A problem of pursuing a group of evaders by a group of pursuers with equal capabilities for all the participants is considered in a finite-dimensional Euclidean space $\mathbb R^k$. In a given time scale $T$, the problem is described by a system
$$ z_i^{\Delta}=u_i-v,$$
where $f^{\Delta}$ is the $\Delta$-derivative of $f$ in the time scale $T$. The set of admissible controls is a ball of unit radius centered at the origin. The terminal sets are the origin. In addition, it is assumed that all the evaders use the same control and, during the game, stay within a convex polyhedral set with nonempty interior. Sufficient conditions are obtained for the solvability of the problem of capturing at least one evader. The method of resolving functions is used as a basis of this research.
Keywords: differential game, pursuer, evader, group pursuit, time scale
Received March 29, 2021
Revised May 11, 2021
Accepted Junе 7, 2021
Funding Agency: This research was funded by the Ministry of Science and Higher Education of the Russian Federation in the framework of state assignment No. 075-00232-20-01, project FEWS-2020-0010 and under grant 20-01-00293 from the Russian Foundation for Basic Research.
Nikolai Nikandrovich Petrov, Dr. Phys.-Math. Sci., Prof., Udmurt State University, Izhevsk, 426034 Russia, e-mail: kma3@list.ru
REFERENCES
1. Krasovskii N.N., Subbotin A.I. Game-theoretical control problems. New York: Springer, 1988, 517 p. ISBN: 978-1-4612-8318-8 . Original Russian text published in Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial’nye igry. Moscow: Nauka Publ., 1974, 458 p.
2. Chikrii A.A. Conflict-controlled processes. Boston; London; Dordrecht: Kluwer Acad. Publ., 1997, 424 p. doi: 10.1007/978-94-017-1135-7 . Original Russian text published in Chikrii A.A. Konfliktno upravlyamye protsessy. Kiev: Nauk. Dumka, 1992, 384 p.
3. Grigorenko N.L. Matematicheskie metody upravleniya neskol’kimi processami [Mathematical methods for control of several dynamic processes]. Moscow: Moscow State University, 1990, 197 p. ISBN: 5-211-00954-1 .
4. Blagodatskikh A.I., Petrov N.N. Konfliktnoe vzaimodeistvie grupp upravlyaemykh ob’ektov [Conflict interaction of groups of controlled objects]. Izhevsk: Udmurt. State Univ. Publ., 2009. 266 p. ISBN: 978-5-904524-17-3 .
5. Bopardikar S.D., Suri S. k-Capture in multiagent pursuit evasion, or the lion and the gyenas. Teoretical Computer Science, 2014, vol. 522, pp. 13–23. doi: 10.1016/j.tcs.2013.12.001
6. Aulbach B., Hilger S. Linear dynamic processes with inhomogeneous time scale. In: “Nonlinear dynamics and quantum dynamical systems”. Contributions to the international seminar ISAM-90, Gaussig (GDR), March 19-23, 1990, G.A. Leonov, V. Reitmann, W. Timmermann (eds.), Ser. Math. Res., vol. 59. Berlin: Akademie-Verlag, 1990, pp. 9–20. ISBN: 9783055008580 .
7. Hilger S. Analysis on measure chains – a unified approach to continuous and discrete calculus. Results in Mathematics, 1990, vol. 18, pp. 18–56. doi: 10.1007/BF03323153
8. Benchohra M., Henderson J., Ntouyas S. Impulsive differential equations and inclusions. N Y: Hindawi Publ., 2006, 381 p. ISBN: 977-5945-50-X .
9. Bohner M., Peterson A. Advances in dynamic equations on time scales. Boston: Birkhauser, 2003, 348 p. doi: 10.1007/978-0-8176-8230-9
10. Martins N., Torres D. Necessary conditions for linear noncooperative N-player delta differential games on time scales. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, 2011, vol. 31, no. 1, pp. 23–37. doi: 10.7151/dmdico.1126
11. Petrov N. V. The problem of simple group pursuit with phase constraints in time scales. Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 2020, vol. 30, no. 2, pp. 249–258 (in Russian). doi: 10.35634/vm200208
12. Vagin D.A., Petrov N.N. A problem of the pursuit of a group of rigidly connected evaders. J. Comp. Systems Sci. International, 2001, vol. 40, no. 5, pp. 749–753.
13. Machtakova A. I. Persecution of rigidly coordinated evaders in a linear problem with fractional derivatives and a simple matrix. Izv. IMI Udmurtskogo Universiteta, 2019, vol. 54, pp. 45–54 (in Russian). doi: 10.20537/2226-3594-2019-54-04
14. Guseinov G.S. Integration on time scales. J. Math. Anal. Appl., 2003, vol. 285, no 1, pp. 107–127. doi: 10.1016/S0022-247X(03)00361-5
15. Cabada A., Vivero D. R. Expression of the Lebesgue $\Delta$-integral on time scales as a usual Lebesgue integral; application to the calculus of $\Delta$-antiderivatives. Math. Comp. Modelling, 2006, vol. 43, no 1-2, pp. 194–207. doi: 10.1016/j.mcm.2005.09.028
16. Petrov N.N. Controllability of autonomous systems. Differ. Uravn., 1968, vol. 4, no. 4, pp. 606–617 (in Russian).
17. Vinogradova M.N., Petrov N.N., Solov’eva N.A. Capture of two cooperative evaders in linear recurrent differential games. Trudy Inst. Mat. i Mekh. UrO RAN, 2013, vol. 19, no. 1, pp. 41–48 (in Russian).
18. Pshenichnyi B.N. Simple pursuit by several objects. Cybernetics, 1976, vol. 12, no. 3, pp. 484–485. doi: 10.1007/BF01070036
19. Ivanov R.P. Simple pursuit-evasion on a compactum. Dokl. Akad. Nauk SSSR, 1980, vol. 254, no. 6, pp. 1318–1321 (in Russian).
Cite this article as: N.N. Petrov. Matrix resolving functions in a linear problem of group pursuit with multiple capture, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 3, pp. 163–171.
