Rosenbrock's theorem for systems over von Neumann regular rings

Abstract

If (A,B) is a finite system over a commutative von Neumann regular ring R, the problem of searching for a matrix F such that the pencil [sI−A−BF] has some prescribed Smith normal form is reduced to the case where R is a field, a problem which for controllable systems is described by a well-known theorem of Rosenbrock on pole assignment [12], and was then generalized to noncontrollable pairs [14]. It this paper, von Neumann regular rings are characterized as the class of commutative rings for which the solution of the above problem over the ring is equivalent to its solution in each residue field.