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A gradient technique for an optimal control problem governed by a system of nonlinear first order partial differential equations

Published online by Cambridge University Press:  17 February 2009

Mohammad A. Kazemi
Mohammad A. Kazemi
Affiliation:
Department of Mathematics, University of North Carolina at Charlotte, Charlotte, NC 28223, U.S.A.
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Abstract

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In this paper a class of optimal control problems with distributed parameters is considered. The governing equations are nonlinear first order partial differential equations that arise in the study of heterogeneous reactors and control of chemical processes. The main focus of the present paper is the mathematical theory underlying the algorithm. A conditional gradient method is used to devise an algorithm for solving such optimal control problems. A formula for the Fréchet derivative of the objective function is obtained, and its properties are studied. A necessary condition for optimality in terms of the Fréchet derivative is presented, and then it is shown that any accumulation point of the sequence of admissible controls generated by the algorithm satisfies this necessary condition for optimality.

Type
Research Article
Information
The ANZIAM Journal , Volume 36 , Issue 3 , January 1995 , pp. 261 - 273
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Cesari, L., Optimization—Theory and applications (Springer-Verlag, New York, 1983).CrossRefGoogle Scholar
[2]Kazemi, M. A., “Optimal control of systems governed by partial differential equations with integral inequality constraints”, Nonlinear Analysis, Theory, Methods, Applications 8 (1984) 1409–1425.CrossRefGoogle Scholar
[3]Kazemi, M. A., “A method of successive approximations for optimal control of distributed parameter systems”, J. Math. Anal. Appl. 133 (1988) 484–497.CrossRefGoogle Scholar
[4]Kazemi, M. A., “Lp-solution for a class of partial differential equations”, J. Math. Anal. Appl. 171 (1992) 14–26.CrossRefGoogle Scholar
[5]Lee, H. H., Heterogeneous reactor design (Butterworth Publishers, 1985).Google Scholar
[6]Leese, S. J., “Convergence of gradient methods for optimal control problems”, J. Optim. Theory Appl. 21 (1977) 329–337.CrossRefGoogle Scholar
[7]Lukyanov, A. T. and Serovalski, S. Ya., “The method of successive approximations in a problem of optimal control of a non-linear parabolic system”, U.S.S.R. Comput. Math. and Math. Phys. 24 (1984) 23–30.CrossRefGoogle Scholar
[8]Oguny, A. F. and Ray, W. H., “Optimal control policies for tubular reactors experiencing catalyst decay”, A1CHE. J. 17 (1971) 43–51.CrossRefGoogle Scholar
[9]Pshenichny, B. N. and Danilin, Yu. M., Numerical methods in external problems (Mir, Moscow, 1978).Google Scholar
[10]Reid, D. W., “On the computational methods for the optimal control of distributed parameter systems”, Ph. D. Thesis, University of New South Wales, Australia, 1980.Google Scholar
[11]Teo, K. L. and Wu, Z. S., Computational methods for optimizing distributed systems (Academic Press, 1984).Google Scholar
[12]Vasil'ev, O. V. and Srochoko, V., “Optimization of a class of controlled processes with distributed parameters”, Siberian Math. J. 14 (1988) 466–470.Google Scholar
[13]Wu, Z. S. and Teo, K. L., “A conditional gradient method for an optimal control problem involving a class of nonlinear second-order hyperbolic partial differential equations”, J. Math. Anal. Appl. 91 (1983) 376–393.CrossRefGoogle Scholar