Necessary and sufficient conditions on \(Q(=C(I+PC)^{-1})\) for stabilization of the linear feedback system S(P,C) (Q1070198)

We find the following necessary and sufficient conditions for \(Q(:=C(I+PC)^{-1})\) to \({\mathcal H}\)-stabilize the standard linear time- invariant unity feedback system S(P,C) where P has the l.c.f. \((D_{pl},N_{pl})\) and the r.c.f. \((N_{pr},D_{pr})\); and \({\mathcal H}\) is a principal ideal domain. (i) Q must have elements in \({\mathcal H}\), (ii) (resp. (iii)) Q must factorize in \({\mathcal H}\) with \(D_{pr}\) (resp. \(D_{pl})\) as a left (resp. right) factor and (iv) (I-QP) must factor in \({\mathcal H}\) with \(D_{pr}\) as a left factor.