On the internal approach to differential equations 2. The controllability structure
The article concerns the geometrical theory of general systems $Ω$ 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 $Ω^0\subsetΩ^1\subset\cdots\subsetΩ$ 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 for the particular case of underdetermined systems of ordinary differential equations. Then $Ω=Ω^1$ and we have the composition series $Ω^0\subsetΩ^1=Ω$ where $Ω^0$ involves all first integrals of $Ω,$ therefore $Ω^0$ is trivial (absent) in the controllable case. The general composition series $Ω^0\subsetΩ^1\subset\cdots\subsetΩ$ 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.