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 Izv. RAN. Ser. Mat., 2014, Volume 78, Issue 5, Pages 167–190 (Mi izv8107)

Singular regimes in controlled systems with multidimensional control in a polyhedron

L. V. Lokutsievskii

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We study Hamiltonian systems that are affine in a multidimensional control varying in a polyhedron $\Omega$. Quite often, a crucial role in the study of the global behaviour of solutions of such systems is played by special trajectories and the geometry of their neighbourhoods. We prove a theorem on the structure of the output of optimal trajectories to a first-order singular trajectory in a neighbourhood of this trajectory (and of the exit from it) for systems with holonomic control. We also prove that in a neighbourhood of a first-order singular trajectory, a Lagrangian surface is woven in a special way from the trajectories of the system that are singular with respect to the faces of $\Omega$. We suggest a simple way to find explicitly first-order special trajectories with respect to the faces of $\Omega$. As a result, we describe a complete picture of the optimal synthesis obtained by the successive conjugation of first-order singular extremals.

Keywords: optimal control, singular trajectories, multidimensional control, optimal synthesis.

DOI: https://doi.org/10.4213/im8107

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English version:
Izvestiya: Mathematics, 2014, 78:5, 1006–1027

Bibliographic databases:

UDC: 517.97
MSC: 49J15

Citation: L. V. Lokutsievskii, “Singular regimes in controlled systems with multidimensional control in a polyhedron”, Izv. RAN. Ser. Mat., 78:5 (2014), 167–190; Izv. Math., 78:5 (2014), 1006–1027

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv8107
• https://doi.org/10.4213/im8107
• http://mi.mathnet.ru/eng/izv/v78/i5/p167

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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, 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
2. L. V. Lokutsievskiy, “On an optimal flow in a class of nilpotent convex problems”, Proc. Steklov Inst. Math., 291 (2015), 146–169
3. L. Manita, “Optimization problems for WSNs: trade-off between synchronization errors and energy consumption”, International Conference on Computer Simulation in Physics and Beyond 2015, Journal of Physics: Conference Series, 681:1 (2016), 012009, IOP Publishing Ltd
4. L. V. Lokoutsievskiy, V. A. Mirikova, “Optimal Synthesis in a Model Problem with Two-Dimensional Control Lying in an Arbitrary Convex Set”, Math. Notes, 105:1 (2019), 36–55
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