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Control Signals Syst."],"published-print":{"date-parts":[[2023,3]]},"abstract":"Abstract<\/jats:title>\n \n The present note is a successor of\u00a0Ilchmann and Kirchhoff (Math Control Signals Syst 33:359\u2013377, 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\n \n \n $$(E,A,B)\\in \\mathbb {R}^{\\ell \\times }\\times \\mathbb {R}^{\\ell \\times n}\\times \\mathbb {R}^{\\ell \\times m}$$<\/jats:tex-math>\n \n \n \n (<\/mml:mo>\n E<\/mml:mi>\n ,<\/mml:mo>\n A<\/mml:mi>\n ,<\/mml:mo>\n B<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2208<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n \n \u2113<\/mml:mi>\n \u00d7<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n \u00d7<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n \n \u2113<\/mml:mi>\n \u00d7<\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n \u00d7<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n \n \u2113<\/mml:mi>\n \u00d7<\/mml:mo>\n m<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , while for relative genericity we allow the restricted set\n \n \n $$\\Sigma _{\\ell ,n,m}^{\\le r} := \\{(E,A,B)\\in \\mathbb {R}^{\\ell \\times n}\\times \\mathbb {R}^{\\ell \\times n}\\times \\mathbb {R}^{\\ell \\times m} \\,\\big \\vert \\,\\textrm{rk}\\,_{\\mathbb {R}} E \\le r\\}$$<\/jats:tex-math>\n \n \n \n \u03a3<\/mml:mi>\n \n \u2113<\/mml:mi>\n ,<\/mml:mo>\n n<\/mml:mi>\n ,<\/mml:mo>\n m<\/mml:mi>\n <\/mml:mrow>\n \n \u2264<\/mml:mo>\n r<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n :<\/mml:mo>\n =<\/mml:mo>\n \n {<\/mml:mo>\n \n (<\/mml:mo>\n E<\/mml:mi>\n ,<\/mml:mo>\n A<\/mml:mi>\n ,<\/mml:mo>\n B<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2208<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n \n \u2113<\/mml:mi>\n \u00d7<\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n \u00d7<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n \n \u2113<\/mml:mi>\n \u00d7<\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n \u00d7<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n \n \u2113<\/mml:mi>\n \u00d7<\/mml:mo>\n m<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n \n \n |<\/mml:mo>\n <\/mml:mrow>\n \n rk<\/mml:mtext>\n \n \n R<\/mml:mi>\n <\/mml:msub>\n E<\/mml:mi>\n \u2264<\/mml:mo>\n r<\/mml:mi>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , where\n \n \n $$ r\\in {\\mathbb {N}}$$<\/jats:tex-math>\n \n \n r<\/mml:mi>\n \u2208<\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Our main results are characterizations of generic controllability and generic stabilizability in\u00a0\n \n \n $$\\Sigma _{\\ell ,n,m}^{\\le r}$$<\/jats:tex-math>\n \n \n \u03a3<\/mml:mi>\n \n \u2113<\/mml:mi>\n ,<\/mml:mo>\n n<\/mml:mi>\n ,<\/mml:mo>\n m<\/mml:mi>\n <\/mml:mrow>\n \n \u2264<\/mml:mo>\n r<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n in terms of the numbers\u00a0\n \n \n $$\\ell , n, m, r$$<\/jats:tex-math>\n \n \n \u2113<\/mml:mi>\n ,<\/mml:mo>\n n<\/mml:mi>\n ,<\/mml:mo>\n m<\/mml:mi>\n ,<\/mml:mo>\n r<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n .\n <\/jats:p>","DOI":"10.1007\/s00498-022-00332-3","type":"journal-article","created":{"date-parts":[[2022,11,17]],"date-time":"2022-11-17T03:02:56Z","timestamp":1668654176000},"page":"45-76","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Relative genericity of controllablity and stabilizability for differential-algebraic systems"],"prefix":"10.1007","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9120-9368","authenticated-orcid":false,"given":"Achim","family":"Ilchmann","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jonas","family":"Kirchhoff","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,11,17]]},"reference":[{"key":"332_CR1","volume-title":"Analysis II, 1st English edition","author":"H Amann","year":"2008","unstructured":"Amann H, Escher J (2008) Analysis II, 1st English edition. 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