2021, issue 1, p. 5-15
Received 18.03.2021; Revised 24.03.2021; Accepted 25.03.2021
Published 30.03.2021; First Online 03.04.2021
One Game Problem for Oscillatory Systems
G.Ts. Chikrii 1 * , K.I. Rastvorova 2
1 V.M. Glushkov Institute of Cybernetics of NAS of Ukraine, Kyiv
2 The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
The paper concerns the linear differential game of approaching a cylindrical terminal set. We study the case when classic Pontryagin’s condition does not hold. Instead, the modified considerably weaker condition, dealing with the function of time stretching, is used. The latter allows expanding the range of problems susceptible to analytical solution by the way of passing to the game with delayed information. Investigation is carried out in the frames of Pontryagin’s First Direct method that provides hitting the terminal set by a trajectory of the conflict-controlled process at finite instant of time. In so doing, the pursuer’s control, realizing the game goal, is constructed on the basis of the Filippov-Castaing theorem on measurable choice. The outlined scheme is applied to solving the problem of pursuit for two different second-order systems, describing damped oscillations. For this game, we constructed the function of time stretching and deduced conditions on the game parameters, ensuring termination of the game at a finite instant of time, starting from arbitrary initial states and under all admissible controls of the evader.
Keywords: differential game, time-variable information delay, Pontryagin’s condition, Aumann’s integral, principle of time stretching, Minkowski’ difference, damped oscillations.
Cite as: Chikrii G.Ts., Rastvorova K.I. One Game Problem for Oscillatory Systems. Cybernetics and Computer Technologies. 2021. 1. P. 5–15. https://doi.org/10.34229/2707-451X.21.1.1
1. Isaacs R.F. Differential Games. New York-London-Sydney: Wiley Interscience, 1965. 479 p.
2. Pontryagin L.S. Selected Scientific Papers, 2. Moscow: Nauka, 1988. 576 p. (in Russian)
3. Krasovskii N.N. Game Problems on the Encounter of Motions. Moscow: Nauka, 1970. 420 p. (in Russian)
4. Berkovitz L.D. Differential games of generalized pursuit and evasion. SIAM, Control and Optimization. 1986. 24 (53). P. 361–373. https://doi.org/10.1137/0324021
5. Friedman A. Differential Games. New York: Wiley Interscience, 1971. 350 p.
6. Hayek O. Pursuit Games. New York: Academic Press, 1975. 266 p.
7. Pshenitchny B.N. ɛ-strategies in Differential Games, Topics in Differential Games. New York, London, Amsterdam: North Holland Publ. Co., 1973. P. 45–99.
8. Pshenitchnyi B.N., Chikrii A.A., Rappoport J.S. Group pursuit in differential games. J. Leipzig Techn High School. 1982. P.13–27.
9. Nikolskij M.S. L.S. Pontryagin’s First Direct Method in Diffential Games. Izdat. Gos Univ., Moscow. 1984. 65 p. (in Russian)
10. Chikrii A.A., Eidelman S.D. Game problems for fractional quasi-linear systems. Computers and Mathematics with Applications. 2002. 44 (7). P. 835–851. https://doi.org/10.1016/S0898-1221(02)00197-9
11. Dziubenko K.G., Chikrii A.A. An approach problem for a discrete system with random perturbations. Cybernetics and Systems Analysis. 2010. 46 (2). P. 271–281. https://doi.org/10.1007/s10559-010-9204-3
12. Siouris G. Missile Guidance and Control Systems. NewYork: Springer-Verlag, 2004. 666 p.
13. Chikrii G.Ts. On a problem of pursuit under variable information time lag on the availability of a state vector. Dokl. Akad. Nauk Ukrainy. 1979. 10. P. 855–857. (in Russian)
14. Chikrii G.Ts. An approach to solution of linear differential games with variable information delay. Journal of Automation and Information Sciences. 1995. 27 (3–4). P. 163–170.
15. Nikolskij M.S. Application of the first direct method in the linear differential games. Izvestia Akad. Nauk SSSR. 1972. 10 (17). P. 51–56. (in Russian)
16. Chikrii A.A. Conflict-Controlled Processes. Boston, London, Dordrecht: Springer Science & Business Media, 2013. 424 p.
17. Mezentsev A.V. On some class of differential games. Izvestia AN SSSR, Techn. kib. 1971. 6. P. 3–7. (in Russian)
18. Zonnevend D. On one method of pursuit. Doklady Akademii Nauk SSSR. 1972. 204 (6). P. 1296–1299. (in Russian)
19. Chikrii G.Ts. Using impact of information delay for solution of game problems of pursuit. Dopovidi Natsional’noi Akademii Nauk Ukrainy. 1999. 12. P. 107–111.
20. Chikrii G.Ts. Using the effect of information delay in differential pursuit games. Cybernetics and Systems Analysis. 2007. 43 (2). P. 233–245. https://doi.org/10.1007/s10559-007-0042-x
21. Chikrii G.Ts. Principle of time stretching in evolutionary games of approach. Journal of Automation and Information Sciences. 2016. 48 (5). P. 12–26. https://doi.org/10.1615/JAutomatInfScien.v48.i5.20
22. Chikrii G.Ts. Principle of time stretching for Motion Control in Condition of Conflict. Chapter in the book “Advanced Control Systems: Theory an Applications”, River Publishers, 2021. P. 52–82.
23. Kolmogorov A.N., Fomin S.V. Elements of Theory of Functions and Functional Analysis. Moscow: Nauka, 1989. 624 p. (in Russian)
24. Aumann R.J. Integrals of set-valued functions. J. Math. Anal. Appl. 1965. 12. P. 1–12. https://doi.org/10.1016/0022-247X(65)90049-1
25. Filippov A.F. Differential equations with discontinuous righthand sides. Dordrecht, Boston: Kluwer Publishers, 1988. 258 p. https://doi.org/10.1007/978-94-015-7793-9
26. Vasilenko N.V. Theory of Oscillations. Kiev: Vyshcha Shkola, 1992. 430 p. (in Russian)
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