Feedback classification of single-input systems over von Neumann regularrings

TL;DRAbstract

This work deals with linear systems with scalars in a commutative ring R with the property of being “von Neumann regular”, i.e. R is zero-dimensional and has no nonzero nilpotents. We prove that every single-input, n-dimensional system over R is feedback equivalent to a special normal form, whose existence actually characterizes the class of von Neumann regular rings. This normal form, which captures completely the structure of the reachable submodule of the system, is associated to a collection of n principal ideals generated by idempotent elements f1, . . . , fn, each dividing the following one. The normal form can be obtained by an explicit algorithm, which is implemented in PARI-GP in the case R = Z/(dZ), where d is a squarefree integer.

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