×

A discrete method of optimal control based upon the cell state space concept. (English) Zbl 0548.93040

A discrete method of optimal control is proposed. 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 (\(\tau)\), the motion of the system under all possible interval controls (u,\(\tau)\) 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 results in the form of a control table seems to give this method a means to make systems real-time controllable.

MSC:

93C10 Nonlinear systems in control theory
37-XX Dynamical systems and ergodic theory
49L20 Dynamic programming in optimal control and differential games
49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
70G10 Generalized coordinates; event, impulse-energy, configuration, state, or phase space for problems in mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] 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.
[2] Leitmann, G.,An Introduction to Optimal Control, McGraw-Hill, New York, New York, 1966. · Zbl 0196.46302
[3] Takahashi, Y., Rabins, M. J., andAuslander, D. M.,Control and Dynamic Systems, Addison-Wesley Publishing Company, Reading, Massachusetts, 1970.
[4] Leitmann, G.,The Calculus of Variations and Optimal Control, Plenum Press, New York, New York, 1981. · Zbl 0475.49003
[5] Hsu, C. S.,A Theory of Cell-to-Cell Mapping Dynamical Systems, Journal of Applied Mechanics, Vol. 47, pp. 931-939, 1980. · Zbl 0452.58019 · doi:10.1115/1.3153816
[6] 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. · Zbl 0452.58020 · doi:10.1115/1.3153817
[7] 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. · Zbl 0482.70017 · doi:10.1115/1.3157686
[8] 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. · Zbl 0509.70017 · doi:10.1115/1.3162632
[9] 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. · Zbl 0507.70026 · doi:10.1115/1.3162633
[10] 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. · Zbl 0548.58038 · doi:10.1016/0022-247X(84)90079-9
[11] 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. · Zbl 0521.58011 · doi:10.1016/0020-7462(83)90009-4
[12] Hsu, C. S., andPolchai, A.,Characteristics of Singular Entities of Simple Cell Mappings, International Journal of Nonlinear Mechanics, Vol. 19, pp. 19-38, 1984. · Zbl 0534.58012 · doi:10.1016/0020-7462(84)90016-7
[13] Bellman, R. E., andDreyfus, S. E.,Applied Dynamic Programming, Princeton University Press, Princeton, New Jersey, 1962. · Zbl 0106.34901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.