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)∈Rℓ××Rℓ×n×Rℓ×m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(E,A,B)\in \mathbb {R}^{\ell \times }\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m}$$\end{document}, while for relative genericity we allow the restricted set Σℓ,n,m≤r:={(E,A,B)∈Rℓ×n×Rℓ×n×Rℓ×m|rkRE≤r}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma _{\ell ,n,m}^{\le r} := \{(E,A,B)\in \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m} \,\big \vert \,\textrm{rk}\,_{\mathbb {R}} E \le r\}$$\end{document}, where r∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ r\in {\mathbb {N}}$$\end{document}. Our main results are characterizations of generic controllability and generic stabilizability in Σℓ,n,m≤r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma _{\ell ,n,m}^{\le r}$$\end{document} in terms of the numbers ℓ,n,m,r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell , n, m, r$$\end{document}.