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

TL;DR

This paper develops a coordinate-free geometric framework to analyze the controllability structure of general systems of partial differential equations, extending classical results to a broader, more abstract setting.

Contribution

It introduces a novel, coordinate-free approach to describe the composition series of PDE systems, revealing their controllability structure beyond traditional jet theory methods.

Findings

Describes the composition series of PDE systems based on controllability.

Shows the triviality of the initial system in controllable cases.

Provides a multidimensional perspective on controllability in PDEs.

Abstract

The article concerns the geometrical theory of general systems $\Omega$ of partial differential equations in the \emph{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 composition series $\Omega^0\subset\Omega^1\subset\cdots\subset\Omega$ where $\Omega^k$ is the maximal system of differential equations "induced" by $\Omega$ such that the solution of $\Omega^k$ depends on arbitrary functions of $k$ independent variables (on constants if $k=0$). This is a~well--known result for the particular case of underdetermined systems of ordinary differential equations. Then $\Omega=\Omega^1$ and we have the composition series $\Omega^0\subset\Omega^1=\Omega$ where $\Omega^0$ involves all first integrals of $\Omega,$ therefore $\Omega^0$ is trivial (absent) in the controllable case. The general composition series $\Omega^0\subset\Omega^1\subset\cdots\subset\Omega$ 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.