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

Abstract

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.