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Optimal Control

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Operations Research in Transportation Systems

Part of the book series: Applied Optimization ((APOP,volume 20))

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Abstract

The theory of optimal control is a branch of applied mathematics that studies the best ways of executing dynamic controlled (controllable) processes [1]. Among those of considerable interest for most applications, there are ones described by ordinary and partial differential equations and also by functional equations with a discrete variable. In all cases, the functions, called controls, appearing in equations describing the processes under study are to be defined assuming that these controls are chosen from a certain domain determined by a system of constraints. The quality of control is described by a functional depending on both the controls and the system of functions determining the trajectory of the dynamic process variation under the influence of the controls.

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References

  1. Kurzhanskiy, A. B. “Mathematical Theory of Optimum Control.” In Matematicheskaja Entziklopedia (Mathematical Encyclopedia). Moscow: Sovetskaya Entziklopediya, 1984; 4: 37–41 [in Russian].

    Google Scholar 

  2. Boltianskiy, V. G. Mathematical Methods of Optimum Control. New York: Holt, Reinart and Winston, 1971.

    Google Scholar 

  3. Kurzhanskiy, A. B. “Pontryagin Maximum Principle.” In Matematicheskaja Entziklopedia (Mathematical Encyclopedia). Moscow: Sovetskaya Entziklopediya, 1984; 4: 487–89 [in Russian].

    Google Scholar 

  4. Kurzhanskiy, A. B. “Programmable Optimum Control.” In Matematicheskaja Entziklopedia (Mathematical Encyclopedia). Moscow: Sovetskaya Entziklopediya, 1984; 4: 47–51 [in Russian].

    Google Scholar 

  5. Evtushenko, Iu. G. Numerical Optimization Techniques. New York: Optimization Software Inc., Publications Division, 1985.

    Google Scholar 

  6. Chernous’ko, F. L. “Computational Methods of Optimum Control.” In Matematika na Sluzhbe Inzhenera (Mathematics in Engineering). Moscow: Znanie, 1973; 56–73 [in Russian].

    Google Scholar 

  7. Ortega, J. M., and Rheinboldt, W. C. Iterative Solution of Nonlinear Equations in Several Variables. New York: Academic Press, 1970.

    MATH  Google Scholar 

  8. Shatrovskii, L. I. One numerical method of solving problems of optimum control. U.S.S.R. Computational Mathematics and Mathematical Physics. 1962; 2, No. 3: 488–491.

    Google Scholar 

  9. Chernous’ko, F. L., and Kolmanovskiy, V. B. “Computational and Approximate Methods of Optimal Control.” In Matematicheskii Analiz (Mathematical Analysis). Moscow: Izd. VINITI, 1977; 14: 101–167 [in Russian].

    Google Scholar 

  10. Moiseev, N. N. Elementy Teorii Optimal’nykh Sistem (Elements of Optimal Systems Theory). Moscow: Nauka, 1974 [in Russian].

    Google Scholar 

  11. Kurzhanskiy, A. B. “Positional Optimal Control.” In Matematicheskaja Entziklopedia (Mathematical Encyclopedia). Moscow: Sovetskaya Entziklopediya, 1984; 4: 42–47 [in Russian].

    Google Scholar 

  12. Pontriagin, L. S., et al. The Mathematical Theory of Optimal Processes. Oxford, New York: Pergamon Press, 1964.

    Google Scholar 

  13. Roytenberg, Ya. N. Avtomaticheskoe Upravlenie (Automatic Control). Moscow: Nauka, 1971 [in Russian].

    Google Scholar 

  14. Tkachev, A. M. Geometric method for numerical solution of a terminal problem of optimal control. Engineering Cybernetics. 1984; No. 2: 21–26.

    MathSciNet  Google Scholar 

  15. Gabasov, R. F., and Kirillova, F. M. Optimizatzia Lineinykh Sistem (Optimization of Linear Systems). Minsk: Izd. BGU (Belorussia State University), 1973 [in Russian].

    Google Scholar 

  16. Tkachev, A. M. A numerical method for a linear optimal response speed problem. Soviet Journal of Computer and Systems Sciences (Formerly Engineering Cybernetics). 1988; 26, No. 1: 174–177.

