On optimal control for cooperative system governed by heat equation with conjugation conditions
Abstract
The optimal control for 2 × 2 cooperative Dirichlet and Neumann parabolic systems with
conjugation conditions is considered. First the existence and uniqueness of the state are proved;
then the necessary and sufficient conditions for the control to be an optimal is obtained by a set
of inequalities. Also we generalize these discussions to n × n cooperative parabolic systems with
conjugation conditions. The control in our problems is of distributed type and is allowed to be in
a Hilbert space.
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References
[1] L. M. Abd-Elrhman, Boundary Control for heat equation under conjugation conditions,Journal
of Nonlinear Functional Analysis and Applications. Vol.29,No.4, December(2024), pp.969-990.
[2] L. M. Abd-Elrhman, Distributed control of an incompressible Navier-Stokes equations under conjugation conditions, Accepted for publication at: Nonlinear Functional Analysis and Applications.
[3] H. A. El-Saify, H. M. Serag and M. A. Shehata, Time optimal control problem for cooperative
hyperbolic systems involving the Laplace operator, J. of Dynamics and Control Systems, 15(3)
(2009), 405-423.
[4] J. Fleckinger and H. M. Serag, Semilinear cooperative elliptic systems on Rn, Rend. Mat. Appl.,
15(1) (1995), 98-108.
[5] I.M. Gali, 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.
[6] I. M. Gali and H. A. El-Saify, 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.
[7] I. M. Gali and H. A.EL-Saify, Distributed control of a system governed by Dirichlet and Neumann
problems for a self adjoint elliptic operator with an infinite number of variables, J. of Optimization
Theory an Applications, Vol. 39, No. 2, (1983), pp. 293-298.
[8] I. M. Gali and H. M.Serag, Optimal control of cooperative systems defined on Rn, J. Egypt. Math.
Soc, Vol. 3, (1995), pp. 33-39.
[9] H.M. Hassan and H.M. Serag, Boundary control for quasi-static problem with viscous boundary
conditions, Indian .J. pure and Applied Math, Vol.31, No.7, (2000), pp.767-772.
[10] W. Kotarski and H.A. El-Saify, Optimality of the boundary control for n×n parabolic lag system,
J. Math. Anal. Appl. 319, (2006), pp.61-73.
[11] W. Kotarski, H.A. El-Saify and G.Bahaa, Time optimal control of parabolic lag system with
infinite number of variables,J. Egypt, Math.Soc. 15(2007).
16
[12] Lions, J. L., Optimal control of a system governed by partial differential equations, Springer
-Verlag, New York 170, (1971).
[13] Qamlo, A. H., Distributed Control for n×n Cooperative Systems Governed by Hyperbolic Operator
of Infinite Order. Advances in Pure Mathematics, (2020), Vol. 10, pp. 728-738.
[14] 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.
[15] Serag, H. M., On Optimal control for elliptic system with variable coefficients, Revista de Mathematica Aplicadas, Department de Ingeniera Mathematica, Universidad Chile, Vol. 19, (1998),
pp. 45-49.
[16] Serag, H. M., Distributed control for cooperative systems involving parabolic operators with an
Infinite number of Variables, IMA Journal of Mathematical Control and Information,Vol. 24, No.
2,(2007), pp.149-161.
[17] H. M.Serag, S. A. EL-Zahaby and L. M. Abd Elrhman, Distributed control for cooperative
parabolic systems with conjugation conditions, Journal of progressive research in mathematics,
Vol.4, No.3,(2015),pp 348-365.
[18] H. M.Serag, L. M . Abd-Elrhman and A. A. Alsaban, Boundary control for cooperative elliptic
systems under conjugation conditions, Advances in Pure Mathematics, Vol.11.No.5. May (2021),
pp. 457-471.
[19] H. M. Serag, L. M. Abd-Elrhman and A. A. Alsaban,Distributed control for non-cooperative
systems under conjugation conditions, Journal of Progressive Research in Mathematics, Vol. 18,
No. 1, (2021), pp. 55-63.
[20] H. M.Serag, L. M. Abd-Elrhman and A. A. Alsaban, ON optimal control for cooperative Elliptic
systems under conjugation conditions,Journal of Applied Mathematics and Informatics, Vol.41,
No.2, (2023), pp 229-246.
[21] H. M. Serag, A. Hyder and M. El-Badawy, Optimal control for cooperative systems involving fractional Laplace operator, Journal of Inequalities and Applications, Vol, 2021, Article number.196,
(2021).
