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## Archive of Issues

Russia Chelyabinsk
Year
2015
Volume
25
Issue
2
Pages
197-211
 Section Mathematics Title Single-type problem of pulse meeting in fixed time with terminal set in form of a ring Author(-s) Ukhobotov V.I.a, Izmest'ev I.V.a Affiliations Chelyabinsk State Universitya Abstract We consider a linear differential game with the fixed end time $p$. Attainability domains of players are $n$-dimensional balls. The terminal set of a game is determined by a condition for assigning the norm of a phase vector to a segment with positive ends. A set defined by this condition is named in the article as ring. The fact that the terminal set is not convex required an additional theory allowing us to calculate Minkowski sum and difference for a ring and a ball in $n$-dimensional space. Control of the first player has a pulse constraint. Abilities of the first player are determined by the stock of resources that can be used by the player at formation of his control. At certain moments of time the separation of a part of the resources stock is possible, which may implicate an “instantaneous” change of a phase vector, thereby complicating the problem. Control of the second player has geometrical constraints. The aim of the first player is to lead a phase vector to the terminal set at fixed time. The aim of the second player is opposite. The maximal stable bridge leading at fixed time to the terminal set has been constructed. A stable bridge is determined by the functions of internal and external radii, which are calculated explicitly. Keywords pulse control, differential game, stable bridge UDC 517.977.80 MSC 91A23, 49N75 DOI 10.20537/vm150205 Received 28 April 2015 Language Russian Citation Ukhobotov V.I., Izmest'ev I.V. Single-type problem of pulse meeting in fixed time with terminal set in form of a ring, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 2, pp. 197-211. References Krasovskii N.N. Teoriya upravleniya dvizheniem (Theory of motion control), Moscow: Nauka, 1968, 475 p. Krasovskii N.N. On a problem of tracking, Journal of Applied Mathematics and Mechanics, 1963, vol. 27, no. 2, pp. 363-377. Krasovskii N.N., Repin Yu.M., Tret'yakov V.E. Some game situations in theory of control systems, Izvestiya Akademii Nauk SSSR. Tekhnicheskaya Kibernetika, 1965, no. 4, pp. 3-23 (in Russian). Krasovskii N.N., Tret'yakov V.E. To problem about pursuit in case of constraints on pulses of control forces, Differ. Uravn., 1966, vol. 2, no. 5, pp. 587-599 (in Russian). Pozharitskii G.K. Game problem of impulse encounter with an opponent limited in energy, Journal of Applied Mathematics and Mechanics, 1975, vol. 39, no. 4, pp. 555-565. Subbotina N.N., Subbotin A.I. Alternative for the encounter-evasion differential game with constraints on the momenta of the players' controls, Journal of Applied Mathematics and Mechanics, 1975, vol. 39, no. 3, pp. 376-385. Petrov N.N. A problem of group pursuit in the class of impulse strategies of pursuers, Journal of Computer and Systems Sciences International, 2009, vol. 48, no. 2, pp. 199-205. Ukhobotov V.I. On a class of linear differential games with impulse controls, Journal of Applied Mathematics and Mechanics, 1974, vol. 38, no. 4, pp. 550-557. Ukhobotov V.I. About one class of linear differential games, Kibernetika, 1974, no. 1, pp. 127-130 (in Russian). Ukhobotov V.I. A linear differential game with constraints imposed on the control impulses, Journal of Applied Mathematics and Mechanics, 1988, vol. 52, no. 3, pp. 277-283. Ukhobotov V.I. Metod odnomernogo proektirovaniya v lineinykh differentsial'nykh igrakh s integral'nymi ogranicheniyami (Method of one-dimensional projecting in linear differential games with integral constraints), Chelyabinsk: Chelyabinsk State University, 2005, 124 p. Ukhobotov V.I., Zaitseva O.V. A linear problem of pulse encounter at a given time under interference, Proceedings of the Steklov Institute of Mathematics, 2011, vol. 272, suppl. 1, pp. 215-228. Chikrii A.A., Matichin I.I., Chikrii K.A. Conflict control processes with discontinuous trajectories, Kibernetika i sistemnyi analiz, 2004, no. 6, pp. 15-29 (in Russian). Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial'nye igry (Positional differential games), Moscow: Nauka, 1974, 456 p. Pontryagin L.S. Linear differential games, Soviet Mathematics. Doklady, 1967, vol. 8, pp. 910-912. Pshenichnyi B.N., Sagaidak M.I. About differential games with fixed time, Kibernetika, 1970, no. 2, pp.54-63 (in Russian). Ukhobotov V.I. Single-type differential game with terminal set in form of a ring, Nekotorye zadachi dinamiki i upravleniya: sbornik nauchnykh trudov (Some problems of dynamic and control: Transactions), Chelyabinsk State University, Chelyabinsk, 2005, pp. 108-123 (in Russian). Full text