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

This article is cited in 5 scientific papers (total in 5 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

Full text: PDF file (437 kB)
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English version:
Izvestiya: Mathematics, 2015, 79:5, 894–901

Bibliographic databases:

Document Type: Article
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

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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. 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
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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