Abstract
The derivations of the equations used in optimal control theory are given in this chapter. Numerical solutions are considered in Chapters 3 and 4. The simplest problem in the calculus of variations is discussed in Section 2.1. The Euler-Lagrange equations are derived using the classical calculus of variations (References 1–5) approach. The equations are then rederived using dynamic programming (References 6–8).
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References
Gelfand, I. M., and Fomin, S. V., Calculus of Variations, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1963.
Kalman, R. E., The theory of optimal control and the calculus of variations, in Mathematical Optimization Techniques, edited by R. Bellman, University of California Press, Berkeley, California, pp. 309–331, 1963.
Courant, R., and Hilbert, D., Methods of Mathematical Physics, Vol. 1, Wiley-Interscience, New York, 1953.
Miele, A., Introduction to the calculus of variations in one independent variable, in Theory of Optimum Aerodynamic Shapes, edited by A. Miele, Academic Press, New York, pp. 3–19, 1965.
Gottfried, B. S., and Weisman, J., Introduction to Optimization Theory, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1973.
Bellman, R., Dynamic Programming, Princeton University Press, Princeton, New Jersey, 1957.
Dreyfus, S. E., Dynamic Programming and the Calculus of Variations, Academic Press, New York, 1965.
Bellman, R., and Kalaba, R., Dynamic Programming and Modern Control Theory, Academic Press, New York, 1965.
Pontryagin, L. S., et al., The Mathematical Theory of Optimal Processes, Wiley-Interscience, New York, 1962.
Lanczos, C, The Variational Principles of Mechanics, University of Toronto Press, Toronto, 1957.
Kalaba, R. E., A New Approach to Optimal Control and Filtering, University of Southern California Report USCEE-316, Los Angeles, California, 1968.
Bellman, R., and Kalaba, R., On the fundamental equations of invariant imbedding—I, Proceedings of the National Academy of Sciences, Vol. 47, pp. 336–338, 1961.
Kagiwada, H. and Kalaba, R., Derivation and Validation of an Initial-Value Method for Certain Nonlinear Two-Point Boundary-Value Problems, RM-5566-PR, The Rand Corporation, January, 1968.
Kagiwada, H., Kalaba, R., Schumitzky, A., and Sridhar, R., Invariant Imbedding and Sequential Interpolating Filters for Nonlinear Processes, RM-5507-PR, The Rand Corporation, November, 1967.
Kalaba, R., and Sridhar, R., Invariant Imbedding and the Simplest Problem in the Calculus of Variations, RM-5781-PR, The Rand Corporation, October, 1968.
Kagiwada, H., Kalaba, R., Schumitzky, A., and Sridhar, R., Cauchy and Fred-holm methods for Euler equations, Journal of Optimization Theory and Applications, Vol. 2, pp. 157–163, 1968.
Bellman, R., Kagiwada, H., and Kalaba, R., Invariant imbedding and the numerical integration of boundary-value problems for unstable linear systems of ordinary differential equations, Communications of the ACM, Vol. 10, pp. 100–102, 1967.
Bellman, R., and Kalaba, R., On the principle of invariant imbedding and propagation through inhomogeneous media, Proceedings of the National Academy of Sciences, Vol. 42, pp. 629–632, 1956.
Suggested Reading
Axelband, E. I., An approximation technique for the optimal control of linear distributed parameter systems, IEEE Transactions on Automatic Control, Vol. 11, pp. 42–45, January, 1966.
Balakrishnan, A. V., and Neustadt, L. W., editors, Computing Methods in Optimization Problems, Academic Press, New York, 1964.
Larson, R. E., State Increment Dynamic Programming, American Elsevier Publishing Company, Inc., New York, 1968.
Leondes, C. T., editor of the series in Control and Dynamic Systems, Advances in Theory and Applications, Academic Press, New York, Volumes 1–16, 1965–1980.
Theil, H., Principles of Econometrics, John Wiley and Sons, Inc., New York, 1971.
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© 1982 Plenum Press, New York
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Kalaba, R., Spingarn, K. (1982). Optimal Control. In: Control, Identification, and Input Optimization. Mathematical Concepts and Methods in Science and Engineering. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-7662-0_2
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DOI: https://doi.org/10.1007/978-1-4684-7662-0_2
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