Abstract
A discrete method of optimal control is proposed in this paper. The continuum state space of a system is discretized into a cell state space, and the cost function is discretized in a similar manner. Assuming intervalwise constant controls and using a finite set of admissible control levels (u) and a finite set of admissible time intervals (τ), the motion of the system under all possible interval controls (u, τ) can then be expressed in terms of a family of cell-to-cell mappings. The proposed method extracts the optimal control results from these mappings by a systematic search, culminating in the construction of a discrete optimal control table.
The possibility of expressing the optimal control results in the form of a control table seems to give this method a means to make systems real-time controllable.
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References
Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., andMishchenko, E. F.,The Mathematical Theory of Optimal Processes, Interscience Publishers, New York, New York, 1962.
Leitmann, G.,An Introduction to Optimal Control, McGraw-Hill, New York, New York, 1966.
Takahashi, Y., Rabins, M. J., andAuslander, D. M.,Control and Dynamic Systems, Addison-Wesley Publishing Company, Reading, Massachusetts, 1970.
Leitmann, G.,The Calculus of Variations and Optimal Control, Plenum Press, New York, New York, 1981.
Hsu, C. S.,A Theory of Cell-to-Cell Mapping Dynamical Systems, Journal of Applied Mechanics, Vol. 47, pp. 931–939, 1980.
Hsu, C. S., andGuttalu, R. S.,An Unravelling Algorithm for Global Analysis of Dynamical Systems: An Application of Cell-to-Cell Mappings, Journal of Applied Mechanics, Vol. 47, pp. 940–947, 1980.
Hsu, C. S.,A Generalized Theory of Cell-to-Cell Mapping for Nonlinear Dynamical Systems, Journal of Applied Mechanics, Vol. 48, pp. 634–642, 1981.
Hsu, C. S., Guttalu, R. S., andZhu, W. H.,A Method of Analyzing Generalized Cell Mappings, Journal of Applied Mechanics, Vol. 49, pp. 885–894, 1982.
Hsu, C. S.,A Probabilistic Theory of Nonlinear Dynamical Systems Based on the Cell State Space Concept, Journal of Applied Mechanics, Vol. 49, pp. 895–902, 1982.
Hsu, C. S., andLeung, W. H.,Singular Entities and An Index Theory for Cell Functions, Journal of Mathematical Analysis and Applications, Vol. 100, pp. 250–291, 1984.
Hsu, C. S.,Singularities of N-Dimensional Cell Functions and the Associated Index Theory, International Journal of Nonlinear Mechanics, Vol. 18, pp. 199–221, 1983.
Hsu, C. S., andPolchai, A.,Characteristics of Singular Entities of Simple Cell Mappings, International Journal of Nonlinear Mechanics, Vol. 19, pp. 19–38, 1984.
Bellman, R. E., andDreyfus, S. E.,Applied Dynamic Programming, Princeton University Press, Princeton, New Jersey, 1962.
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Dedicated to G. Leitmann
The material is based upon work supported by the National Science Foundation under Grant No. MEA-82-17471. The author is also indebted to Professor G. Leitmann for his many helpful comments.
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Hsu, C.S. A discrete method of optimal control based upon the cell state space concept. J Optim Theory Appl 46, 547–569 (1985). https://doi.org/10.1007/BF00939159
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DOI: https://doi.org/10.1007/BF00939159