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Automation and Remote Control

, Volume 79, Issue 5, pp 919–939 | Cite as

Second Order Methods for the Optimal Control Problems

  • V. A. BaturinEmail author
  • S. V. Cheremnykh
Large Scale Systems Control
  • 18 Downloads

Abstract

The paper considers the iterative improvement algorithms, the efficiency of which substantially depends on the chosen parameters values. The problem of control of these parameters is formulated and discussed. We designed the modified algorithms where parameters are automatically adjusted at each iteration. The original and modified algorithms are applied to solve the problem of optimal control for the ecology-economic system.

Keywords

optimal control iterative improvement algorithm strong improvement weak improvement adaptive parameters adjustment 

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References

  1. 1.
    Baturin, V.A. and Urbanovich, D.E., Priblizhennye metody optimal’nogo upravleniya, osnovannye na printsipe rasshireniya (Approximate Optimal Control Methods Based on the Extension Principle), Novosibirsk: Nauka, 1997.Google Scholar
  2. 2.
    Baturin, V.A. and Cheremnykh, S.V., Control over the Choice of Parameters in Second-Order Algorithms of Weak Improvement for Optimal Control Problems, J. Comput. Syst. Sci. Int., 2006, vol. 45, no. 2, pp. 226–233.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Vasil’ev, O.V. and Tyatyushkin, A.I., A Method for Solving Optimal Control Problems That Is Based on the Maximum Principle, Zh. Vychisl. Mat. Mat. Fiz., 1981, vol. 21, no. 6, pp. 1376–1384.MathSciNetzbMATHGoogle Scholar
  4. 4.
    Gurman, V.I., Baturin, V.A., and Rasina, I.V., Priblizhennye metody optimal’nogo upravleniya (Approximate Optimal Control Methods), Irkutsk: Irkutsk. Univ., 1983.Google Scholar
  5. 5.
    Gurman, V.I. and Rasina, I.V., Practical Applications of Conditions Sufficient for a Strong Relative Minimum, Autom. Remote Control, 1979, vol. 40, no. 10, part 1, pp. 1410–1415.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Krotov, V.F. and Gurman, V.I., Metody i zadachi optimal’nogo upravleniya (Methods and Problems of Optimal Control), Moscow: Nauka, 1973.zbMATHGoogle Scholar
  7. 7.
    Krotov, V.F. and Fel’dman, I.N., Iterative Method to Solve the Optimal Control Problems, Izv. Akad. Nauk SSSR, Tekh. Kibern., 1983, no. 2, pp. 160–168.Google Scholar
  8. 8.
    Lyubushin, A.A. and Chernous’ko, F.L., The Method of Successive Approximation for Computation of Optimal Control, Izv. Akad. Nauk SSSR, Tekhn. Kibern., 1983, no. 2, pp. 147–159.zbMATHGoogle Scholar
  9. 9.
    Modeli upravleniya prirodnymi resursami (Models of Natural Resource Control), Gurman, V.I., Ed., Moscow: Nauka, 1981.Google Scholar
  10. 10.
    Gurman, V.I. et al., Novye metody uluchsheniya upravlyaemykh protsessov (New Refinement Methods for Control Processes), Novosibirsk: Nauka, 1987.zbMATHGoogle Scholar
  11. 11.
    Cheremnykh, S.V., Controlling the Choice of Parameters in Algorithms of Strong Improvement, Proc. IX Chetaev Conf. “Analytical Mechanics, Stability and Control of Motion,” Irkutsk—Lake Baikal, 12–16 June, 2007, vol. 3, pp. 268–279.Google Scholar
  12. 12.
    Cheremnykh, S.V., Automating the Selection of Parameter Values in the Second Order Improving Algorithms, Proc. Int. XIV Baikal Workshop “Optimization Methods and Their Applications,” Irkutsk— Severobaikal’sk, July 2–8, 2008, vol. 2, pp. 211–221.Google Scholar
  13. 13.
    Gurman, V.I., Vikulov, V.Ye., Danilina, Ye.V., et al., Ekologo-ekonomicheskaya strategiya razvitiya regiona. Matematicheskoe modelirovanie i sistemnyi analiz na primere Baikal’skogo regiona (Ecological and Economic Strategy of the Region Development: Mathematical Modeling and System Analysis on the Example of the Baikal Region), Novosibirsk: Nauka, 1990.Google Scholar
  14. 14.
    Baturin, V.A., First-Order Improvement Method for the Problems of Optimal Control of Logic-Dynamic Systems, Nonlin. Anal., 2006, vol. 64, no. 2, pp. 288–294.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jacobson, D.H., New Second-Order and First-Order Algorithms for Determining Optimal Control. A Differential Programming Approach, J. Optimiz. Theory Appl., 1968, vol. 2, no. 4, pp. 441–440.MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ortega, J.M. and Rheinboldt, W.C., Iterative Solution of Nonlinear Equations in Several Variables, New York: Academic, 1970.zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Institute of Systems Dynamics and Control TheorySiberian Branch of the Russian Academy of SciencesIrkutskRussia
  2. 2.Irkutsk Euro-Asian Linguistic Institute of the Federal State Government-Funded Educational Organization of Higher Professional Education “Moscow State Linguistic University,”IrkutskRussia

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