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Mathematica Slovaca

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Volume 67, Issue 4


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
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/ Václav Tryhuk
  • Brno University of Technology Faculty of Civil Engineering AdMaS Centre Purkyňova 139, 612 00 Brno Czech Republic
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Published Online: 2017-07-14 | DOI: https://doi.org/10.1515/ms-2017-0029


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”.


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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.

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