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

2014/09/20 by Chrastinová, Veronika, Tryhuk, Václav
#35A30 #58A17 #58J99 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1409.5904

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 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 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.

Related

• Report on the absolute differential equations I
• On the internal approach to differential equations 1. The involutiveness\n and standard basis
• Notes on exterior differential systems
• Explicit differential characterization of PDE systems pointwise equivalent to Y_X^j1X^j2=0, 1≤ j₁,j₂≤ n≥ 2
• Cylindricality and autonomy of integrals and last multipliers of multidimensional differentional systems