    MathSciNet  MATH  Google Scholar 

  17. Kiselev, Yu. N. “Methods for solving a smooth linear time-optimal problem.” In Proceedings of the Steklov Institute of Mathematics. Optimal Control and Differential Games. Edited by Pontryagin L. S. American Mathematical Society, 1990; 185, No. 2: 121–132.

    Google Scholar 

  18. Samsonov, S. P. “An optimal control problem with various quality functionals.” In Proceedings of the Steklov Institute of Mathematics. Optimal Control and Differential Games. Edited by Pontryagin L. S. American Mathematical Society, 1990; 185, No. 2: 241–248.

    MATH  Google Scholar 

  19. Tarakanov, A. F. The maximum principle for certain minimax control problems for connected sets. Soviet Journal of Computer and Systems Sciences (Formerly Engineering Cybernetics). 1989; 27, No. 2: 142–146.

    MathSciNet  MATH  Google Scholar 

  20. Butkovskiy, A. G. Distributed Control Systems. New York: American Elsevier Pub. Co., 1969.

    MATH  Google Scholar 

  21. Yegorov, Yu. V. “Optimum control of systems with distributed parameters.” In Matematika na Sluzhbe Inzhenera (Mathematics in Engineering). Moscow: Znanie, 1973; 187–99 [in Russian].

    Google Scholar 

  22. Boltianskiy, V. G. Optimal Control of Discrete Systems. New York: John Wiley and Sons Publ. Co., 1978.

    Google Scholar 

  23. Boltianskiy, V. G. Discrete maximum principle (method of local sections). Differential Equations. 1972; VIII, No. 11: 1497–1503.

    Google Scholar 

  24. Iliutovich, A. E. “Decomposition of a procedure of choosing a possible control in the problem of distributed resources.” In Sbornik Trudov VNIISI (Proceedings of All-Union Institute of System Studies). Moscow: Izd. VNIISI (All-Union Institute of System Studies), 1987; No. 3: 28–37 [in Russian].

    Google Scholar 

  25. Kolmanovskii, V. B. Optimal control in certain systems involving small parameters. Differential Equations. 1975; 11, No. 8: 1181–1189.

    MathSciNet  Google Scholar 

  26. Akulenko, L. D., and Chernous’ko, F. L. The averaging method in optimal control problems. U.S.S.R. Computational Mathematics and Mathematical Physics. 1975; 15, No. 4: 54–67.

    Google Scholar 

  27. Bellman, R. E. Dynamic Programming. Princeton, New Jersey: Princeton University Press, 1957.

    Google Scholar 

  28. Kolmanovskii, V. B. The approximate synthesis of some stochastic quasilinear systems. Automation and Remote Control. 1975; 36, No. 1: 44–50.

    MATH  Google Scholar 

  29. Boltianskiy, V. G. “The method of local sections and the supporting principle.” In Matematika na Sluzhbe Inzhenera (Mathematics in Engineering). Moscow: Znanie, 1973; 140–164 [in Russian].

    Google Scholar 

  30. Rozov, N. Kh. The local section method for systems with refraction of trajectories. Soviet Mathematics. 1972; 13, No. 1: 146–151.

    MATH  Google Scholar 

  31. Blagodatskikh, V. I. Sufficient optimality conditions for differential embeddings. Izvestiya AN SSSR. Seriya Matematika. 1974; 8, No. 3: 621–630.

    Article  Google Scholar 

  32. Karulina, N. I. “A sufficient condition for optimality for differential inclusions.” Proceedings of the Steklov Institute of Mathematics. Optimal Control and Differential Games. Edited by Pontryagin L. S. American Mathematical Society, 1990; 185, No. 2: 95–98.

    Google Scholar 

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Authors
  1. Alexander S. Belenky
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© 1998 Springer Science+Business Media Dordrecht

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Belenky, A.S. (1998). Optimal Control. In: Operations Research in Transportation Systems. Applied Optimization, vol 20. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6075-0_6

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  • DOI: https://doi.org/10.1007/978-1-4757-6075-0_6

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-4803-8

  • Online ISBN: 978-1-4757-6075-0

  • eBook Packages: Springer Book Archive

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