Computational aspects of systems over rings - reachability and stabilizability

In this paper, computational methods to test the reach ability and stabilizability of a system over a (polynomial) ring are derived. For a system S = (A, B) both reachability and stabilizability can be restated as right-invertibility conditions on the matrix (zI - A I B) over different rings. After the introduction of a polynomial ideal I related to the system, both properties can be studied simultaneously. We derive methods to compute a Gröbner basis of the ideal I and also characterize its variety. In this way we obtain algorithms to verify the reachability of a system over a polynomial ring. The corresponding stabilizability tests are mainly derived for the particular application of time-delay systems with point delays.