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On the theory of singular optimal controls in dynamic systems with control delay

  • M. J. Mardanov
  • T. K. Melikov
Article

Abstract

An optimal control problem with a control delay is considered, and a more broad class of singular (in classical sense) controls is investigated. Various sequences of necessary conditions for the optimality of singular controls in recurrent form are obtained. These optimality conditions include analogues of the Kelley, Kopp–Moyer, R. Gabasov, and equality-type conditions. In the proof of the main results, the variation of the control is defined using Legendre polynomials.

Keywords

singular control optimal control variation transformation method Legendre polynomial necessary optimality conditions 

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References

  1. 1.
    A. Halanay, “Optimal controls for systems with time lag,” SIAM J. Control 6 (2), 215–234 (1968).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    G. L. Haratišvili and T. A. Tadumadze, “Nonlinear optimal control systems with variable time lags,” Math. USSR-Sb. 36 (6), 863–881 (1979).CrossRefzbMATHGoogle Scholar
  3. 3.
    T. A. Tadumadze, Topics in Qualitative Theory of Optimal Control (Tbilis. Gos. Univ., Tbilisi, 1983) [in Russian].zbMATHGoogle Scholar
  4. 4.
    M. J. Mardanov, “Necessary optimality conditions in systems with lags and phase constraints,” Math. Notes 42 (5), 880–887 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A. V. Arutyunov and M. J. Mardanov, “On the theory of maximum principle in problems with delays,” Differ. Uravn. 25 (12), 2048–2058 (1989).MathSciNetzbMATHGoogle Scholar
  6. 6.
    M. J. Mardanov, Investigation of Optimal Processes with Delay under Constraints (Elm, Baku, 2009) [in Russian].Google Scholar
  7. 7.
    A. S. Matveev, “Optimal control problems with delays of general form and with phase constraints,” Math. USSR-Izv. 33 (3), 521–552 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    V. A. Srochko, “Optimality of singular controls in systems with aftereffect,” Differ. Uravn. 12 (12), 2275–2278 (1976).Google Scholar
  9. 9.
    K. T. Akhmedov, T. K. Melikov, and K. K. Gasanov, “On optimality of singular controls in delay systems,” Dokl. Akad. Nauk Az. SSR 31 (7), 7–10 (1975).zbMATHGoogle Scholar
  10. 10.
    T. K. Melikov, Candidate’s Dissertation in Mathematics and Physics (Baku, 1976).Google Scholar
  11. 11.
    L. T. Ashchepkov and D. S. Eppel’, “Analogue of Kelley’s condition in optimal delay systems,” Differ. Uravn. 10 (4), 591–597 (1974).Google Scholar
  12. 12.
    M. J. Mardanov, “On optimality conditions for singular controls,” Dokl. Akad. Nauk SSSR 253 (4), 815–818 (1980).MathSciNetGoogle Scholar
  13. 13.
    K. B. Mansimov, “Multipoint necessary conditions for optimality of singular (in the classical sense) controls in delay control systems,” Differ. Uravn. 21 (3), 527–530 (1985).zbMATHGoogle Scholar
  14. 14.
    K. B. Mansimov, Singular Controls in Delay Systems (Elm, Baku, 1999) [in Russian].Google Scholar
  15. 15.
    K. K. Gasanov and B. M. Usifov, “Induction analysis of singular control in systems with delay,” Autom. Remote Control 43 (6), 732–736 (1982).Google Scholar
  16. 16.
    T. K. Melikov, “Recurrent optimality conditions for singular controls in delay systems,” Dokl. Akad. Nauk 322 (5), 843–846 (1992).Google Scholar
  17. 17.
    T. K. Melikov, “An analogue of the Kelley condition for optimal systems with aftereffect of neutral type,” Comput. Math. Math. Phys. 38 (9), 1429–1438 (1998).MathSciNetzbMATHGoogle Scholar
  18. 18.
    T. K. Melikov, “Optimality of singular controls in systems with aftereffects of neutral type,” Comput. Math. Math. Phys. 41 (9), 1267–1278 (2001).MathSciNetzbMATHGoogle Scholar
  19. 19.
    T. K. Melikov, Singular Controls in Systems with Aftereffect (Elm, Baku, 2002) [in Russian].Google Scholar
  20. 20.
    L. E. Zabello, “Investigation of singular controls in delay systems with the help of matrix pulses,” Differ. Uravn., No. 8, 1332–1341 (1984).Google Scholar
  21. 21.
    M. J. Mardanov and T. K. Melikov, “Recurrent optimality conditions of singular controls in delay control systems”, The 3rd International Conference on Problems of Cybernetics and informatics, PCI 2010, September 6–8, 2010 (Baku, 2010), pp. 3–5.Google Scholar
  22. 22.
    M. J. Mardanov, “Legendre necessary conditions in delay control optimization problems,” Dokl. Akad. Nauk SSSR 297 (4), 795–797 (1987).Google Scholar
  23. 23.
