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Time Optimality for Systems with Multidimensional Control and Vector Moment Min-Problem

  • V. I. KorobovEmail author
Article
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Abstract

The linear time-optimal problem for some classes of linear systems with a multidimensional control is considered. The general position condition, which guarantees the uniqueness of the optimal control, is not assumed to be satisfied. We introduce a vector moment min-problem, which is a further development of the moment min-problem proposed by V.I. Korobov and G.M. Sklyar in 1987 for solving linear time-optimal problems with a one-dimensional control. In the paper the case of the time-optimal problem for linear systems with two-dimensional control is thoroughly studied by use of the vector moment min-problem.

Keywords

Linear time-optimal control problem Vector moment min-problem Systems with multidimensional control Non-uniqueness of the optimal control 

Mathematics Subject Classification (2010)

49J15 

Notes

References

  1. 1.
    Pontryagin L S, Boltyanskii V G, Gamkrelidze R V, Mishchenko E F. 1962. The mathematical theory of optimal processes. Moskva Nauka 1961, Engl. transl.: Interscience Publishers John Wiley & Sons, Inc., New York-London.Google Scholar
  2. 2.
    Feldbaum A A. Basis of optimal automatic systems theory (Russian). Nauka: Moskva; 1966.Google Scholar
  3. 3.
    Gamkrelidze R V. On the theory of optimal processes in linear systems (Russian). Dokl Akad Nauk SSSR (N.S.) 1957;116:9–11.MathSciNetzbMATHGoogle Scholar
  4. 4.
    Gamkrelidze R V. Theory of processes in linear systems which are optimal with respect to rapidity of action (Russian). Izv Akad Nauk SSSR Ser Mat 1958;22(4):449–474.MathSciNetGoogle Scholar
  5. 5.
    Gamkrelidze R V, Vol. 7. Principles of optimal control theory (Russian). Izd-vo Tbilisskogo un-ta, Tbilisi (1977); Engl transl.: Mathematical Concepts and Methods in Science and Engineering. New York: Plenum Press; 1978.Google Scholar
  6. 6.
    Tchebichef P. Sur les valeurs limites des integrales. Journal de mathematiques pures et appliquees 2e serie 1874;19:157–160.zbMATHGoogle Scholar
  7. 7.
    Krein M G, Nudel’man A A. 1977. The Markov moment problem and extremal problems. Ideas and problems of P.L. Čebyšev and A.A. Markov and their further development. Moskva, Nauka (1973); Engl. transl.: Translations of Mathematical Monographs, 50, AMS, Providence R. I.Google Scholar
  8. 8.
    Karlin S, Stadden W. 1966. Tchebycheff systems: With applications in analysis and statistics. New York: Interscience Publishers John Wiley & Sons, v Pure and Applied Mathematics XV. Wiley.Google Scholar
  9. 9.
    Korobov V I, Sklyar G M. 1991. The Markov moment min-problem and time optimality (Russian) Sibirsk. Mat. Zh., 32, 60–71 (1991); Engl. transl, in: Siberian Math. J., 32, 46–55.Google Scholar
  10. 10.
    Korobov V I, Sklyar G M. 1990. The Markov moment problem on a minimally possible segment (Russian), Dokl. Akad. Nauk SSSR, 308, 525–528 (1989); Engl. transl, in: Soviet Math. Dokl., 40, 334–337.Google Scholar
  11. 11.
    Korobov V I, Sklyar G M. 1989. Time-optimality and the power moment problem (Russian), Mat. Sb. (N.S.), 134(176), 186–206 (1987); Engl. transl. in: Math. USSR-Sb., 62, 185–206.Google Scholar
  12. 12.
    Korobov V I, Sklyar G M. 1990. Time-optimality and the trigonometric moment problem (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 53, 868–885 (1989); Engl. transl. in: Math. USSR-Izv., 35, 203–220.Google Scholar
  13. 13.
    Korobov V I, Sklyar G M. 1988. Exact solution of an n-dimensional time-optimality problem (Russian), Dokl. Akad. Nauk SSSR, 298, 1304–1308 (1988); Engl. transl. in: Soviet Math. Dokl., 37, 247–250.Google Scholar
  14. 14.
    Korobov V I, Sklyar G M. Markov power min-problem with periodic gaps. J Math Sci 1996;80(1):1559–1581.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Korobov V I, Sklyar G M. 1992. The generating function method in the problem of moments with periodic gaps (Russian), Dokl. Akad. Nauk SSSR, 318(1), 32–35 (1991); Engl. transl. in: Soviet Math. Dokl., 43(3), 657–660.Google Scholar
  16. 16.
    Korobov V I, Bugaevskaya A N. The solution of time-optimal problem on the basis of the Markov moment min-problem with even gaps. Mat Fiz Anal Geom 2003;10: 505–523.MathSciNetzbMATHGoogle Scholar
  17. 17.
    Korobov V I, Bugaevskaya A N. Almost power sum systems. Math Comp 2016; 85:717–736.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Krein M G. 1962. L-moment problem in the abstract linear normed space, in N.I. Akhiezer, M.G. Krein, Some questions in the theory of moments (Russian), Kharkov, Nauchno-Tekhnich. Izdat. Ukrainy, 1938; Engl. transl. in: Translations of Mathematical Monographs 2. AMS, Providence, R.I.Google Scholar
  19. 19.
    Krasovsky N N. Theory of control of motion (Russian). Nauka: Moskva; 1968.Google Scholar
  20. 20.
    Butkovsky A G. Methods of controlling the distributed parameter systems (Russian). Nauka: Moskva; 1975.Google Scholar
  21. 21.
    Boltyansky V G. Mathematical methods of optimal control (Russian). Nauka: Moskva; 1969.Google Scholar
  22. 22.
    Korobov V I, Sklyar G M, Florinskii V V. A polynomial of minimal degree for determining all switching moments in a time optimal problem (Russian). Mat Fiz Anal Geom 2000;7(3):308–320.MathSciNetzbMATHGoogle Scholar
  23. 23.
    Korobov V I, Sklyar G M, Florinskii V V. 2002. A minimal polynomial for finding the switching times and the support vector to the controllability domain (Russian), Differ. Uravn., 38(1), 16–19 (2002); Engl. transl. in: Differ. Equ., 38(1), 15–18.Google Scholar
  24. 24.
    Korobov V I, Sklyar G M, Ignatovich SYU. Solving of the polynomial systems arising in the linear time-optimal control problem. Commun Math Anal conference, Conf 2011;3:153–171.MathSciNetzbMATHGoogle Scholar
  25. 25.
    Luenberger D. Canonical forms for linear multivariable systems. IEEE Trans Autom Cont 1967;12(3):290–293.MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.V. N. Karazin Kharkiv National UniversityKharkivUkraine

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