Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

6 Issues per year


IMPACT FACTOR 2017: 0.314
5-year IMPACT FACTOR: 0.462

CiteScore 2017: 0.46

SCImago Journal Rank (SJR) 2017: 0.339
Source Normalized Impact per Paper (SNIP) 2017: 0.845

Mathematical Citation Quotient (MCQ) 2017: 0.26

Online
ISSN
1337-2211
See all formats and pricing
More options …
Volume 67, Issue 4

Issues

On the internal approach to differential equations 2. The controllability structure

Veronika Chrastinová
  • Brno University of Technology Faculty of Civil Engineering Department of Mathematics Veveří 331/95, 602 00 Brno Czech Republic
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Václav Tryhuk
  • Brno University of Technology Faculty of Civil Engineering AdMaS Centre Purkyňova 139, 612 00 Brno Czech Republic
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2017-07-14 | DOI: https://doi.org/10.1515/ms-2017-0029

Abstract

The article concerns the geometrical theory of general systems Ω of partial differential equations in the absolute sense, i.e., without any additional structure and subject to arbitrary change of variables in the widest possible meaning. The main result describes the uniquely determined composition series Ω0 ⊂ Ω1 ⊂ … ⊂ Ω where Ωk is the maximal system of differential equations “induced” by Ω such that the solution of Ωk depends on arbitrary functions of k independent variables (on constants if k = 0). This is a well-known result only for the particular case of underdetermined systems of ordinary differential equations. Then Ω = Ω1 and we have the composition series Ω0 ⊂ Ω1 = Ω where Ω0 involves all first integrals of Ω, therefore Ω0 is trivial (absent) in the controllable case. The general composition series Ω0 ⊂ Ω1 ⊂ … ⊂ Ω may be regarded as a “multidimensional” controllability structure for the partial differential equations.

Though the result is conceptually clear, it cannot be included into the common jet theory framework of differential equations. Quite other and genuinely coordinate-free approach is introduced.

MSC 2010: 58A17; 58J99; 35A30

Keywords: diffiety; controllability; Cauchy characteristics

This paper was elaborated with the financial support of the European Uniony’s “Operational Programme Research and Development for Innovations”, No. CZ.1.05/2.1.00/03.0097, as an activity of the regional Centre AdMaS “Advanced Materials, Structures and Technologies”.

References

  • [1]

    Bryant, R.—Chern, S. S.—Goldschmidt, H.—Griffiths, P. A.: Exterior Differential Systems. Math. Sci. Res. Inst. Publ., No. 18, Springer-Verlag, 1991.Google Scholar

  • [2]

    Cartan, É.: Les Systémes Différentiels Extérieurs et Leurs Applications Géometriques. Actualités scientifiques et industrielles, No. 994, Paris: Hermann, 1971.Google Scholar

  • [3]

    Cartan, É.: Les sous-groupes des groupes continus de transformations, Ann. de l’É c. Norm. (3), (French) 25 (1908), 57–194.Google Scholar

  • [4]

    Cartan, É.: La Structure des Groupes Infinis. Seminaire de Math., exposé G, 1er mars 1937, reprinted in Elie Cartan, Oeuvres complétes, Vol. II, Editions du CNRS, 1984.Google Scholar

  • [5]

    Cartan, É.: Lecons Sur Les Invariants Intégraux, 3. ed. (French), Paris: Hermann X, 1971.Google Scholar

  • [6]

    Chrastina, J.: What the differential equations should be. In: Proceedings of the conference on differential geometry and its applications, Part 2, Univ. J. E. Purkyně, Brno, 1984, pp. 41–50.Google Scholar

  • [7]

    Chrastina, J.: The Formal Theory of Differential Equations. Folia Fac. Sci. Natur. Univ. Masaryk. Brun. Math., No. 6, Masaryk University, Brno, 1998.Google Scholar

  • [8]

    Chrastinová, V.—Tryhuk, V.: On the internal approach to differential equations 1. The involutiveness and standard basis, Math. Slovaca 66 (2016), 999–1018, .CrossrefWeb of ScienceGoogle Scholar

  • [9]

    Krasil’shchik, I. S.—Lychagin, V. V.—Vinogradov, A. M.: Geometry of Jet Spaces and Nonlinear Partial Differential Equations. Advanced Studies in Contemporary Mathematics, Gordon and Breach Science Publishers, New York, 1986.Google Scholar

  • [10]

    Kumpera, A.: On the Lie and Cartan theory of invariant differential equations, J. Math. Sci. Univ. Tokyo 6 (1999), 229–314.Google Scholar

  • [11]

    Mitropol’skij, Y. A.—Prikarpatskij, A. E.—Samoilenko, V. G.: Integrability of ideals in Grassman algebras on differentiable manifolds and some of its applications, Ukrainian Math. J. 36 (1984), 365–369.Google Scholar

  • [12]

    Montgomery, R.: A Tour to Subriemannian Geometries. Math. Surveys Monogr., No. 91, AMS, Providence, RI, USA, 2002.Google Scholar

  • [13]

    Olver, P. J.: Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics, No. 107, Springer-Verlag, New York, 1986.Google Scholar

  • [14]

    Ovsyannikov, L. V.: Group Analysis of Differential Equations, Acad. Press, New York, 1982.Google Scholar

  • [15]

    Stephani, H.: Differential Equations: Their Solution Using Symmetries, Cambridge University Press, 1989.Google Scholar

  • [16]

    Tryhuk, V.—Chrastinová, V.: Automorphisms of curves, J. Nonlinear Math. Phys. 16 (2009), 259–281.Google Scholar

  • [17]

    Tryhuk V.—Chrastinová V.: The symmetry reduction of variational integrals, Math. Bohemica (to appear).Google Scholar

  • [18]

    Vinogradov, A. M.: The category of differential equations and its significance for physics. In: Proceedings of the conference on differential geometry and its applications, Part 2, Univ. J. E. Purkyně, Brno, 1984, pp. 289–301.Google Scholar

About the article


Received: 2014-09-20

Accepted: 2015-06-03

Published Online: 2017-07-14

Published in Print: 2017-08-28


Citation Information: Mathematica Slovaca, Volume 67, Issue 4, Pages 1011–1030, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0029.

Export Citation

© 2017 Mathematical Institute Slovak Academy of Sciences.Get Permission

Comments (0)

Please log in or register to comment.
Log in