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

Abstract

AbstractThe article concerns the geometrical theory of general systems Ω of partial differential equations in theabsolute 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 Ωkis the maximal system of differential equations “induced” by Ω such that the solution of Ωkdepends on arbitrary functions ofkindependent variables (on constants ifk= 0). This is a well-known result only for the particular case of underdetermined systems of ordinary differential equations. Then Ω = Ω1and we have the composition series Ω0⊂ Ω1= Ω where Ω0involves all first integrals of Ω, therefore Ω0is 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.