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A Nonlinear Control Problem with Several Goal Sets

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Abstract

A problem is considered on detour of a manifold group moving in a geometrical coordinate space, performed by a nonlinear controlled object in a prescribed order. The quality of the process is evaluated by a sum of terminal criteria calculated on these manifolds. The necessary optimality conditions for the control of the nonlinear object and the moments of rendezvous in the form of Pontryagin maximum principle are obtained without time decomposition.

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REFERENCES

  1. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, Mathematical Theory of Optimal Processes [in Russian], Nauka, Moscow (1961).

    Google Scholar 

  2. N. N. Krasovskii, Theory of Motion Control [in Russian], Nauka, Moscow (1968).

    Google Scholar 

  3. J. Warga, Optimal Control of Differential and Functional Equations [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  4. E. B. Lee and L. Markus, Foundations of Optimal Control Theory [Russian translation], Nauka, Moscow (1972).

    Google Scholar 

  5. B. N. Phenichnyi, “Linear differential games,” Avtom. Telemekh., No. 1, 65-78 (1968).

  6. A. B. Kurzhanskii and Yu. S. Osipov, “On a control problem with bounded phase coordinates,” Prikl. Mat. Mekh., 32, Issue 2, 194-203 (1968).

    Google Scholar 

  7. A. B. Kurzhanskii and Yu. S. Osipov, “On a control problem under restrained constraints,” Prikl. Mat. Mekh., 33, Issue 4, 705-719 (1969).

    Google Scholar 

  8. Yu. I. Berdyshev and A. G. Chentsov, “Optimization of a weighted criterion function in one control problem,” Kibernetika, No. 1, 67-75 (1986).

  9. Yu. I. Berdyshev and A. G. Chentsov, “Reduction of some linear optimal control problems with integral constraints,” Kibernetika, No. 4, 59-64 (1990).

  10. Yu. I. Berdyshev and A. G. Chentsov, “On some problems of sequential optimization of controlled systems,” Deposited in VINITI 05.01.83, No. 109- 83 Dep., Sverdlovsk (1983).

    Google Scholar 

  11. S. I. Morina and A. G. Chentsov, “Problem of sequential approach with combined constraints on a choice of control,” Deposited in VINITI 02.09.87, 6461-B87, Sverdlovsk (1987).

  12. V. A. Troitskii, “Variational problems of optimization of control processes with functionals depending on intermediate values of coordinates,” Prikl. Mat. Mekh., No. 6, 1003-1011 (1962).

  13. G. K. Zakharov, “Optimization of stepwise control systems,” Avtom. Telemekh., No. 8, 2-9 (1981).

  14. V. A. Medvedev and V. N. Rozova, “Optimal control of stepwise functions,” Avtom. Telemekh., No. 3, 15-23 (1972).

  15. A. N. Krasovskii and N. N. Krasovskii, Control under Lack of Information, Birkhauser, USA (1995).

    Google Scholar 

  16. N. N. Krasovskii and N. Yu. Lukoyanov, “A conflict-controlled problem with hereditary information,” Prikl. Mat. Mekh., 60, Issue 6, 885-900 (1996).

    Google Scholar 

  17. A. A. Chikrii, Conflict-Controlled Processes [in Russian], Kiev, Naukova Dumka (1992).

    Google Scholar 

  18. Yu. I. Berdyshev, “Problem of successive optimization without time decomposition,” Kibernetika, No. 4, 32-35 (1987).

  19. A. V. Arutyunov, Extremum Conditions. Abnormal and Degenerate Problems [in Russian], Faktorial (1997).

  20. N. N. Krasovskii and A. I. Subbotin, Positional Differential Games [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  21. R. V. Gamkrelidze, Fundamentals of Optimal Control [in Russian], Izd. Tbilis. Univ., Tbilisi (1977).

    Google Scholar 

  22. A. F. Filippov, “Differential equations with the discontinuous right-hand side,” in: Mat. Sb., 51 (93), Issue 1, 76-85 (1960).

    Google Scholar 

  23. A. I. Subbotin and A. G. Chentsov, Optimization of a Guarantee in Control Problems [in Russian], Nauka, Moscow (1981).

    Google Scholar 

  24. A. G. Chentsov, Applications of the Theory of Measure to Control Problems [in Russian], Sredne-Ural. Knizh. Izd., Sverdlovsk (1985).

    Google Scholar 

  25. V. D. Batukhtin and N. N. Krasovskii, “Extreme aiming in a nonlinear game of approach” in: Extremal Strategy in Positional Differential Games, UNTs, IMM AN SSSR, Sverdlovsk (1974), pp. 26-76.

    Google Scholar 

  26. N. Danford and J. T. Schwarz, Linear Operators. General Theory [Russian translation], Izd. Inostr. Lit., Moscow (1962).

    Google Scholar 

  27. A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  28. R. Gabasov and F. M. Kirillova, Maximum Principle in Optimal Control Theory [in Russian], Nauka i Tekhn., Minsk (1974).

    Google Scholar 

  29. E. A. Barbashin, Introduction to the Theory of Stability [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  30. V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations [in Russian], Gostekhizdat, Moscow-Leningrad (1949).

    Google Scholar 

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Authors and Affiliations

  1. Ural Department of the Institute of Mathematics and Mechanics, Russian Academy of Sciences, Ekaterinburg, Russia

    Yu. I. Berdyshev

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  1. Yu. I. Berdyshev
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Berdyshev, Y.I. A Nonlinear Control Problem with Several Goal Sets. Cybernetics and Systems Analysis 38, 558–567 (2002). https://doi.org/10.1023/A:1021158202855

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