Persecution of rigidly coordinated evaders in a linear problem with fractional derivatives and a simple matrix

A. I. Machtakova

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 54 (2019), pp. 45-54.

Résumé

In the finite-dimensional Euclidean space, the problem of pursuit of a group of evaders by a group of pursuers is considered, which is described by a system of the form $$D^{(\alpha)} z_{ij} = a z_{ij} + u_i - v,$$ where $D^{(\alpha)} f$ is the Caputo derivative of the order $\alpha \in (0,1)$ of the function $f$. It is assumed that all evaders use the same control. The goal of the pursuers is to catch at least one of the evaders. The evaders use piecewise-program strategies, and the pursuers use piecewise-program counterstrategies. Every pursuer catches not more than one evader. The set of admissible controls is a ball of unit radius with the center at the origin, the target sets are the origin. In terms of initial positions and game parameters, a sufficient conditions for the capture are obtained.

Détail

Keywords: differential game, group persecution, pursuer, evader, fractional derivatives.

Bibliographie

[1] Pshenichnyi B. N., “Simple pursuit by several objects”, Cybernetics, 12:3 (1976), 484–485 | MR

[2] Chernous'ko F.L., “A problem of evasion from many pursuers”, Journal of Applied Mathematics and Mechanics, 40:1 (1976), 11–20 | DOI | Zbl

[3] Rikhsiev B. B., Differential games with simple mothion, Fan, Tashkent, 1992

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

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

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

[7] Satimov N., Mamatov M.Sh., “On problems of pursuit and evasion away from meeting and differential games between groups of pursuers and evaders”, Dokl. Akad. Nauk UzSSR, 1983, no. 4, 3–6 (in Russian)

[8] Petrov N. N., “Simple pursuit after a group of evaders”, Automation and Remote Control, 58:12 (1997), 1914–1919 | MR | Zbl

[9] Vagin D. A., Petrov N. N., “A problem of the pursuit of a group of rigidly connected evaders”, Journal of Computer and Systems Sciences International, 40:5 (2001), 749–753 | MR | Zbl

[10] Vagin D. A., Petrov N. N., “A problem of group pursuit with phase constraints”, Journal of Applied Mathematics and Mechanics, 66:2 (2002), 225–232 | DOI | MR | Zbl

[11] Vagin D. A., Petrov N. N., “Pursuit of a group of evaders in the Pontryagin example”, Journal of Applied Mathematics and Mechanics, 68:4 (2004), 555–560 | DOI | MR | Zbl

[12] Blagodatskikh A. I., “Two non-stationary pursuit problems of a rigidly connected evaders”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2008, no. 1, 47–60 (in Russian) | DOI

[13] Blagodatskikh A. I., “Multiple capture of rigidly coordinated evaders”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 26:1 (2016), 46–57 (in Russian) | DOI | Zbl

[14] Petrov N. N., Solov'eva N. A., “Problem of pursuit of a group of coordinated evaders in linear recurrent differential games”, Journal of Computer and Systems Sciences International, 51:6 (2012), 770–778 | DOI | MR | Zbl

[15] Vinogradova M. N., “On the capture of two evaders in a simple pursuit–evasion problem with phase restrictions”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2011, no. 4, 3–8 (in Russian) | DOI

[16] Vinogradova M. N., Petrov N. N., Solov'eva N. A., “Capture of two cooperative evaders in linear recurrent differential games”, Trudy Instituta Matematiki i Mekhaniki URO RAN, 19, no. 1 (2013), 41–48 (in Russian)

[17] Petrov N. N., Prokopenko V. A., “On a problem of the pursuit of a group of evaders”, Differ. Uravn., 23:4 (1987), 725–726 (in Russian)

[18] Sakharov D. V., “On two differential games of simple group pursuit”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, no. 1, 50–59 (in Russian) | DOI

[19] Petrov N. N., Solov'eva N. A., “Problem of group pursuit in linear recurrent differential games”, Journal of Mathematical Sciences, 230:5 (2018), 732–736 | DOI | Zbl

[20] Petrov N. N., “On a group pursuit problem”, Automation and Remote Control, 56:6 (1996), 808–813

[21] Petrov N. N., “About controllability of autonomous systems”, Differ. Uravn., 4:4 (1968), 606–617 (in Russian) | Zbl

[22] Popov A.Yu., Sedletskii A. M., “Distribution of roots of Mittag-Leffler functions”, Journal of Mathematical Sciences, 190:2 (2011), 209–409 | DOI | MR

[23] Dzhrbashyan M. M., Integral transformation and representation functions in complex area, Nauka, M., 1966

[24] Bichurina A. I., “Persecution of a group of rigidly coordinated evaders in a linear problem with fractional derivatives”, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 50 (2017), 13–20 (in Russian) | DOI

[25] Petrov N. N., “One problem of group pursuit with fractional derivatives and phase constraints”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 27:1 (2017), 54–59 (in Russian) | DOI | Zbl

[26] Chikrii A. A., Machikhin 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) | Zbl