Abstract:
It is well known that any control that steers the trajectory of a control system to the boundary of the reachable set satisfies the Pontryagin maximum principle. This fact is valid for systems with pointwise constraints on the control. We consider a system with quadratic integral constraints on the control. The system is nonlinear in the state variables and linear in the control. It is shown that any admissible control that steers the system to the boundary of its reachable set is a local solution of some optimal control problem with integral quadratic functional if the corresponding linearized system is completely controllable. The proof of this fact is based on the Graves theorem on covering mappings. This implies the maximum principle for the controls that steer the trajectories to the boundary of the reachable set. We also discuss an algorithm for constructing the reachable set based on the maximum principle.
Keywords:
control system, integral constraints, reachable set, maximum principle.
Citation:
M. I. Gusev, I. V. Zykov, “On extremal properties of the boundary points of reachable sets for control systems with integral constraints”, Trudy Inst. Mat. i Mekh. UrO RAN, 23, no. 1, 2017, 103–115; Proc. Steklov Inst. Math. (Suppl.), 300, suppl. 1 (2018), 114–125
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\by M.~I.~Gusev, I.~V.~Zykov
\paper On extremal properties of the boundary points of reachable sets for control systems with integral constraints
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2017
\vol 23
\issue 1
\pages 103--115
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\crossref{https://doi.org/10.21538/0134-4889-2017-23-1-103-115}
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\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2018
\vol 300
\issue , suppl. 1
\pages 114--125
\crossref{https://doi.org/10.1134/S0081543818020116}
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Linking options:
https://www.mathnet.ru/eng/timm1387
https://www.mathnet.ru/eng/timm/v23/i1/p103
This publication is cited in the following 27 articles:
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M. S. Nikol'skii, “On the continuity of the optimal time as a function of the initial state for linear controlled objects with integral constraints on controls”, Proc. Steklov Inst. Math. (Suppl.), 325, suppl. 1 (2024), S147–S154
Valerii S. Patsko, Georgii I. Trubnikov, Andrei A. Fedotov, “Mnozhestvo dostizhimosti mashiny Dubinsa s integralnym ogranicheniem na upravlenie”, MTIP, 15:2 (2023), 89–104
Mikhail I. Gusev, “Computing the reachable set bounda for an abstract control system: revisited”, Ural Math. J., 9:2 (2023), 99–108
Ivan O. Osipov, “Convexity of reachable sets of quasilinear systems”, Ural Math. J., 9:2 (2023), 141–156
Nesir Huseyin, “Approximation of the image of the L p
ball under Hilbert-Schmidt integral operator”, Demonstratio Mathematica, 56:1 (2023)
Nesir HÜSEYİN, “Urysohn Tür İntegral Denklem ile Verilen Kontrol Sistemin Yörüngeler Kümesinin Özellikleri Üzerine”, Düzce üniversitesi Bilim ve Teknoloji Dergisi, 11:4 (2023), 1772
Anar HUSEYİN, “İntegrallenebilir Yörüngeleri ve Kontrol KaynaklarıK{\i}s{\i}tlıolan Kontrol Sistemin Yörüngeler Kümesinin Özellikleri Üzerine”, Kafkas üniversitesi Fen Bilimleri Enstitüsü Dergisi, 16:1 (2023), 24
V. S. Patsko, G. I. Trubnikov, A. A. Fedotov, “Reachable Set of the Dubins Car with an Integral Constraint on Control”, Dokl. Math., 108:S1 (2023), S34
N. Huseyin, A. Huseyin, Kh. G. Guseinov, “On the properties of the set of trajectories of nonlinear control systems with integral constraints on the control functions”, Tr. IMM UrO RAN, 28, no. 3, 2022, 274–284
M. I. Gusev, I. O. Osipov, “O zadache lokalnogo sinteza dlya nelineinykh sistem s integralnymi ogranicheniyami”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 32:2 (2022), 171–186
Anar Huseyin, “On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation”, Open Mathematics, 20:1 (2022), 1101
V. N. Ushakov, V. I. Ukhobotov, A. V. Ushakov, I. V. Izmest'ev, “Control Systems of Variable Structure. Attainability Sets and Integral Funnels”, J Math Sci, 260:6 (2022), 820
N. Huseyin, A. Huseyin, Kh. G. Guseinov, “On the Robustness Property of a Control System Described by an Urysohn Type Integral Equation”, Tr. IMM UrO RAN, 27, no. 3, 2021, 263–270
V. N. Ushakov, A. V. Ushakov, “O navedenii integralnoi voronki upravlyaemoi sistemy
na tselevoe mnozhestvo v fazovom prostranstve”, Izv. IMI UdGU, 56 (2020), 79–101
N. Huseyin, “On the properties of the set ofp-integrable trajectories of the control system with limited control resources”, Int. J. Control, 93:8 (2020), 1810–1816
A. Huseyin, N. Huseyin, Kh. G. Guseinov, “Approximation of the integral funnel of a nonlinear control system with limited control resources”, Minimax Theory Appl., 5:2, SI (2020), 327–346
Mikhail Gusev, 2020 15th International Conference on Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy's Conference) (STAB), 2020, 1
A. A. Ershov, A. V. Ushakov, V. N. Ushakov, “An approach problem for a control system and a compact set in the phase space in the presence of phase constraints”, Sb. Math., 210:8 (2019), 1092–1128
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