Relative genericity of controllablity and stabilizability for differential-algebraic systems

The present note is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359-377, 2021) on generic controllability and stabilizability of linear differential-algebraic equations. We resolve the drawback that genericity is considered in the unrestricted set of system matrices (E, A, B) is an element of R-lx x R-l x n x R-l x m, while for relative genericity we allow the restricted set Sigma(<= r)(l,n,m) := {(E, A, B) is an element of R(lxn )x R-lxn x R-lxm vertical bar rk(R)E <= r}, where r is an element of N. Our main results are characterizations of generic controllability and generic stabilizability in Sigma(l, n,) (<= r)(m) in terms of the numbers l, n, m, r.