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

Veronika Chrastinová; Václav Tryhuk

doi:10.1515/ms-2017-0029

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

Veronika Chrastinová and Václav Tryhuk

Published/Copyright: July 14, 2017

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From the journal Mathematica Slovaca Volume 67 Issue 4

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 Ω^k is the maximal system of differential equations “induced” by Ω such that the solution of Ω^k 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”.

(Communicated by Andras Ronto)

Appendix

Let us first of all informally mention the well-known concept of the infinitesimal symmetry of a “geometrical object 𝓐” on a space M. Such infinitesimal symmetry Z ∈ 𝓣(M) is defined by the property that the Lie derivative 𝓛_Z “does not change 𝓐”. As a result, there appears a Lie algebra over ℝ of such vector fields Z.

A slight change of this idea provides the Adj-module [7]. Let us suppose that even all Lie derivatives 𝓛_fZ (f ∈ 𝓕(M)) do not change 𝓐. Then the geometrical intuition suggests that the object is “represented by the orbits of Z”. Alternatively saying, 𝓐 can be “expressed in terms of functions f ∈ 𝓕(M)” constant along the orbits. In other words, if Adj𝓐 ⊂ Φ(M) is the submodule generated by differentials df then 𝓗(Adj𝓐) ⊂ 𝓣(M) is generated by vector fields Z.

Examples

If 𝓐 ⊂ Φ(M) is a subset of differential forms, vector fields Z ∈ 𝓗(Adj𝓐) satisfy 𝓛_fZφ = 0 (φ ∈ 𝓐). If 𝓐 ⊂ Φ(M) is a submodule, we require 𝓛_fZ 𝓐 ⊂ 𝓐. Instead of differential forms, we may take tensors as well. For the exterior systems, the Adj-module describes just the classical Cauchy characteristics.

The Adj-modules frequently appear already in early E. Cartan’s articles, see especially [2], [5] and we also refer to the recent article [11] for quite other approach and useful review of the classical literature. All these authors however deal with finite-dimensional spaces M. In our infinite-dimensional space M, certain caution is necessary since the vector fields Z need not generate any group and therefore “do not produce” any orbits. In order to obtain the “economical variables for 𝓐”, it is necessary to introduce the Cauchy submodule 𝓒 of module Adj. On this occasion, we refer to the following result [7: VII. 6].

Proposition

Let Ω ⊂ Φ(M) be a diffiety with a good filtration Ω_*. Let 𝓒(Ω) ⊂ 𝓗(Ω) be the submodule of all vector fields Z such thatLZkΩl⊂Ωl+c(Z)for all (equivalently: for some) l large enough. Then there exists a basis of Ω expressible in terms of functions f ∈ 𝓕(M) such that Zf = 0 (Z ∈ 𝓒(Ω)).

Alternatively saying, the orbits of vector fields Z ∈ 𝓒(Ω) exist and may be regarded for the absolute Cauchy characteristics of the diffiety Ω. The Proposition remains true for the prediffieties [7: VIII. 3]. In this way, the uniquely determined underlying space N of flat submodules 𝓡⊂Ω without any “parasite variables” appears.

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Received: 2014-9-20

Accepted: 2015-6-3

Published Online: 2017-7-14

Published in Print: 2017-8-28

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https://doi.org/10.1515/ms-2017-0029

Keywords for this article

diffiety; controllability; Cauchy characteristics