Relative genericity of controllablity and stabilizability for differential-algebraic systems

Abstract

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)in mathbb {R}^{ell times }times mathbb {R}^{ell times n}times mathbb {R}^{ell times m}, while for relative genericity we allow the restricted set Sigma _{ell ,n,m}^{le r} := {(E,A,B)in mathbb {R}^{ell times n}times mathbb {R}^{ell times n}times mathbb {R}^{ell times m} ,big vert ,textrm{rk},_{mathbb {R}} E le r}, where rin {mathbb {N}}. Our main results are characterizations of generic controllability and generic stabilizability in Sigma _{ell ,n,m}^{le r} in terms of the numbers ell , n, m, r.