Geodesic
Group pursuit of two evaders in a linear game with a simple matrix
Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 50 (2017), pp. 21-28.

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We consider a linear nonstationary problem of pursuit of a group of two evaders by a group of pursuers with equal dynamic capabilities of all participants. The laws of motion of the participants have the form $\dot z(t)+a(t)z=u(t).$ For $t=t_0$ initial conditions are given. Phase state restrictions are imposed on the evaders state under the assumption that they use the same control. Geometric constraints on the controls are strictly convex compact set with a smooth boundary, terminal sets are the origin of coordinates. It is assumed that in the process of the game the evaders do not leave the limits of the halfspace $D=\{y\colon y\in \mathbb{R}^k, \langle p_1, y\rangle \leqslant~0\},$ where $p_1$ is a unit vector. The aim of the pursuers is to capture both evaders, the aim of the group of evaders is the opposite. It is also assumed that all evaders use tightly coordinated management, which is determined at each point in time, taking into account the positions of other players in the game. The capture occurs if there are times $\tau_1,$ $\tau_2$ such that the coordinates of the pursuers and evaders coincide, and the capture times may not coincide. For a nonstationary simple pursuit problem in terms of initial positions and game parameters, sufficient conditions for catching two evaders are obtained. An example is given that illustrates the results obtained.
Keywords: differential game, phase restrictions, piece-program strategy, counterstrategy. 
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M. N. Vinogradova. Group pursuit of two evaders in a linear game with a simple matrix. Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 50 (2017), pp. 21-28. http://geodesic.mathdoc.fr/item/IIMI_2017_50_a2/

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

[2] Chernous'ko F. L., “A problem of evasion from many pursuers”, J. Appl. Math. Mech., 40:1 (1976), 11–20 | DOI | MR | Zbl

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

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

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

[6] Vagin D. A., Petrov N. N., “A problem of group pursuit with phase constraints”, J. Appl. Math. Mech., 66:2 (2002), 225–232 | DOI | MR | Zbl

[7] Prokopovich P. V., Chikrii A. A., “A linear evasion problem for interacting groups of objects”, J. Appl. Math. Mech., 58:4 (1994), 583–591 | DOI | MR | Zbl

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

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

[10] Grigorenko N. L., “Pursuit of two evaders by several controlled objects”, Sov. Math., Dokl., 31 (1985), 550–553 | MR | Zbl

[11] Vinogradova M. N., “On the capture of two evaders in a simple pursuit-evasion problem with phase restrictions”, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2011, no. 4, 3–8 (in Russian) | DOI

[12] Vinogradova M. N., “On the capture of two escapees in the non-stationary problem of simple pursuit”, Mat. Teor. Igr Pril., 4:1 (2012), 21–31 (in Russian)

[13] Vinogradova M. N., Petrov N. N., Solov'eva N. A., “Capture of two cooperative evaders in linear recurrent differential games”, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 19, no. 1, 2013, 41–48 (in Russian)

[14] Vinogradova M. N., Petrov N. N., “Soft capture of two coordinated evaders”, J. Comput. Syst. Sci. Int., 52:6 (2013), 949–954 | DOI | DOI | MR | MR | Zbl

[15] Vinogradova M. N., “On the capture of two evaders in a non-stationary pursuit–evasion problem with phase restrictions”, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 25:1 (2015), 12–20 (in Russian) | DOI

[16] Bannikov A. S., “About a problem of positional capture of one evader by group of pursuers”, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2011, no. 1, 3–7 (in Russian) | DOI

[17] Petrov N. N., Solov'eva N. A., “Problem of pursuit of a group of coordinated evaders in linear recurrent differential games”, J. Comput. Syst. Sci. Int., 51:6 (2012), 770–778 | DOI | MR | Zbl

[18] Petrov N. N., “Non-stationary Pontryagin example with phase restrictions”, Journal of Automation and Information Sciences, 32:10 (2000), 11–17 | DOI | MR

[19] Blagodatskikh A. I., “Multiple capture of rigidly coordinated evaders”, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 26:1 (2016), 46–57 (in Russian) | DOI | Zbl

[20] Blagodatskikh A. I., “Two non-stationary pursuit problems of a rigidly connected evaders”, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2008, no. 1, 47–60 (in Russian) | DOI

[21] Petrov N. N., “On certain problems on a group pursuit with phase constraints”, Russian Mathematics, 38:4 (1994), 21–26 | MR | Zbl