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Numerical solution of an optimal control problem with variable time points in the objective function

Published online by Cambridge University Press:  17 February 2009

K. L. Teo ,
Y. Liu ,
W. R. Lee ,
L. S. Jennings  and
S. Wang
K. L. Teo
Affiliation:
Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Koowloon, Hong Kong.
Y. Liu
Affiliation:
Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Koowloon, Hong Kong.
W. R. Lee
Affiliation:
Department of Mathematics and Statistics, Curtin University of Technology, GPO Box U 1987, Perth, WA 6845, Australia.
L. S. Jennings
Affiliation:
Center for Applied Dynamics and Optimization, The University of Western Australia, WA 6907, Australia.
S. Wang
Affiliation:
Center for Applied Dynamics and Optimization, The University of Western Australia, WA 6907, Australia.
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Abstract

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In this paper, we consider the numerical solution of a class of optimal control problems involving variable time points in their cost functions. The control enhancing transform is first used to convert the optimal control problem with variable time points into an equivalent optimal control problem with fixed multiple characteristic time (MCT). Using the control parametrization technique, the time horizon is partitioned into several subintervals. Let the partition points also be taken as decision variables. The control functions are approximated by piecewise constant or piecewise linear functions in accordance with these variable partition points. We thus obtain a finite dimensional optimization problem. The control parametrization enhancing control transform (CPET) is again used to convert approximate optimal control problems with variable partition points into equivalent standard optimal control problems with MCT, where the control functions are piecewise constant or piecewise linear functions with pre-fixed partition points. The transformed problems are essentially optimal parameter selection problems with MCT. The gradient formulae for the objective function as well as the constraint functions with respect to relevant decision variables are obtained. Numerical examples are solved using the proposed method.

Type
Research Article
Information
The ANZIAM Journal , Volume 43 , Issue 4 , April 2002 , pp. 463 - 478
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Clarke, F. H. and Vinter, R. B., “Applications of optimal multiprocesses“, SIAM J. Control Optim. 27 (1989) 1048–1071.CrossRefGoogle Scholar
[2]Clarke, F. H. and Vinter, R. B., “Optimal multiprocesses”, SIAM J. Control Optim. 27 (1989) 1072–1091.CrossRefGoogle Scholar
[3]Craven, B. D., Control and Optimization (Chapman & Hall, London, 1995).CrossRefGoogle Scholar
[4]Jennings, L. S., Fisher, M. E., Teo, K. L. and Goh, C. J., MISER3 Optimal Control Software: Theory and User Manual (EMCOSS, 1990).Google Scholar
[5]Jennings, L. S., Fisher, M. E., Teo, K. L. and Goh, C. J., “MISER3: Solving optimal control problems—an update”, Advanced Engineering Software 13 (1991) 190–196.Google Scholar
[6]Jennings, L. S., Fisher, M. E., Teo, K. L. and Goh, C. J., MISER3—Version 2.0 (1997) Optimal Control Software: Theory and User Manual (EMCOSS, 1997).Google Scholar
[7]Lee, H. W. J., Teo, K. L., Jennings, L. S. and Rehbock, V., “Control parameterization enhancing technique for time optimal control problems”, Dynamic Systems and Applications 6 (1997) 243–262.Google Scholar
[8]Martin, R. and Teo, K. L., Optimal Control of Drug Administration in Cancer Chemotherapy (World Scientific, Singapore, 1996).Google Scholar
[9]Miele, A., “Gradient algorithms for optimization of dynamic systems”, in Control and Dynamic Systems: Advances in Theory and Applications (ed. Leondes, C. T.), (Academic Press, New York, 1980) 1–52.Google Scholar
[10]Sakawa, Y. and Shindo, Y., “On the global convergence of an algorithm for optimal control”, IEEE Transactions on Automatic Control AC-25 (1980) 1149–1153.CrossRefGoogle Scholar
[11]Teo, K. L., Goh, C. J. and Wong, K. H., A Unified Computational Approach to Optimal Control Problems (Longman, New York, 1991).Google Scholar
[12]Teo, K. L., Jennings, L. S., Lee, H. W. J. and Rehbock, V., “The control parametrization enhancing transform for constrained optimal control problems”, J. Austral. Math. Soc. Ser. B 40 (1999) 314–335.CrossRefGoogle Scholar