Abstract
In this paper we consider a heat flow in an inhomogeneous body without internal source. There exists special initial and boundary conditions in this system and we intend to find a convenient coefficient of heat conduction for this body so that body cool off as much as possible after definite time. We consider this problem in a general form as an optimal control problem which coefficient of heat conduction is optimal function. Then we replace this problem by another in which we seek to minimize a linear form over a subset of the product of two measures space defined by linear equalities. Then we construct an approximately optimal control.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Fakharzadeh,Shapes, measure and elliptic equations Ph. D. Thesis Mathematics, University of Leeds, Leeds, England, 1996.
M. H. Farahi, J. E. Rubio and D. A. Wilson,The global control of a nonlinear wave equation, International J. of Control65(1) (1996), 1–15.
M. H. Farahi,The boundary control of the wave equation, PhD Thesis, School of Mathematics, University of Leeds, Leeds, England, 1996.
A. Friedman and L-S. Jiang,Nonlinear optimal control problems in heat equation, SIAM Journal of Control and Optimization21(6) (1983), 940–952.
A. Friedman,Nonlinear optimal control problems for parabolic equations, SIAM J. of Control and Optimization22(5) (1984), 805–816.
A. V. Kamyad,Boundary control problem for the multidimensional diffusion equation, PhD Thesis, School of Mathematics, University of Leeds, Leeds, England, 1987.
A. V. Kamyad, J. E. Rubio and D. A. Wilson,Optimal control of multidimensional diffusion equation, Journal of Optimization Theory and Applications,70 (1991), 191–209.
A. V. Kamyad, J. E. Rubio and D. A. Wilson,An Optimal control problem for the multidimensional diffusion equation with a generalized control variable, Journal of Optimization Theory and Applications75 (1992), 101–132.
A. V. Kamyad and A. H. Borzabadi,Strong controllability and optimal control of the heat equation with a thermal source, J. Appl. Math. & Computing(old:KJCAM)7(3) (2000), 555–568.
V. P. Mikhailov,Partial differential equation, Mir, Moscow, 1978.
J. E. Rubio,Control and optimization; the linear treatment of non-linear problems, Manchester, U. K., Manchester University Press, 1986.
J. E. Rubio,The global control of nonlinear elliptic equation, Journal of Franklin Institute, Pergamon Press Ltd330(1) (1994), 29–35.
J. E. Rubio,The global control of nonlinear diffusion equation, SIAM Journal of Control and Optimization33(1) (1995), 308–322.
H. Rudolph,Global solution in optimal via SILP, in Lecture Notes in Control and Inform. Sci.143, Springer-Verlag, New York, 1990.
E. Zeidler,Applied functional analysis, application to mathematical physics, Springer-Verlag, New York,108, 1995.
Author information
Authors and Affiliations
Corresponding author
Additional information
Akbar Hashemi Borzabadi received his B. Sc from Birjand University, Iran and his M. Sc from Ferdowsi University of Mashhad. He is a lecturer in Damghan University of Basic Sciences and at the moment he is Ph. D student in Ferdowsi University of Mashhad. His research interests center on optimal control of distributed parameter systems and optimal path planning.
Ali Vahidian Kamyad received his B. Sc from Ferdowsi University of Mashhad, Mashhad, Iran, M.Sc from Institute of Mathematics Tehran, Iran and Ph. D at Leeds University, Leeds, England under supervisior of J. E. Rubio. Since 1972 he has been at the Ferdowsi University of Mashhad, he is a professor at the College of Mathematics, Ferdowsi University of Mashhad, Iran and his research interests are mainly on optimal control of distributed parameter systems and applications of fuzzy theory.
Mohammad Hadi Farahi received his B. Sc from Ferdowsi University of Mashhad, Mashhad, Iran, M. Sc from Brunel University, U.K. and Ph. D at Leeds University, U.K. At the moment he is an associate professor at the College of Mathematics, Ferdowsi University of Mashhad, Iran and his research areas are optimal control, optimization, approximation theory and numerical analysis.
Rights and permissions
About this article
Cite this article
Borzabadi, A.H., Kamyad, A.V. & Farahi, M.H. Optimal control of the heat equation in an inhomogeneous body. JAMC 15, 127–146 (2004). https://doi.org/10.1007/BF02935750
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02935750