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## Archive of Issues

Russia Chelyabinsk; Yekaterinburg
Year
2020
Volume
30
Issue
3
Pages
429-443
 Section Mathematics Title On one control problem with disturbance and vectograms depending linearly on given sets Author(-s) Ukhobotov V.I.ab, Ushakov V.N.b Affiliations Chelyabinsk State Universitya, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesb Abstract A control problem with a given end time is considered, in which the control vectograms and disturbance depend linearly on the given convex compact sets. A multivalued mapping of the phase space of the control problem to the linear normed space $E$ is given. The goal of constructing a control is that at the end of the control process the fixed vector of the space $E$ belongs to the image of the multivalued mapping for any admissible realization of the disturbance. A stable bridge is defined in terms of multivalued functions. The presented procedure constructs, according to a given multivalued function which is a stable bridge, a control that solves the problem. Explicit formulas are obtained that determine a stable bridge in the considered control problem. Conditions are found under which the constructed stable bridge is maximal. Some problems of group pursuit can be reduced to the considered control problem with disturbance. The article provides such an example. Keywords control problem, disturbance, stable bridge UDC 517.977 MSC 49N70, 49N75, 91A23, 91A24 DOI 10.35634/vm200306 Received 24 July 2020 Language Russian Citation Ukhobotov V.I., Ushakov V.N. On one control problem with disturbance and vectograms depending linearly on given sets, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 3, pp. 429-443. References Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial'nye igry (Positional differential games), Moscow: Nauka, 1974. Krasovskii N.N. Upravlenie dinamicheskoi sistemoi (Control of dynamical system), Moscow: Nauka, 1985. Pontryagin L.S. Linear differential games. I, Soviet Mathematics. Doklady, 1967, vol. 8, pp. 769-771. https://zbmath.org/?q=an:0157.16304 Pontryagin L.S. Linear differential games. II, Soviet Mathematics. Doklady, 1967, vol. 8, pp. 910-912. https://zbmath.org/?q=an:0157.16401 Satimov N.Yu. One pursuit method in linear differential games, Dokl. Akad. Nauk UzSSR, 1981, no. 6, pp. 5-7 (in Russian). Chikrii A.A. An analytical method in dynamic pursuit games, Proceedings of the Steklov Institute of Mathematics, 2010, vol. 271, pp. 69-85. https://doi.org/10.1134/S0081543810040073 Ukhobotov V.I. A stable bridge in game with vectograms that depend linearly on given sets, Soviet Mathematics, 1988, vol. 32, no. 2, pp. 89-92. https://zbmath.org/?q=an:0677.90099 Matviychuk A.R., Ukhobotov V.I., Ushakov A.V., Ushakov V.N. The approach problem of a nonlinear controlled system in a finite time interval, Journal of Applied Mathematics and Mechanics, 2017, vol. 81, issue 2, pp. 114-128. https://doi.org/10.1016/j.jappmathmech.2017.08.005 Ushakov V.N., Ukhobotov V.I., Lipin A.E. An addition to the definition of a stable bridge and an approximating system of sets in differential games, Proceedings of the Steklov Institute of Mathematics, 2019, vol. 304, pp. 268-280. https://doi.org/10.1134/S0081543819010206 Ukhobotov V.I. Synthesis of guaranteed control based on approximating scheme, Proceedings of the Steklov Institute of Mathematics, 2000, suppl. 1, pp. 254-260. https://zbmath.org/?q=an:1116.93330 Kumkov S.S., Le Menec S., Patsko V.S. Zero-sum pursuit-evasion differential games with many objects: survey of publications, Dynamic Games and Applications, 2017, vol. 7, pp. 609-633. https://doi.org/10.1007/s13235-016-0209-z Izmest'ev I.V., Ukhobotov V.I. Game problem of convergence of a group of objects with different types of dynamics, and target, 2018 International Conference “Global Smart Industry Conference” (GloSIC), IEEE, 2018. https://doi.org/10.1109/GloSIC.2018.8570102 Petrov N.N., Narmanov A.Ya. Multiple capture of a given number of evaders in the problem of a simple pursuit, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 2, pp. 193-198 (in Russian). https://doi.org/10.20537/vm180205 Kolmogorov A.N., Fomin S.V. Elementy teorii funktsii i funktsional'nogo analiza (Elements of theory of functions and functional analysis), Moscow: Nauka, 1972. Pshenichnyi B.N., Sagaidak M.I. About differential games with fixed time, Kibernetika, 1970, no. 2, pp. 54-63 (in Russian). Ukhobotov V.I. Stable property of the programmed absorption operator in games with simple motion and with a convex target in spaces with incomplete linear structure, Vestnik Chelyabinskogo Gosudarstvennogo Universiteta. Matematika, Mekhanika, Informatika, 2003, issue 8, pp. 181-189 (in Russian). http://mi.mathnet.ru/eng/vchgu133 Full text