Skip to main content
SpringerLink
Log in

Convex hulls of surfaces with boundaries and corners and singularities of transitivity zone in ℝ3

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

Generic singularities of the boundary of the local transitivity set of a control system on two- and three-dimensional manifolds are classified. The indicatrices of the system are assumed to be given by generic equations and inequalities.

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. V. I. Arnold, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Maps, 2nd ed. (MCCME, Moscow, 2004) [in Russian].

    Google Scholar 

  2. V. I. Arnold, “Catastrophe Theory,” in Dynamical Systems-5 (VINITI, Moscow, 1986), Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Fundam. Napravl. 5, pp. 219–277; Engl. transl. in Dynamical Systems V (Springer, Berlin, 1994), Encycl. Math. Sci. 5, pp. 207–264.

    Google Scholar 

  3. P. J. Giblin and V. M. Zakalyukin, “Singularities of Centre Symmetry Sets,” Proc. London Math. Soc., Ser. 3, 90, 132–166 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  4. A. B. Givental’, “Singular Lagrangian Manifolds and Their Lagrangian Maps,” Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Noveishie Dostizheniya 33, 55–112 (1988) [J. Sov. Math. 52 (4), 3246–3278 (1990)].

    MathSciNet  Google Scholar 

  5. V. V. Goryunov and V. M. Zakalyukin, “Simple Symmetric Matrix Singularities and the Subgroups of Weyl Groups A µ, D µ, E µ,” Moscow Math. J. 3(2), 507–530 (2003).

    MATH  MathSciNet  Google Scholar 

  6. V. V. Goryunov and V. M. Zakalyukin, “On Stability of Projections of Lagrangian Varieties,” Funkts. Anal. Prilozh. 38(4), 13–21 (2004) [Funct. Anal. Appl. 38, 249–255 (2004)].

    MathSciNet  Google Scholar 

  7. A. A. Davydov, Qualitative Theory of Control Systems (Am. Math. Soc., Providence, RI, 1994), Transl. Math. Monogr. 141.

    MATH  Google Scholar 

  8. A. A. Davydov, “Singularities of Fields of Limiting Directions of Two-Dimensional Control Systems,” Mat. Sb. 136(4), 478–499 (1988) [Math. USSR, Sb. 64, 471–493 (1989)].

    MathSciNet  Google Scholar 

  9. J. Damon, “A-Equivalence and the Equivalence of Sections of Images and Discriminants,” in Singularity Theory and Its Applications, Warwick, 1989 (Springer, Berlin, 1991), Part 1, Lect. Notes Math. 1462, pp. 93–121.

    Chapter  Google Scholar 

  10. V. M. Zakalyukin, “Singularities of Convex Hulls of Smooth Manifolds,” Funkts. Anal. Prilozh. 11(3), 76–77 (1977) [Funct. Anal. Appl. 11, 225–227 (1977)].

    MATH  Google Scholar 

  11. V. M. Zakalyukin and A. N. Kurbatskii, “Envelope Singularities of Families of Planes in Control Theory,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 262, 73–86 (2008) [Proc. Steklov Inst. Math. 262, 66–79 (2008)].

    MathSciNet  Google Scholar 

  12. V. M. Zakalyukin and A. N. Kurbatskii, “Convex Hulls of Curves and Singularities of Transitivity Sets in R3,” in Modern Problems of Mathematics and Mechanics (Moscow State Univ., Moscow, 2009), Vol. IV, Issue 2, pp. 3–23 [in Russian].

    Google Scholar 

  13. M. É. Kazaryan, “Singularities of the Boundary of Fundamental Systems, Flat Points of Projective Curves, and Schubert Cells,” Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Noveishie Dostizheniya 33, 215–234 (1988) [J. Sov. Math. 52 (4), 3338–3349 (1990)].

    MATH  MathSciNet  Google Scholar 

  14. V. D. Sedykh, “Stabilization of Singularities of Convex Hulls,” Mat. Sb. 135(4), 514–519 (1988) [Math. USSR, Sb. 63 (2), 499–505 (1989)].

    Google Scholar 

  15. O. P. Shcherbak, “Projectively Dual Space Curves and Legendre Singularities,” Tr. Tbilis. Univ. 232–233(13–14), 280–336 (1982).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991, Russia

    V. M. Zakalyukin & A. N. Kurbatskii

  2. University of Liverpool, Liverpool, L69 7ZL, UK

    V. M. Zakalyukin

Authors
  1. V. M. Zakalyukin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. N. Kurbatskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2010, Vol. 268, pp. 284–303.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zakalyukin, V.M., Kurbatskii, A.N. Convex hulls of surfaces with boundaries and corners and singularities of transitivity zone in ℝ3 . Proc. Steklov Inst. Math. 268, 274–293 (2010). https://doi.org/10.1134/S0081543810010190

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543810010190

Keywords

Search

Navigation