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IMA Journal of Mathematical Control and Information Advance Access originally published online on March 10, 2007
IMA Journal of Mathematical Control and Information 2008 25(1):49-57; doi:10.1093/imamci/dnm003
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© The author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Optimality conditions for n x n infinite-order parabolic coupled systems with control constraints and general performance index

G. M. Bahaa{dagger}

Mathematics Department, Faculty of Science, Beni Suef University, Beni Suef, Egypt

W. Kotarski{ddagger}

Institute of Informatics, Silesian University, Bedzinska 60, 41-200 Sosnowiec, Poland

{dagger} Email: bahaa_gm{at}hotmail.com

{ddagger} Email: kotarski{at}gate.math.us.edu.pl

Received on March 14, 2006; Accepted on December 20, 2006

A distributed control problem for n x n parabolic coupled systems involving operators with infinite order is considered. The performance index is more general than the quadratic one and has an integral form. Constraints on controls are imposed. Making use of the Dubovitskii–Milyutin theorem, the necessary and sufficient conditions of optimality are derived for the Dirichlet problem. Yet, the problem considered here is more general than the problems in El-Saify & Bahaa (2002, Optimal control for n x n hyperbolic systems involving operators of infinite order. Math. Slovaca, 52, 409–424), El-Zahaby (2002, Optimal control of systems governed by infinite order operators. Proceeding (Abstracts) of the International Conference of Mathematics (Trends and Developments) of the Egyptian Mathematical Society, Cairo, Egypt, 28–31 December 2002. J. Egypt. Math. Soc. (submitted)), Gali & El-Saify (1983, Control of system governed by infinite order equation of hyperbolic type. Proceeding of the International Conference on Functional-Differential Systems and Related Topics, vol. III. Poland, pp. 99–103), Gali et al. (1983, Distributed control of a system governed by Dirichlet and Neumann problems for elliptic equations of infinite order. Proceeding of the International Conference on Functional-Differential Systems and Related Topics, vol. III. Poland, pp. 83–87) and Kotarski et al. (200b, Optimal control problem for a hyperbolic system with mixed control-state constraints involving operator of infinite order. Int. J. Pure Appl. Math., 1, 241–254).

Keywords: optimal control problems; n x n parabolic coupled systems with infinite order; Dirichlet problem; Dubovitskii–Milyutin theorem; conical approximations; optimality conditions; Weierstrass theorem.


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