phone +7 (3412) 91 60 92

Archive of Issues

Russia Chelyabinsk
Section Mathematics
Title Synthesis of controls in a single-type game problem of pulse meeting at fixed time with a terminal set in the form of a ring
Author(-s) Ukhobotov V.I.a, Izmest'ev I.V.a
Affiliations Chelyabinsk State Universitya
Abstract We consider a linear differential game with a pulse control of the first player. The abilities of the first player are determined by the stock of resources that the player can use when forming his control. At certain instants of time a separation of part of the resources stock is possible, which may implicate an “instantaneous” change of a phase vector, resulting in the complication of the problem. The control of the second player has geometrical constraints. The vectograms of the players are described by the same ball with different time-dependent radii. The terminal set of the game is determined by the condition of belonging the norm of a phase vector to a segment with positive ends. In this paper, a set defined by this condition is called a ring. The aim of the first player is to lead a phase vector to the terminal set at fixed time. The aim of the second player is opposite. With the maximal stable bridge, which has been defined by the authors previously, optimal controls of players are constructed.
Keywords pulse control, optimal control, differential game
UDC 517.977.80
MSC 91A23, 49N75
DOI 10.20537/vm170107
Received 12 December 2016
Language Russian
Citation Ukhobotov V.I., Izmest'ev I.V. Synthesis of controls in a single-type game problem of pulse meeting at fixed time with a terminal set in the form of a ring, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 1, pp. 69-85.
  1. Krasovskii N.N. Teoriya upravleniya dvizheniem (Theory of motion control), Moscow: Nauka, 1968, 475 p.
  2. Krasovskii N.N. On a problem of tracking, Journal of Applied Mathematics and Mechanics, 1963, vol. 27, issue 2, pp. 363-377. DOI: 10.1016/0021-8928(63)90006-6
  3. Krasovskii N.N., Repin Yu.M., Tret'yakov V.E. Some game situations in theory of control systems, Izvestiya Akademii Nauk SSSR. Tekhnicheskaya Kibernetika, 1965, no. 4, pp. 3-23 (in Russian).
  4. Krasovskii N.N., Tret'yakov V.E. On a pursuit problem in the case of restrictions on the impulses of control forces, Differ. Uravn., 1966, vol. 2, no. 5, pp. 587-599 (in Russian).
  5. Pozharitskii G.K. Game problem of impulse encounter with an opponent limited in energy, Journal of Applied Mathematics and Mechanics, 1975, vol. 39, issue 4, pp. 555-565. DOI: 10.1016/0021-8928(75)90056-8
  6. Subbotina N.N., Subbotin A.I. Alternative for the encounter-evasion differential game with constraints on the momenta of the players' controls, Journal of Applied Mathematics and Mechanics, 1975, vol. 39, issue 3, pp. 376-385. DOI: 10.1016/0021-8928(75)90002-7
  7. Petrov N.N. A problem of group pursuit in the class of impulse strategies of pursuers, Journal of Computer and Systems Sciences International, 2009, vol. 48, issue 2, pp. 199-205. DOI: 10.1134/S106423070902004X
  8. Ukhobotov V.I. On a class of linear differential games with impulse controls, Journal of Applied Mathematics and Mechanics, 1974, vol. 38, issue 4, pp. 550-557. DOI: 10.1016/0021-8928(74)90002-1
  9. Kumkov S.I., Patsko V.S. Information sets in the pulse control problem, Automation and Remote Control, 1997, vol. 58, no. 7, part 2, pp. 1224-1234.
  10. Ukhobotov V.I. A linear differential game with constraints imposed on the control impulses, Journal of Applied Mathematics and Mechanics, 1988, vol. 52, issue 3, pp. 277-283. DOI: 10.1016/0021-8928(88)90078-0
  11. Ukhobotov V.I. Metod odnomernogo proektirovaniya v lineinykh differentsial'nykh igrakh s integral'nymi ogranicheniyami (Method of one-dimensional projecting in linear differential games with integral constraints), Chelyabinsk: Chelyabinsk State University, 2005, 124 p.
  12. Ukhobotov V.I., Zaitseva O.V. A linear problem of pulse encounter at a given time under interference, Proceedings of the Steklov Institute of Mathematics, 2011, vol. 272, suppl. 1, pp. 215-228. DOI: 10.1134/S0081543811020167
  13. Chikrii A.A., Matichin I.I., Chikrii K.A. Conflict control processes with discontinuous trajectories, Kibernetika i Sistemnyi Analiz, 2004, no. 6, pp. 15-29 (in Russian).
  14. Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial'nye igry (Positional differential games), Moscow: Nauka, 1974, 456 p.
  15. Ukhobotov V.I., Izmest'ev I.V. Single-type problem of pulse meeting in fixed time with terminal set in form of a ring, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2015, vol. 25, issue 2, pp. 197-211 (in Russian). DOI: 10.20537/vm150205
  16. Chernous'ko F.L., Melikyan A.A. Igrovye zadachi upravleniya i poiska (Game problems of control and search), Moscow: Nauka, 1978, 270 p.
  17. Lyusternik L.A., Sobolev V.I. Elementy funktsional'nogo analiza (Elements of functional analysis), Moscow: Nauka, 1965, 520 p.
  18. Ukhobotov V.I., Izmest'ev I.V. Single-type differential games with terminal set in form of a ring, Dinamika sistem i processy upravleniya: Trudy Mezhdunarodnoi konferentsii (System dynamic and control processes: Proceedings of Int. Conf. Dedicated to the 90th Anniversary of Academician N.N. Krasovskii), Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, 2015, pp. 325-332 (in Russian).
  19. Kolmogorov A.N., Fomin S.V. Elementy teorii funktsii i funktsional'nogo analiza (Elements of functions theory and functional analysis), Moscow: Nauka, 1972, 496 p.
Full text
<< Previous article
Next article >>