N.N. Petrov. A multiple capture in a group pursuit problem with fractional derivatives ... P. 156-164

In a finite-dimensional Euclidean space, we consider a problem of pursuing one evader by a group of pursuers with equal capabilities of all participants. The dynamics of the problem is described by the system
$$ D^{(\alpha)}z_i=az_i+u_i-v,\quad u_i,v\in V, $$
where $D^{(\alpha)}f$ is the Caputo derivative of order $\alpha\in(1,2)$ of the function $f$. The set of admissible controls $V$ is a strictly convex compact set and $a$ is a real number. The aim of the group of pursuers is to catch the evader by at least $m$ different pursuers, possibly at different times. The terminal sets are the origin. The pursuers use quasi-strategies. We obtain sufficient conditions for the solvability of the pursuit problem in terms of the initial positions. The investigation is based on the method of resolving functions, which allows us to obtain sufficient conditions for the termination of the approach problem in some guaranteed time.

Keywords: differential game, group pursuit, multiple capture, pursuer, evader.

The paper was received by the Editorial Office on September 25, 2017.

Funding Agency:

Russian Foundation for Basic Research (project no. 16-01-00346);

Ministry of Education and Science of the Russian Federation (project no. 1.5211.2017/8.9).

Nikolai Nikandrovich Petrov, Dr. Phys.-Math. Sci., Prof., Institute of Mathematics, Information
Technology and Physics Udmurt State University, Izhevsk, 426034 Russia, e-mail: kma3@list.ru .

REFERENCES

1. Krasovskii N.N., Subbotin A.I. Game-theoretical control problems. N Y, 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, 456 p.

2. Chikrii A.A. Conflict-controlled processes. Boston, London, Dordrecht, Kluwer Acad. Publ., 1997, 403 p. doi: 10.1007/978-94-017-1135-7 . Original Russian text published in Chikrii A.A. Konfliktno upravlyaemye protsessy. Kiev, Naukova Dumka, 1992, 384 p.

3. Grigorenko N.L. Matematicheskie metody upravleniya neskol’kimi dinamicheskimi protsessami. [Mathematical methods for control of several dynamic processes]. Moscow, Mosk. Gos. Univ. Publ., 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 University Publ., 2009, 266 p. ISBN: 978-5-904524-17-3 .

5. Eidel’man S.D., Chikrii A.A. Dynamic game problems of approach for fractional-order equations. Ukr. Math. J., 2000, vol. 52, no. 11, pp. 1787–1806. doi: 10.1023/A:1010439422856 .

6. Chikrii A.A., Matichin I.I. Game problems for fractional-order linear systems. Proc. Steklov Inst. Math., 2010, vol. 268, suppl. 1, pp. 54–70. doi: 10.1134/S0081543810050056 .

7. Chikrii A.A., Matichin I.I. On linear conflict-controlled processes with fractional derivatives. Tr. Inst. Mat. Mekh. UrO RAN, 2011, vol. 17, no. 2, pp. 256–270 (in Russian).

8. Grigorenko N.L. A game of simple pursuit – evasion for a group of pursuers and one evader. Vestn. Mosk. Univ., Ser. XV, 1983, no. 1, pp. 41-47 (in Russian).

9. Blagodatskikh A.I. Simultaneous multiple capture in a simple pursuit problem. J. Appl. Math. Mech., 2009, vol 73, no. 1, pp. 36–40. doi: 10.1016/j.jappmathmech.2009.03.010 .

10. Petrov N.N. Multiple capture in Pontryagin’s example with phase constraints. J. Appl. Math. Mech., 1997, vol. 61, no. 5, pp. 725–732. doi: 10.1016/S0021-8928(97)00095-6 .

11. Blagodatskikh A.I. Multiple capture in a Pontriagin’s problem. Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2009, no. 2, pp. 3–12.

12. Petrov N.N., Solov’eva N.A. Multiple capture in Pontryagin’s recurrent example with phase constraints. Proc. Steklov Inst. Math., 2016, vol. 293, no. 1, suppl. 1, pp. 174–182. doi: 10.1134/S0081543816050163 .

13. Petrov N. N., Solov’eva N.A. Multiple Capture in Pontryagin’s Recurrent Example. Automation and Remote Control, 2016, vol. 77, no. 5, pp. 855–861. doi: 10.1134/S0005117916050088 .

14. Blagodatskikh A.I. Simultaneous multiple capture in a conflict-controlled process. J. Appl. Math. Mech., 2013, vol. 77, no. 3, pp. 314–320. doi: 10.1016/j.jappmathmech.2013.09.007 .

15. Petrov N.N., Solov’eva N.A. A multiple capture of an evader in linear recursive differential games. Tr. Inst. Mat. Mekh. UrO RAN, 2017, vol. 23, no. 1, pp. 212–218. doi: 10.21538/0134-4889-2017-23-1-212-218 .

16. Blagodatskikh A.I. Multiple capture of rigidly coordinated evaders. Vestn. Udmurt. Univ. Mat. Mekh. Komp. Nauki, 2016, vol. 26, no. 1, pp. 46–57 (in Russian).

17. Petrov N.N. One problem of group pursuit with fractional derivatives and phase constraints. Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2017, vol. 27, no. 1, pp. 54–59. doi: 10.20537/vm170105 .

18. Caputo M. Linear model of dissipation whose q is almost frequency independent-II. Geophys. J. R. Astr. Soc., 1967, vol. 13, no. 5, pp. 529–539. doi: 10.1111/j.1365-246X.1967.tb02303.x .

19. Popov A.Y., Sedletskii A.M. Distribution of roots of Mittag-Leffler functions. J. Math. Sci., 2013, vol. 190, no. 3, pp. 209–409. doi: 10.1007/s10958-013-1255-3 .

20. Dzhrbashyan M.M. Integral’nye preobrazovaniya i predstavleniya funktsii v kompleksnoi oblasti [Integral transforms and representations of functions in the complex domain]. Moscow, Nauka Publ., 1966, 672 p.

21. Chikrii A.A., Matichin I.I. An analog of the Cauchy formula for linear systems of arbitrary fractional order. Dokl. NAN Ukrainy, 2007, no. 1, pp. 50–55.