Skip to main content
SpringerLink
Log in

Singularities of the attainable set on an orientable surface with boundary

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

We classify generic singularities of attainable sets of smooth control systems on a compact connected smooth orientable surface with boundary under the assumption that the range of the control parameter is a compact smooth manifold or the disjoint union of such manifolds and the starting set lies in the interior of the attainable set. Bibliography: 10 titles. Illustrations: 8 figures.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. N. Petrov, “Local controlability of autonomous systems” [in Russian], Differ. Uravn. 4, No. 7, 1218–1232 (1968); English transl.: Differ. Equ. 4, 632–639 (1972).

    Google Scholar 

  2. A. A. Davydov, “Singularities of fields of limiting directions of two-dimensional control systems” Mat. Sb. 136, No. 4, 478–499 (1988); English transl.: Math. USSR, Sb. 64, No. 2, 471–493 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  3. A. A. Davydov, “Structural stability of control systems on orientable surfaces” [in Russian], Mat. Sb. 182, No. 1, 3–35 (1991); English transl.: Math. USSR, Sb. 72, No. 1, 1–28 (1992).

    Article  MathSciNet  Google Scholar 

  4. A. A. Davydov, Qualitative Theory of Control Systems, Am. Math. Soc., Providence, RI (1994).

    MATH  Google Scholar 

  5. Hy Dú’c Manh, “Stability of local transitivity of a generic control system on a surface with boundary” [in Russian], Tr. Steklov MIAN 278, 269–275 (2012): English transl.: Proc. Steklov Inst. Math. 278, No. 1, 260–266 (2012).

    Article  Google Scholar 

  6. V. I. Arnol’d, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Maps: Critical Points, Caustics and Wave Fronts [in Russian], Nauka, Moscow (1982); English transl.: Birkhäuser, Boston, MA (1985).

    Google Scholar 

  7. M. M. Peixoto. “Structural stability on two-dimensional manifolds,” Topology 1, 101–120 (1960).

    Article  MathSciNet  Google Scholar 

  8. J. Palis Jr., W. de Melo, Geometry Theory of Dynamical Systems, An Introduction, Springer, New York etc. (1982).

    Book  Google Scholar 

  9. V. I. Arnol’d, Supplementary Chapters to the Theory of Ordinary Differential Equations [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  10. A. A. Davydov, “Normal form of a differential equation, not solvable for the derivative, in a neighborhood of a singular point” Funkts. Anal. Prilozh. 19, No. 2. 1–10 (1985); English transl.: Func. Anal. Appl. 19, 81–89 (1985).

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Vladimir State University, 87, ul. Gor’kogo, Vladimir, 600000, Russia

    A. A. Davydov & Hy Duc Manh

  2. International Institute for Applied Systems Analysis, A-2361, Laxenburg, Austria

    A. A. Davydov

Authors
  1. A. A. Davydov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hy Duc Manh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Davydov.

Additional information

Translated from Problemy Matematicheskogo Analiza 67, November 2012, pp. 13–22.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Davydov, A.A., Manh, H.D. Singularities of the attainable set on an orientable surface with boundary. J Math Sci 188, 185–196 (2013). https://doi.org/10.1007/s10958-012-1117-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-012-1117-4

Keywords

Search

Navigation