Restricted invariant factor assignment under state-feedback (Q1307227)

Let \({\mathbb F}\) denote an arbitrary field and \({\mathbb F}[ \lambda ]\) the polynomial ring over \({\mathbb F}\). The problem investigated is the following: If \((A,B) \in {\mathbb F}^{n \times n} \times {\mathbb F}^{n \times m}\) (not necessarily controllable) and \(P_{1} (\lambda) \in {\mathbb F}[ \lambda ] ^ {t \times s},\) find necessary and sufficient conditions for the existence of a matrix \(F \in {\mathbb F} ^ {m \times n}\) and a square polynomial matrix \(P( \lambda)\) such that (i) \(\lambda I_{n} - (A + BF)\) is extended equivalent to \(P( \lambda)\) and (ii) \(P_{1} ( \lambda)\) is a principal submatrix of \(P( \lambda)\). Such conditions, involving invariant factors and controllability indices are indeed found, in several cases, with different requirements on the prescribed submatrix. The role of controllability of the systems is investigated.