Section
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Mathematics
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Title
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On reducing the motion of a controlled system to a Lebesgue set of a Lipschitz function
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Author(-s)
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Ushakov V.N.ab,
Ershov A.A.ab,
Parshikov G.V.a
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Affiliations
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Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa,
Ural Federal Universityb
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Abstract
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We consider a nonlinear controlled system in a finite-dimensional Euclidean space defined on a finite time interval. One of the main problems of mathematical control theory is studied: the problem of approaching a phase vector of a controlled system with a compact target set in the phase space at a fixed time instant. In this paper, a Lebesgue set of a scalar Lipschitz function defined on the phase space is a target set. The mentioned approach problem is closely connected with many important and key problems of control theory and, in particular, with the problem of optimally reducing a controlled system to a target set. Due to the complexity of the approach problem for nontrivial controlled systems, an analytical representation of solutions is impossible even for relatively simple controlled systems. Therefore, in the present work, we study first of all the issues related to the construction of an approximate solution of the approach problem. The construction of an approximate solution by the method described in the paper is primarily concerned with the design of the integral funnel of the controlled system, presented in the so-called “reverse” time. To date, there are several algorithms for constructing a resolving program control in the approach problem. This paper presents an algorithm for constructing a control based on the maximum attraction of the system's motion to the solvability set of the approach problem. Examples are provided.
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Keywords
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control system, Lebesgue set, solvability set, optimal control
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UDC
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517.977.58
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MSC
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49M25
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DOI
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10.20537/vm180405
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Received
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13 October 2018
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Language
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Russian
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Citation
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Ushakov V.N., Ershov A.A., Parshikov G.V. On reducing the motion of a controlled system to a Lebesgue set of a Lipschitz function, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 4, pp. 489-512.
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References
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- Kurzhanskii A.B. Izbrannye trudy (Selected works), Moscow: Moscow State University, 2009, 756 p.
- Kryazhimskiy A.V., Osipov Yu.S. On the solvability of problems of guaranteeing control for partially observable linear dynamical systems, Proceedings of the Steklov Institute of Mathematics, 2012, vol. 277, no. 1, pp. 144-159. DOI: 10.1134/S0081543812040104
- Krasovskii N.N. Igrovye zadachi o vstreche dvizhenii (Game problems about the meeting of movements), Moscow: Nauka, 1970, 420 p.
- 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 (in Russian). DOI: 10.20537/vm150205
- Khalil Kh.K. Nelineinye sistemy (Nonlinear systems), Izhevsk: Regular and Chaotic Dynamics, 2009, 812 p.
- Ushakov A.V. On one version of approximate permitting control calculation in a problem of approaching, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 4, pp. 94-107 (in Russian). DOI: 10.20537/vm120408
- Ganebnyi S.A., Kumkov S.S., Patsko V.S. Extremal aiming in problems with an unknown level of dynamic disturbance, Journal of Applied Mathematics and Mechanics, 2009, vol. 73, issue 4, pp. 411-420. DOI: 10.1016/j.jappmathmech.2009.08.010
- Ushakov V.N., Ukhobotov V.I., Ushakov A.V., Parshikov G.V. On solving approach problems for control systems, Proceedings of the Steklov Institute of Mathematics, 2015, vol. 291, issue 1, pp. 263-278. DOI: 10.1134/S0081543815080210
- Ushakov V.N., Ershov A.A. On the solution of control problems with fixed terminal time, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 4, pp. 543-564. DOI: 10.20537/vm160409
- Ershov A.A., Ushakov V.N. An approach problem for a control system with an unknown parameter, Sbornik: Mathematics, 2017, vol. 208, no. 9, pp. 1312-1352. DOI: 10.1070/SM8761
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