Distributed Control for Non-Cooperative Systems Under Conjugation Conditions

  • A. A. Alsaban Department of Mathematics, Faculty of Science, Ibb University, Ibb, Yemen
  • H. M. Serag Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City (11884), Cairo, Egypt
  • Lamiaa Mohamed Abdelrhman Department of Mathematics, Faculty of Science, Al-Azhar University [for girls], Nasr City, Cairo, Egypt
Keywords: Non cooperative elliptic systems - Conjugation conditions - Dirichlet and Neumann conditions - Existence and uniqueness of solutions - Distributed control

Abstract

In this paper, the distributed control for non-cooperative elliptic systems under conjugation conditions is established. First, the existence and uniqueness of the state for these systems with Dirichlet and conjugation conditions is proved, then the set of equations and inequalities that characterizes the distributed control of these systems is found. The non-cooperative Neumann systems with conjugation conditions is also discussed.

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References

[1] El-Saify , H. A. , Serag. , H. M. , and Shehata, M. A. , Time optimal control problem for cooperative hyperbolic systems involving the Laplace operator, J. of Dynamics and Control Systems, Vol. 15, No. 3, (2009), pp. 405-423.
[2] Fleckinger J., and Serag, H. M., Semilinear cooperative elliptic systems on Rn , Rend. Mat. Appl., Vol. 15 , No.1, (1995), pp. 89-108.
[3] Gali, I.M., Optimal control of system governed by elliptic operators of infinite order, Ordinary and Partial Diff. Eqns., Proc. Dundee Scotland ,Springer-Verlag Ser. Lecture Notes in Maths., Vol. 964, (1982), pp. 263-272.
[4] Gali, I. M. and El-Saify, H. A. , Optimal control of a system governed by hyperbolic operator with an infinite number of variables, J. of Mathematical Analysis and Applications, Vol. 85, No. 1, (1982), pp.24-30.
[5] Gali, I. M and EL-Saify H. A., Distributed control of a system governed by Dirichlet and Neumann problems for a self adjoint elliptic operator with an innite number of variables, J. of Optimization Theory an Applications, Vol. 39, No. 2, (1983), pp. 293-298.
[6] Gali, I. M. and Serag, H. M. , Optimal control of cooperative elliptic systems dened on Rn , J. of the Egyptian Mathematical Society, Vol. 3, (1995), pp.33-39.
[7] Hassan, H.M. and Serag, H.M., Boundary control of quasi-static problem with viscous boundary conditions, Indian .J. pure and Applied Math., Vol.31, No.7, (2000), pp.767-772.
[8] Khafagy, S. and Serag, H. M. , Stability results of positive weak solution for singular p-Laplacian nonlinear system, J. Appl. Math. Inf. 36, 173-179 (2018).
[9] Khafagy, S. and Serag, H. M. , On the existence of positive weak solution for nonlinear system with singular weights, Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), Vol. 55, No.4, (2020), pp. 259-267.
[10] Kotarski, W. and El- Saify, H.A., Optimality of the boundary control problem for nn parabolic lag system, J. Math. Anal. Appl. 319, (2006), pp.61-73.
[11] Lions, J. L., Optimal control of a system governed by partial differential equations, Springer- Verlag, New York 170, (1971).
[12] Serag, H. M. , Optimal control of systems involving Schrodinger operators, Int. J. of Control and Intelligent Systems ,Canada , Vol. 32, No. 3, (2004), pp.154-159.
[13] Serag, H. M. , Distributed control for cooperative systems involving parabolic operators with an infinite number of variables, IMA J. of Mathematical Control and Information, Vol. 24, No. 2,(2007), pp. 149-161.
[14] Serag, H. M. and Qamlo, A, H. , On elliptic systems involving Schrodinger operator, The Mediter-ranean J. of Measurement and control, Vol. 1, No. 2, (2005), pp.91-96.
[15] Serag, H.M. and Khafagy, S., On non - homogeneousn nn elliptic systems involving p- Laplacian with different weights, J. of Advanced Research in Differential Equations, Vol. 33, (2009), pp.1-13.
[16] Serag, H.M., EL-Zahaby, S.A. and Abd Elrhman, L. M., Distributed control for cooperative parabolic systems with conjugation conditions Journal of progressive research in mathematics ,Vol.4, No.3, (2015), pp. 348-365.
[17] Serag, H. M., and Qamlo, A. H., Boundary control of non-cooperative elliptic system, Advances in Modeling Analysis, Vol. 38, No. 3,(2001), pp. 31-42.
[18] Sergienko I. V., Deineka V.S. , The Dirichlet and Neumann problems for elliptical equations with conjugation conditions and high-precision algorithms of their discretization, Cybernetics and Systems Analysis, Vol. 37, No. 3,(2001), pp. 323-347.
[19] Sergienko I. V and Deineka V. S, Optimal control of an elliptic system with conjugation conditions and Neumann boundary conditions, Cybernetics and Systems Analysis, Vol. 40, No. 6, (2004), pp.865-882.
[20] Sergienko, I. V. and Deineka, V. S., Optimal control of distributed systems with conjugation conditions, Springer, (2005).
Published
2021-03-03
How to Cite
Alsaban, A. A., Serag, H. M., & Abdelrhman, L. (2021). Distributed Control for Non-Cooperative Systems Under Conjugation Conditions. Journal of Progressive Research in Mathematics, 18(1), 55-63. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/2022
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Articles