Abstract
This paper contains general transformation techniques useful to convert minimax problems of optimal control into the Mayer-Bolza problem of the calculus of variations [Problem (P)]. We consider two types of minimax problems: minimax problems of Type (Q), in which the minimax function depends on the state and does not depend on the control; and minimax problems of Type (R), in which the minimax function depends on both the state and the control. Both Problem (Q) and Problem (R) can be reduced to Problem (P).
For Problem (Q), we exploit the analogy with a bounded-state problem in combination with a transformation of the Jacobson type. This requires the proper augmentation of the state vectorx(t), the control vectoru(t), and the parameter vector π, as well as the proper augmentation of the constraining relations. As a result of the transformation, the unknown minimax value of the performance index becomes a component of the parameter vector being optimized.
For Problem (R), we exploit the analogy with a bounded-control problem in combination with a transformation of the Valentine type. This requires the proper augmentation of the control vectoru(t) and the parameter vector π, as well as the proper augmentation of the constraining relations. As a result of the transformation, the unknown minimax value of the performance index becomes a component of the parameter vector being optimized.
In a subsequent paper (Part 2), the transformation techniques presented here are employed in conjunction with the sequential gradient-restoration algorithm for solving optimal control problems on a digital computer; both the single-subarc approach and the multiple-subarc approach are discussed.
Similar content being viewed by others
References
Miele, A., Mohanty, B. P., Venkataraman, P., andKuo, Y. M.,Numerical Solution of Minimax Problems of Optimal Control, Part 1, Introduction, Rice University, Aero-Astronautics Report No. 150, 1980.
Miele, A., Mohanty, B. P., Venkataraman, P., andKuo, Y. M.,Numerical Solution of Minimax Problems of Optimal Control, Part 2, Examples, Rice University, Aero-Astronautics Report No. 151, 1980.
Miele, A., Mohanty, B. P., Venkataraman, P., andKuo, Y. M.,Numerical Solution of Minimax Problems of Optimal Control, Part 3, Algorithms, Rice University, Aero-Astronautics Report No. 152, 1980.
Miele, A., Mohanty, B. P., Venkataraman, P., andKuo, Y. M.,Numerical Solution of Minimax Problems of Optimal Control, Part 4, Single-Subarc Approach, Rice University, Aero-Astronautics Report No. 153, 1981.
Miele, A., Mohanty, B. P., Venkataraman, P., andKuo, Y. M.,Numerical Solution of Minimax Problems of Optimal Control, Part 5, Single-Subarc Approach, Rice University, Aero-Astronautics Report No. 154, 1981.
Miele, A., Mohanty, B. P., Venkataraman, P., andKuo, Y. M.,Numerical Solution of Minimax Problems of Optimal Control, Part 6, Multiple-Subarc Approach, Rice University, Aero-Astronautics Report No. 155, 1981.
Miele, A., Mohanty, B. P., Venkataraman, P., andKuo, Y. M.,Numerical Solution of Minimax Problems of Optimal Control, Part 7, Multiple-Subarc Approach, Rice University, Aero-Astronautics Report No. 156, 1981.
Bliss, G. A.,Lectures on the Calculus of Variations, The University of Chicago Press, Chicago, Illinois, 1946.
Hestenes, M. R.,Calculus of Variations and Optimal Control Theory, John Wiley and Sons, New York, New York, 1966.
Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., andMishchenko, E. F.,The Mathematical Theory of Optimal Processes, John Wiley and Sons (Interscience Publishers), New York, New York, 1962.
Miele, A., Pritchard, R. E., andDamoulakis, J. N.,Sequential Gradient-Restoration Algorithm for Optimal Control Problems, Journal of Optimization Theory and Applications, Vol. 5, No. 4, pp. 235–282, 1970.
Miele, A., Damoulakis, J. N., Cloutier, J. R., andTietze, J. L.,Sequential Gradient-Restoration Algorithm for Optimal Control Problems with Nondifferential Constraints, Journal of Optimization Theory and Applications, Vol. 13, No. 2, pp. 218–255, 1974.
Gonzalez, S., andMiele, A.,Sequential Gradient-Restoration Algorithm for Optimal Control Problems with General Boundary Conditions, Journal of Optimization Theory and Applications, Vol. 26, No. 3, pp. 395–425, 1978.
Miele, A.,Recent Advances in Gradient Algorithms for Optimal Control Problems, Journal of Optimization Theory and Applications, Vol. 17, Nos. 5/6, pp. 361–430, 1975.
Miele, A.,Gradient Algorithms for the Optimization of Dynamic Systems, Control and Dynamic Systems, Advances in Theory and Application, Edited by C. T. Leondes, Academic Press, New York, New York, Vol. 16, pp. 1–52, 1980.
Johnson, C. D.,Optimal Control with Chebyshev Minimax Performance Index, Journal of Basic Engineering, Vol. 89, No. 2, pp. 251–262, 1967.
Michael, G. J.,Computation of Chebyshev Optimal Control, AIAA Journal, Vol. 9, No. 5, pp. 973–975, 1971.
Warga, J.,Minimax Problems and Unilateral Curves in the Calculus of Variations, SIAM Journal on Control, Vol. 3, No. 1, pp. 91–105, 1965.
Powers, W. F.,A Chebyshev Minimax Technique Oriented to Aerospace Trajectory Optimization Problems, AIAA Journal, Vol. 10, No. 10, pp. 1291–1296, 1972.
Holmaker, K.,A Minimax Optimal Control Problem, Journal of Optimization Theory and Applications, Vol. 28, No. 3, pp. 391–410, 1979.
Holmaker, K.,A Property of an Autonomous Minimax Optimal Control Problem, Journal of Optimization Theory and Applications, Vol. 32, No. 1, pp. 81–87, 1980.
Jacobson, D. H., andLele, M. M.,A Transformation Technique for Optimal Control Problems with a State Variable Inequality Constraint, IEEE Transactions on Automatic Control, Vol. AC-14, No. 5, pp. 457–464, 1969.
Miele, A., Wu, A. K., andLiu, C. T.,A Transformation Technique for Optimal Control Problems with Partially Linear State Inequality Constraints, Journal of Optimization Theory and Applications, Vol. 28, No. 2, pp. 185–212, 1979.
Valentine, F. A.,The Problem of Lagrange with Differential Inequalities as Added Side Conditions, Contributions to the Calculus of Variations, 1933–1937. The University of Chicago Press, Chicago, Illinois, pp. 403–447, 1937.
Miele, A., Mohanty, B. P., Venkataraman, P., andKuo, Y. M.,Numerical Solution of Minimax Problems of Optimal Control, Part 2, Journal of Optimization Theory and Applications, Vol. 38, No. 1, pp. 111–135, 1982.
Author information
Authors and Affiliations
Additional information
This research was supported by the National Science Foundation, Grant No. ENG-79-18667, and by Wright-Patterson Air Force Base, Contract No. F33615-80-C3000. This paper is a condensation of the investigations reported in Refs. 1–7. The authors are indebted to E. M. Coker and E. M. Sims for analytical and computational assistance.
Rights and permissions
About this article
Cite this article
Miele, A., Mohanty, B.P., Venkataraman, P. et al. Numerical solution of minimax problems of optimal control, part 1. J Optim Theory Appl 38, 97–109 (1982). https://doi.org/10.1007/BF00934325
Issue Date:
DOI: https://doi.org/10.1007/BF00934325