    M. J. Mardanov, “Necessary second-order conditions for optimality in problems with delays in controls,” Russ. Math. Surv. 43 (4), 221–222 (1988).CrossRefzbMATHGoogle Scholar
  24. 24.
    V. Yu. Guliev, Candidate’s Dissertation in Mathematics and Physics (Baku, 1985).Google Scholar
  25. 25.
    R. K. Gabasov, “On the theory of necessary conditions for optimality of singular controls,” Dokl. Akad. Nauk SSSR 183 (2), 300–302 (1968).MathSciNetGoogle Scholar
  26. 26.
    H. J. Kelley, “A second variation test for singular extremals,” AIAA J. 2 (8), 1380–1382 (1964).MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    H. J. Kelley, R. E. Kopp, and H. G. Moyer, “Singular extremals,” in Topics in Optimization, Ed. by G. Leitman (Academic, New York, 1967), pp. 63–101.CrossRefGoogle Scholar
  28. 28.
    M. Yu. Kiseleva and V. I. Smagin, “Control with a prognostic model with allowance for control delay,” Vestn. Tomsk. Gos. Univ. Upr., Vychisl. Tekh. Inf. 11 (2), 5–12 (2010).Google Scholar
  29. 29.
    G. Sansone, Ordinary Differential Equations (Inostrannaya Literatura, Moscow, 1954), Vol. 2 [in Russian].Google Scholar
  30. 30.
    L. I. Rozonoer, “Pontryagin maximum principle in the theory of optimal systems III,” Avtom. Telemekh. 20 (12), 1561–1578 (1959).Google Scholar
  31. 31.
    R. E. Kopp and H. G. Moyer, “Necessary condition for singular extremals,” AIAA J. 3 (8), 1439–1444 (1965).CrossRefzbMATHGoogle Scholar
  32. 32.
    B. S. Goh, “Necessary conditions for singular extremals involving multiple control variables,” SIAM J. Control 4, 716–731 (1966).MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    A. A. Bolonkin, “Singular extremals in optimal control problems,” Izv. Akad. Nauk SSSR, Tekh. Kibern., No. 2, 187–198 (1969).MathSciNetzbMATHGoogle Scholar
  34. 34.
    D. H. Jacobson, “A new necessary condition of optimality for singular control problems,” SIAM J. Control 7 (4), 578–595 (1969).MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    R. Gabasov and F. M. Kirillova, Singular Optimal Controls (Nauka, Moscow, 1973) [in Russian].zbMATHGoogle Scholar
  36. 36.
    A. A. Agrachev and R. V. Gamkrelidze, “A second-order optimality principle for a time optimal control problem,” Math. USSR-Sb. 29 (4), 547–576 (1976).CrossRefzbMATHGoogle Scholar
  37. 37.
    I. B. Vapnyarskii, “An existence theorem for optimal control in the Bolza problem, some its applications, and the necessary conditions for the optimality of moving and singular systems,” USSR Comput. Math. Math. Phys. 7 (2), 22–54 (1967).MathSciNetCrossRefGoogle Scholar
  38. 38.
    V. A. Srochko, “Method of variation transformation in the theory of singular controls,” Differ. Integr. Uravn. Irkutsk. Gos. Univ., No. 2, 70–80 (1973).Google Scholar
  39. 39.
    V. A. Srochko, “Study of the second variation under singular controls,” Differ. Uravn. 10 (6), 1050–1066 (1974).Google Scholar
  40. 40.
    V. A. Srochko, “Multipoint optimality conditions for singular controls,” in Numerical Methods in Analysis (Applied Mathematics) (Sib. Energ. Inst. Sib. Otd. Akad. Nauk SSSR, Irkutsk, 1976), pp. 43–50 [in Russian].Google Scholar
  41. 41.
    A. Krener, “The high order maximal order maximal principle and its application to singular extremals,” SIAM J. Control Optim. 15 (2), 256–293 (1977).MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    S. L. Kaganovich, “On an inductive method for studying singular extremals,” Autom. Remote Control 37 (11), 1651–1660 (1976).zbMATHGoogle Scholar
  43. 43.
    V. V. Gorokhovik and S. Ya. Gorokhovik, “Different forms of the generalized Legendre–Clebsch conditions,” Autom. Remote Control 43 (7), 860–865 (1982).MathSciNetzbMATHGoogle Scholar
  44. 44.
    E. E. Barbashina, “Kopp–Moyer type necessary conditions for Goursat–Darboux systems,” Differ. Uravn. 25 (6), 1045–1047 (1989).zbMATHGoogle Scholar
  45. 45.
    T. K. Melikov, “Recurrent conditions for the optimality of singular controls in Goursat–Darboux systems,” Dokl. Akad. Nauk Az. SSR 14 (8), 6–10 (1990).zbMATHGoogle Scholar
  46. 46.
    T. K. Melikov, “Necessary conditions for high-order optimality,” Comput. Math. Math. Phys. 35 (7), 907–911 (1995).MathSciNetzbMATHGoogle Scholar
  47. 47.
    A. D. Kudryavtsev, A Course of Calculus (Vysshaya Shkola, Moscow, 1981), Vol. 2 [in Russian].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Institute of Mathematics and MechanicsNational Academy of Sciences of AzerbaijanBakuAzerbaijan
  2. 2.Institute of Control SystemsNational Academy of Sciences of AzerbaijanBakuAzerbaijan

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