Geodesic
One problem of group pursuit with fractional derivatives and phase constraints
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 27 (2017) no. 1, pp. 54-59 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In the finite-dimensional Euclidean space, we consider the problem of persecution of one evader by the group of pursuers, which is described by the system $$D^{(\alpha)}z_i = a z_i + u_i - v,$$ where $D^{(\alpha)}f$ is the Caputo derivative of order $\alpha \in (0, 1)$ of the function $f$. It is further assumed that the evader does not leave the convex polyhedron with nonempty interior. The evader uses piecewise-program strategies, and the pursuers use piecewise-program counterstrategies. The set of admissible controls is a convex compact, the target sets are the origin of coordinates, and $a$ is a real number. In terms of the initial positions and the parameters of the game, sufficient conditions for the solvability of the pursuit problem are obtained.
Keywords: differential game, group pursuit, phase restrictions, pursuer, evader. 
@article{VUU_2017_27_1_a4,
     author = {N. N. Petrov},
     title = {One problem of group pursuit with fractional derivatives and phase constraints},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {54--59},
     year = {2017},
     volume = {27},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VUU_2017_27_1_a4/}
}
TY  - JOUR
AU  - N. N. Petrov
TI  - One problem of group pursuit with fractional derivatives and phase constraints
JO  - Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
PY  - 2017
SP  - 54
EP  - 59
VL  - 27
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VUU_2017_27_1_a4/
LA  - ru
ID  - VUU_2017_27_1_a4
ER  - 
%0 Journal Article
%A N. N. Petrov
%T One problem of group pursuit with fractional derivatives and phase constraints
%J Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
%D 2017
%P 54-59
%V 27
%N 1
%U http://geodesic.mathdoc.fr/item/VUU_2017_27_1_a4/
%G ru
%F VUU_2017_27_1_a4
N. N. Petrov. One problem of group pursuit with fractional derivatives and phase constraints. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 27 (2017) no. 1, pp. 54-59. http://geodesic.mathdoc.fr/item/VUU_2017_27_1_a4/

[1] Krasovskii N. N., Subbotin A. I., Positional differential games, Nauka, M., 1974, 456 pp.

[2] Petrosyan L. A., Differential games of pursuit, Leningrad State University, L., 1977, 222 pp.

[3] Rikhsiev B. B., Differential games with simple motion, Fan, Tashkent, 1989, 232 pp.

[4] Chikrii A. A., Conflict-controlled processes, Springer Netherlands, 1997, 404 pp. | DOI | MR

[5] Grigorenko N. L., Mathematical methods of control over multiple dynamic processes, Moscow State University, M., 1990, 197 pp.

[6] Blagodatskikh A. I., Petrov N. N., Conflict interaction of groups of controlled objects, Udmurt State University, Izhevsk, 2009, 266 pp.

[7] Eidel'man S. D., Chikrii A. A., “Dynamic game problems of approach for fractional-order equations”, Ukrainian Mathematical Journal, 52:11 (2000), 1787–1806 | DOI | MR | Zbl

[8] Chikrii A. A., Matichin I. I., “Game problems for fractional-order linear systems”, Proceedings of the Steklov Institute of Mathematics, 268, suppl. 1 (2010), 54–70 | DOI | MR | Zbl

[9] Chikrii A. A., Matichin I. I., “On linear conflict-controlled processes with fractional derivatives”, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 17, no. 2, 2011, 256–270 (in Russian)

[10] Petrov N. N., “To a nonstationary group pursuit problem with phase constraints”, Automation and Remote Control, 75:8 (2014), 1525–1531 | DOI | MR

[11] Blagodatskikh A. I., Petrov N. N., “Group pursuit with state constraints in Pontryagin's almost periodic example”, Differential Equations, 51:3 (2015), 391–398 | DOI | DOI | MR | Zbl

[12] Petrov N. N., Solov'eva N. A., “Multiple capture in Pontryagin's recurrent example with phase constraints”, Proceedings of the Steklov Institute of Mathematics, 293, Suppl. 1 (2016), 174–182 | DOI | MR

[13] Caputo M., “Linear models of dissipation whose $Q$ is almost frequency independent-II”, Geophysical Journal International, 13:5 (1967), 529–539 | DOI

[14] Chikrii A. A., Matichin I. I., “On an analogue of the Cauchy formula for linear systems of any fractional order”, Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky, 2007, no. 1, 50–55 (in Russian)

[15] Popov A. Yu., Sedletskii A. M., “Distribution of roots of Mittag–Leffler functions”, Journal of Mathematical Sciences, 190:2 (2013), 209–409 | DOI | MR

[16] Dzhrbashyan M. M., Integral transformations and representations of functions in the complex domain, Nauka, M., 1966, 672 pp.