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Tr. Mat. Inst. Steklova, 2012, Volume 277, Pages 74–90 (Mi tm3381)  

This article is cited in 10 scientific papers (total in 10 papers)

Geometry of neighborhoods of singular trajectories in problems with multidimensional control

M. I. Zelikina, L. V. Lokutsievskiya, R. Hildebrandb

a Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia
b Laboratoire Jean Kuntzmann, Université Joseph Fourier, Grenoble, France

Abstract: It is shown that the order of a singular trajectory in problems with multidimensional control is described by a flag of linear subspaces in the control space. In terms of this flag, we construct necessary conditions for the junction of a nonsingular trajectory with a singular one in affine control systems. We also give examples of multidimensional problems in which the optimal control has the form of an irrational winding of a torus that is passed in finite time.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2012, 277, 67–83

Bibliographic databases:

Document Type: Article
UDC: 517.97
Received in May 2011

Citation: M. I. Zelikin, L. V. Lokutsievskiy, R. Hildebrand, “Geometry of neighborhoods of singular trajectories in problems with multidimensional control”, Mathematical control theory and differential equations, Collected papers. In commemoration of the 90th anniversary of Academician Evgenii Frolovich Mishchenko, Tr. Mat. Inst. Steklova, 277, MAIK Nauka/Interperiodica, Moscow, 2012, 74–90; Proc. Steklov Inst. Math., 277 (2012), 67–83

Citation in format AMSBIB
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\by M.~I.~Zelikin, L.~V.~Lokutsievskiy, R.~Hildebrand
\paper Geometry of neighborhoods of singular trajectories in problems with multidimensional control
\inbook Mathematical control theory and differential equations
\bookinfo Collected papers. In commemoration of the 90th anniversary of Academician Evgenii Frolovich Mishchenko
\serial Tr. Mat. Inst. Steklova
\yr 2012
\vol 277
\pages 74--90
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm3381}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3052265}
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\transl
\jour Proc. Steklov Inst. Math.
\yr 2012
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\pages 67--83
\crossref{https://doi.org/10.1134/S0081543812040062}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. M. I. Zelikin, D. D. Kiselev, L. V. Lokutsievskii, “Optimal control and Galois theory”, Sb. Math., 204:11 (2013), 1624–1638  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. Hildebrand R. Lokutsievskiy L.V. Zelikin M.I., “Generic Fractal Structure of Finite Parts of Trajectories of Piecewise Smooth Hamiltonian Systems”, Russ. J. Math. Phys., 20:1 (2013), 25–32  crossref  mathscinet  zmath  isi  elib  scopus
    3. Hildebrand R., Lokutsievskiy L.V., Zelikin M.I., “Generic Fractal Structure of the Optimal Synthesis in Problems with Affine Multi-Dimensional Control”, 2013 European Control Conference, IEEE, 2013, 3203–3208  isi
    4. L. V. Lokutsievskii, “Singular regimes in controlled systems with multidimensional control in a polyhedron”, Izv. Math., 78:5 (2014), 1006–1027  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. M. I. Zelikin, L. V. Lokutsievskii, R. Hildebrand, “Typicality of chaotic fractal behavior of integral vortices in Hamiltonian systems with discontinuous right hand side”, Journal of Mathematical Sciences, 221:1 (2017), 1–136  mathnet  crossref
    6. L. V. Lokutsievskiy, “On an optimal flow in a class of nilpotent convex problems”, Proc. Steklov Inst. Math., 291 (2015), 146–169  mathnet  crossref  crossref  isi  elib  elib
    7. M. I. Zelikin, “Fractal theory of Saturn's ring”, Proc. Steklov Inst. Math., 291 (2015), 87–101  mathnet  crossref  crossref  isi  elib
    8. D. D. Kiselev, “On a dense winding of the 2-dimensional torus”, Sb. Math., 207:4 (2016), 581–589  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    9. L'Afflitto A., Haddad W.M., “Abnormal Optimal Trajectory Planning of Multi-Body Systems in the Presence of Holonomic and Nonholonomic Constraints”, J. Intell. Robot. Syst., 89:1-2, SI (2018), 51–67  crossref  isi  scopus
    10. D. D. Kiselev, “Galois theory, the classification of finite simple groups and a dense winding of a torus”, Sb. Math., 209:6 (2018), 840–849  mathnet  crossref  crossref  adsnasa  isi  elib
  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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