A.H. (2001) Boundary Control of Non-Cooperative Elliptic System. Advances in Modeling Analysis, 38, 31-42.
- El-Saify, H.A., Serag, H.M. and Shehata, M.A. (2009) Time Optimal Control Problem for Cooperative Hyperbolic Systems Involving the Laplace Operator. Journal of Dynamical and Control Systems, 15, 405-423. https://doi.org/10.1007/s10883-009-9067-y
- Gali, I.M. and Serag, H.M. (1995) Optimal Control of Cooperative Elliptic Systems Defined on R n . Journal of the Egyptian Mathematical Society, 3, 33-39.
- Serag, H.M. (2007) Distributed Control for Cooperative Systems Involving Parabolic Operators with an Infinite Number of Variables. The IMA Journal of Mathematical Control and Information, 24, 149-161. https://doi.org/10.1093/imamci/dnl018
- Serag, H.M., EL-Zahaby, S.A. and Abd Elrhman, L.M. (2015) Distributed Control for Cooperative Parabolic Systems with Conjugation Conditions. Journal of Progressive Research in Mathematics, 4, 348-365.
- Qamlo, A.H. and Bedaiwi, G.M. (2017) Distributed Control for 2 × 2 Coupled Infinite Order Hyperbolic Systems. Advances in Differential Equations and Control Processes, 18, 201-227. https://doi.org/10.17654/DE018040201
- Qamlo, A.H. (2021) Boundary Control Problems for 2 × 2 Cooperative Hyperbolic Systems with Infinite Order Operators. Open Journal of Optimization, 10, 1-12. https://doi.org/10.4236/ojop.2021.101001
- El-Saify, H.A. and Serag, H.M. (1991) Boundary Control of a Systems Governed by Non-Linear Neumann Problem. Journal of Institute of Mathematics & Computer Sciences, 2, 41-48.
- Khafagy, S. and Serag, H.M. (2018) Stability Results of Positive Weak Solution for Singular p-Laplacian Nonlinear System. Journal of Applied Mathematics & Informatics, 36, 173-179.
- Khafagy, S. and Serag, H.M. (2020) On the Existence of Positive Weak Solution for Nonlinear System with Singular Weights. Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 55, 259-267.
- Serag, H.M. and Khafagy, S. (2008) Existence of Weak Solution for n × n Nonlinear Systems Involving Different Degenerated p-Laplacian Operators. New Zealand Journal of Mathematics, 38, 75-86.