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Izv. RAN. Ser. Mat., 2015, Volume 79, Issue 5, Pages 39–46 (Mi izv8413)  

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

Calculus of variations in the large, existence of trajectories in a domain with boundary, and Whitney's inverted pendulum problem

S. V. Bolotin, V. V. Kozlov

Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: For non-autonomous Lagrangian systems we introduce the notion of a dynamically convex domain with respect to the Lagrangian. We establish the solubility of boundary-value problems in compact dynamically convex domains. If the Lagrangian is time-periodic, then such a domain contains a periodic trajectory. The proofs use the Hamilton principle and known tools of the calculus of variations in the large. Our general results are applied to Whitney's problem on the existence of motions of an inverted pendulum without falls.

Keywords: Lagrangian system, dynamically convex domain, Hamilton principle, Palais–Smale condition, Whitney's problem.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
The work is supported by the Russian Science Foundation under grant 14-50-00005.

DOI: https://doi.org/10.4213/im8413

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English version:
Izvestiya: Mathematics, 2015, 79:5, 894–901

Bibliographic databases:

UDC: 531.01+517.974
MSC: 37C60, 37J45
Received: 21.05.2015

Citation: S. V. Bolotin, V. V. Kozlov, “Calculus of variations in the large, existence of trajectories in a domain with boundary, and Whitney's inverted pendulum problem”, Izv. RAN. Ser. Mat., 79:5 (2015), 39–46; Izv. Math., 79:5 (2015), 894–901

Citation in format AMSBIB
\by S.~V.~Bolotin, V.~V.~Kozlov
\paper Calculus of variations in the large, existence of trajectories in a~domain with boundary, and Whitney's inverted pendulum problem
\jour Izv. RAN. Ser. Mat.
\yr 2015
\vol 79
\issue 5
\pages 39--46
\jour Izv. Math.
\yr 2015
\vol 79
\issue 5
\pages 894--901

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Polekhin I., “A Topological View on Forced Oscillations and Control of An Inverted Pendulum”, Geometric Science of Information, Gsi 2017, Lecture Notes in Computer Science, 10589, eds. Nielsen F., Barbaresco F., Springer International Publishing Ag, 2017, 329–335  crossref  mathscinet  zmath  isi  scopus
    2. I. Polekhin, “On topological obstructions to global stabilization of an inverted pendulum”, Syst. Control Lett., 113 (2018), 31–35  crossref  mathscinet  zmath  isi  scopus
    3. S. Ozana, M. Schlegel, “Computation of reference trajectories for inverted pendulum with the use of two-point BvP with free parameters”, IFAC PAPERSONLINE, 51:6 (2018), 408–413  crossref  isi  scopus
    4. I. Polekhin, “On motions without falling of an inverted pendulum with dry friction”, J. Geom. Mech., 10:4 (2018), 411–417  crossref  isi
    5. I. Yu. Polekhin, “On the impossibility of global stabilization of the Lagrange top”, Mech. Sol., 53:2 (2018), S71–S75  mathnet  crossref  crossref  isi  elib  scopus
    6. R. Srzednicki, “On periodic solutions in the whitney's inverted pendulum problem”, Discret. Contin. Dyn. Syst.-Ser. S, 12:7 (2019), 2127–2141  crossref  isi
    7. I. Yu. Polekhin, “Remarks on Forced Oscillations in Some Systems with Gyroscopic Forces”, Rus. J. Nonlin. Dyn., 16:2 (2020), 343–353  mathnet  crossref  mathscinet
    8. Ivan Yu. Polekhin, “The Method of Averaging for the Kapitza – Whitney Pendulum”, Regul. Chaotic Dyn., 25:4 (2020), 401–410  mathnet  crossref  mathscinet
    9. Ivan Yu. Polekhin, “Some Results on the Existence of Forced Oscillations in Mechanical Systems”, Proc. Steklov Inst. Math., 310 (2020), 250–261  mathnet  crossref  crossref  isi  elib
    10. N. A. Stepanov, M. A. Skvortsov, “Lyapunov exponent for Whitney's problem with random drive”, JETP Letters, 112:6 (2020), 376–382  mathnet  crossref  crossref  isi  elib
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