Abstract:
The paper is devoted to the problem of constructing external estimates for the reachable set of a multidimensional control system by means of vector estimators. A system is considered that permits a decomposition into several independent subsystems with simple structure (for example, linear subsystems), which are connected to each other by means of nonlinear interconnections. For each of the subsystems, an external estimate of the reachable set is assumed to be known; this estimate is representable in the form of a level set of some function satisfying a differential inequality. An estimate for the reachable set of the united system is constructed with the use of estimates for subsystems. The method of deriving the estimates is based on constructing comparison systems for analogs of vector Lyapunov functions (cost functions).
Keywords:
control system, reachable set, comparison principle, vector Lyapunov function.
Citation:
M. I. Gusev, “Estimates of reachable sets of multidimensional control systems with nonlinear interconnections”, Trudy Inst. Mat. i Mekh. UrO RAN, 15, no. 4, 2009, 82–94; Proc. Steklov Inst. Math. (Suppl.), 269, suppl. 1 (2010), S134–S146
\Bibitem{Gus09}
\by M.~I.~Gusev
\paper Estimates of reachable sets of multidimensional control systems with nonlinear interconnections
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2009
\vol 15
\issue 4
\pages 82--94
\mathnet{http://mi.mathnet.ru/timm428}
\elib{https://elibrary.ru/item.asp?id=12952757}
\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2010
\vol 269
\issue , suppl. 1
\pages S134--S146
\crossref{https://doi.org/10.1134/S008154381006012X}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84962376495}
Linking options:
https://www.mathnet.ru/eng/timm428
https://www.mathnet.ru/eng/timm/v15/i4/p82
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V. N. Ushakov, A. A. Ershov, A. V. Ushakov, “Control Systems Depending on a Parameter: Reachable Sets and Integral Funnels”, Mech. Solids, 57:7 (2022), 1672
Vladimir N. Ushakov, Aleksandr A. Ershov, Andrey V. Ushakov, Oleg A. Kuvshinov, “Control system depending on a parameter”, Ural Math. J., 7:1 (2021), 120–159
V. N. Ushakov, A. V. Ushakov, O. A. Kuvshinov, “O konstruirovanii razreshayuschego upravleniya v zadache o sblizhenii v fiksirovannyi moment vremeni”, Izv. IMI UdGU, 58 (2021), 73–93
V. N. Ushakov, A. A. Ershov, “Reachable sets and integral funnels of differential inclusions depending on a parameter”, Dokl. Math., 104:1 (2021), 200–204
Ushakov V.N., Ukhobotov V.I., Matviychuk A.R., Parshikov G.V., “On Some Nonlinear Control System Problems on a Finite Time Interval”, IFAC PAPERSONLINE, 51:32 (2018), 832–837
Ushakov V.N. Ukhobotov V.I. Ushakov A.V. Parhsikov G.V., “On Game Approach Problems on a Finite Time Interval”, AIP Conference Proceedings, 2040, ed. Simos T. Kalogiratou Z. Monovasilis T., Amer Inst Physics, 2018, 050003
G. V. Parshikov, “O priblizhennom vychislenii mnozhestva razreshimosti v zadache o sblizhenii statsionarnoi upravlyaemoi sistemy na konechnom promezhutke vremeni”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 27:2 (2017), 210–221
Matviychuk A.R., Ukhobotov V.I., Ushakov A. V. and Ushakov V.N., “The Approach Problem of a Nonlinear Controlled System in a Finite Time Interval”, Pmm-J. Appl. Math. Mech., 81:2 (2017), 114–128
V. N. Ushakov, V. I. Ukhobotov, A. V. Ushakov, G. V. Parshikov, “On solving approach problems for control systems”, Proc. Steklov Inst. Math., 291 (2015), 263–278
V. N. Ushakov, A. R. Matviichuk, “K resheniyu zadach upravleniya nelineinymi sistemami na konechnom promezhutke vremeni”, Izv. IMI UdGU, 2015, no. 2(46), 202–215
Sinyakov V.V., “Method For Computing Exterior and Interior Approximations To the Reachability Sets of Bilinear Differential Systems”, Differ. Equ., 51:8 (2015), 1097–1111
V. N. Ushakov, N. G. Lavrov, A. V. Ushakov, “Konstruirovanie reshenii v zadache o sblizhenii statsionarnoi upravlyaemoi sistemy”, Tr. IMM UrO RAN, 20, no. 4, 2014, 277–286
M. I. Gusev, “O metode shtrafnykh funktsii v zadache postroeniya mnozhestv dostizhimosti upravlyaemykh sistem s fazovymi ogranicheniyami”, Tr. IMM UrO RAN, 19, no. 1, 2013, 81–86
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V. N. Ushakov, A. R. Matviychuk, G. V. Parshikov, “A method for constructing a resolving control in an approach problem based on attraction to the solvability set”, Proc. Steklov Inst. Math. (Suppl.), 284, suppl. 1 (2014), 135–144
P. D. Lebedev, A. A. Uspenskii, V. N. Ushakov, “Algoritmy nailuchshei approksimatsii ploskikh mnozhestv ob'edineniyami krugov”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2013, no. 4, 88–99
P. D. Lebedev, D. S. Bukharov, “Approksimatsiya mnogougolnikov nailuchshimi naborami krugov”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 6:3 (2013), 72–87
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