{
  "abstract": "We study a constrained basis-selection problem for an ordered block matrix $B=[ B_1\\mid\\cdots\\mid B_n ].$ The successive tail matrices determine a nested family of column spaces, and we consider bases of representatives which are compatible with this induced flag. Our main linear-algebraic result shows that, among bases of representatives, compatibility with the entire flag is characterized by the intrinsic block quotas $r_k=w_k-w_{k+1}$, where $w_k$ is the rank of the $k$-th tail matrix. We then show that the same structure arises naturally in finite-horizon linear control. For the reachability matrix $[A^{N-1}G\\mid \\cdots\\mid G]$, requiring every actuator to be used exactly once turns an actuator schedule into a representative system. Preservation of the complete reachability profile is then equivalent to compatibility with the tail-space flag. This correspondence leads to an exact depth-first search with a future-rank pruning criterion. For an explicit structured family and a fixed search ordering, the baseline search visits $3\\cdot2^m+m-2$ nodes before the first solution, whereas future-rank pruning reduces this number to $3m+1$. Exact-arithmetic experiments reproduce these formulas for all tested instances.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03353",
  "author_total": 2,
  "authors": [
    {
      "name": "Dawid E. Kędzierski"
    },
    {
      "name": "Piotr Krasoń"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3hxx53b1qa3vj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03353",
  "fulltext_url": null,
  "id": "3hxx53b1qa3vj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03353",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2207|Advanced Control Systems Optimization"
    ]
  ],
  "openalex": "W7220444495",
  "pdf_url": "https://arxiv.org/pdf/2610.03353",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3hxx53b1qa3vj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.03353",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03353"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Compatible Bases and Actuator Scheduling",
  "title_text": "Compatible Bases and Actuator Scheduling",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/3hxx53b1qa3vj",
  "venue": "arXiv",
  "versions": []
}