Feedback cyclization for rings with finite stable range

For systems over commutative rings, we introduce a property called FCs (s > 0) which means “feedback cyclization with s inputs”: given a controllable system (A, B), there exist a feedback matrix K and a matrix U with s columns such that (A + BK, BU) is controllable. Clearly, FC1 is the usual FC property. The main result of this paper is the following: for a ring R with the GCU property (whenever (A, B) is controllable, there exists a vector u with Bu unimodular), R satisfies a strong form of the FCs property if and only if R is s-stable, i.e. R has s in its stable range. This generalizes the known facts that 1-stable GCU rings have the FC property, and principal ideal domains, which are 2-stable GCU rings, satisfy an analogous cyclization property with 2 inputs. Examples are given of FCs rings (for s > 1) which are not FC rings.