[22] 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.
[23] 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.
[24] Sergienko I. V. and Deineka V. S., Optimal control of distributed systems with conjugation conditions, Springer, (2005).
of Nonlinear Functional Analysis and Applications. Vol.29,No.4, December(2024), pp.969-990.
[2] L. M. Abd-Elrhman, Distributed control of an incompressible Navier-Stokes equations under conjugation conditions, Accepted for publication at: Nonlinear Functional Analysis and Applications.
[3] H. A. El-Saify, H. M. Serag and M. A. Shehata, Time optimal control problem for cooperative
hyperbolic systems involving the Laplace operator, J. of Dynamics and Control Systems, 15(3)
(2009), 405-423.
[4] J. Fleckinger and H. M. Serag, Semilinear cooperative elliptic systems on Rn, Rend. Mat. Appl.,
15(1) (1995), 98-108.
[5] I.M. Gali, 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.
[6] I. M. Gali and H. A. El-Saify, 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.
[7] I. M. Gali and H. A.EL-Saify, Distributed control of a system governed by Dirichlet and Neumann
problems for a self adjoint elliptic operator with an infinite number of variables, J. of Optimization
Theory an Applications, Vol. 39, No. 2, (1983), pp. 293-298.
[8] I. M. Gali and H. M.Serag, Optimal control of cooperative systems defined on Rn, J. Egypt. Math.
Soc, Vol. 3, (1995), pp. 33-39.
[9] H.M. Hassan and H.M. Serag, Boundary control for quasi-static problem with viscous boundary
conditions, Indian .J. pure and Applied Math, Vol.31, No.7, (2000), pp.767-772.
[10] W. Kotarski and H.A. El-Saify, Optimality of the boundary control for n×n parabolic lag system,
J. Math. Anal. Appl. 319, (2006), pp.61-73.
[11] W. Kotarski, H.A. El-Saify and G.Bahaa, Time optimal control of parabolic lag system with
infinite number of variables,J. Egypt, Math.Soc. 15(2007).
16
[12] Lions, J. L., Optimal control of a system governed by partial differential equations, Springer
-Verlag, New York 170, (1971).
[13] Qamlo, A. H., Distributed Control for n×n Cooperative Systems Governed by Hyperbolic Operator
of Infinite Order. Advances in Pure Mathematics, (2020), Vol. 10, pp. 728-738.
[14] 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.
[15] Serag, H. M., On Optimal control for elliptic system with variable coefficients, Revista de Mathematica Aplicadas, Department de Ingeniera Mathematica, Universidad Chile, Vol. 19, (1998),
pp. 45-49.
[16] Serag, H. M., Distributed control for cooperative systems involving parabolic operators with an
Infinite number of Variables, IMA Journal of Mathematical Control and Information,Vol. 24, No.
2,(2007), pp.149-161.
[17] H. M.Serag, S. A. EL-Zahaby and L. M. Abd Elrhman, Distributed control for cooperative
parabolic systems with conjugation conditions, Journal of progressive research in mathematics,
Vol.4, No.3,(2015),pp 348-365.
[18] H. M.Serag, L. M . Abd-Elrhman and A. A. Alsaban, Boundary control for cooperative elliptic
systems under conjugation conditions, Advances in Pure Mathematics, Vol.11.No.5. May (2021),
pp. 457-471.
[19] H. M. Serag, L. M. Abd-Elrhman and A. A. Alsaban,Distributed control for non-cooperative
systems under conjugation conditions, Journal of Progressive Research in Mathematics, Vol. 18,
No. 1, (2021), pp. 55-63.
[20] H. M.Serag, L. M. Abd-Elrhman and A. A. Alsaban, ON optimal control for cooperative Elliptic
systems under conjugation conditions,Journal of Applied Mathematics and Informatics, Vol.41,
No.2, (2023), pp 229-246.
[21] H. M. Serag, A. Hyder and M. El-Badawy, Optimal control for cooperative systems involving fractional Laplace operator, Journal of Inequalities and Applications, Vol, 2021, Article number.196,
(2021).
[22] 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.
[23] 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.
[24] Sergienko I. V. and Deineka V. S., Optimal control of distributed systems with conjugation conditions, Springer, (2005).
Published
2025-09-30
Section
Advanced Computational Methods for Fractional Calculus
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