phone +7 (3412) 91 60 92
mail imi@udsu.ru
ru-RU Russian
Home » Issues » Archive of Issues » 2021 - 4 issue » pp. 613-628

Archive of Issues

Russia Yekaterinburg
Year
2021
Volume
31
Issue
4
Pages
613-628
<<
>>
Section Mathematics
Title On guarantee optimization in control problem with finite set of disturbances
Author(-s) Gomoyunov M.I.ab, Serkov D.A.ab
Affiliations Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa, Ural Federal Universityb
Abstract In this paper, we deal with a control problem under conditions of disturbances, which is stated as a problem of optimization of the guaranteed result. Compared to the classical formulation of such problems, we assume that the set of admissible disturbances is finite and consists of piecewise continuous functions. In connection with this additional functional constraint on the disturbance, we introduce an appropriate class of non-anticipative control strategies and consider the corresponding value of the optimal guaranteed result. Under a technical assumption concerning a property of distinguishability of the admissible disturbances, we prove that this result can be achieved by using control strategies with full memory. As a consequence, we establish unimprovability of the class of full-memory strategies. A key element of the proof is a procedure of recovering the disturbance acting in the system, which allows us to associate every non-anticipative strategy with a full-memory strategy providing a close guaranteed result. The paper concludes with an illustrative example.
Keywords control problem under disturbances, optimal guaranteed result, non-anticipative strategy, full-memory strategy, recovery of disturbances, unimprovability
UDC 517.977
MSC 49N30, 49N35, 93C15
DOI 10.35634/vm210406
Received 4 October 2021
Language English
Citation Gomoyunov M.I., Serkov D.A. On guarantee optimization in control problem with finite set of disturbances, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, vol. 31, issue 4, pp. 613-628.
References
  1. Isaacs R. Differential games, New York: John Wiley and Sons, 1965. https://zbmath.org/?q=an:0125.38001
  2. Pontrjagin L.S. Linear differential games of pursuit, Mathematics of the USSR-Sbornik, 1981, vol. 40, no. 3, pp. 285-303. https://doi.org/10.1070/SM1981v040n03ABEH001815
  3. Krasovskii N.N., Subbotin A.I. Game-theoretical control problems, New York: Springer, 1988. https://www.springer.com/gp/book/9781461283188
  4. Krasovskii A.N., Krasovskii N.N. Control under lack of information, Boston: Birkhäuser, 1995. https://doi.org/10.1007/978-1-4612-2568-3
  5. Kryazhimskii A.V. The problem of optimization of the ensured result: unimprovability of full-memory strategies, Constantin Caratheodory: An International Tribute, Teaneck, New Jersey: World Scientific, 1991, vol. 1, pp. 636-675. https://doi.org/10.1142/9789814350921_0037
  6. Serkov D.A. On a dynamic game problem with an indecomposable set of disturbances, Ural Mathematical Journal, 2019, vol. 5, no. 2, pp. 72-79. https://doi.org/10.15826/umj.2019.2.007
  7. Gomoyunov M.I., Serkov D.A. Non-anticipative strategies in guarantee optimization problems under functional constraints on disturbances, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 4, pp. 553-571. https://doi.org/10.35634/vm200402
  8. Barron E.N. Differential games with Lipschitz control functions and fixed initial control positions, Journal of Differential Equations, 1977, vol. 26, issue 2, pp. 161-180. https://doi.org/10.1016/0022-0396(77)90188-7
  9. Nikol'skij M.S. A crossing problem with possible stop of engine, Differential Equations, 1993, vol. 29, issue 11, pp. 1681-1684. https://zbmath.org/?q=an:0816.90147
  10. Ledyaev Yu.S. Program-predictive feedback control for systems with evolving dynamics, IFAC-PapersOnLine, 2018, vol. 51, issue 32, pp. 723-726. https://doi.org/10.1016/j.ifacol.2018.11.461
  11. Ukhobotov V.I. On a control problem under disturbance and possible breakdown, Proceedings of the Steklov Institute of Mathematics, 2019, vol. 307, suppl. 1, pp. 159-171. https://doi.org/10.1134/S0081543819070137
  12. Ushakov V.N., Ershov A.A. On recovering of unknown constant parameter by several test controls, Ufa Mathematical Journal, 2020, vol. 12, no. 4, pp. 99-113. https://doi.org/10.13108/2020-12-4-99
  13. Roxin E. Axiomatic approach in differential games, Journal of Optimization Theory and Applications, 1969, vol. 3, issue 3, pp. 153-163. https://doi.org/10.1007/BF00929440
  14. Elliott R.J., Kalton N.J. The existence of value in differential games of pursuit and evasion, Journal of Differential Equations, 1972, vol. 12, issue 3, pp. 504-523. https://doi.org/10.1016/0022-0396(72)90022-8
  15. Serkov D.A. On the unimprovability of full-memory strategies in problems of guaranteed result optimization, Proceedings of the Steklov Institute of Mathematics, 2015, vol. 291, suppl. 1, pp. 157-172. https://doi.org/10.1134/S0081543815090114
  16. Vinter R.B. Minimax optimal control, SIAM Journal on Control and Optimization, 2005, vol. 44, issue 3, pp. 939-968. https://doi.org/10.1137/S0363012902415244
Full text
/doc/issues-2021/21-04-06.pdf
<< Previous article
Next article >>