{
  "abstract": "In the 2004 collection \\emph{Unsolved Problems in Mathematical Systems and Control Theory}, Erik Verriest posed the problem of characterizing Riccati stability ``without invoking additional matrices.'' We give such a characterization through a scalar invariant of the resolvent family. The invariant is defined by covariance balances using at most $n^2+1$ frequency--direction pairs. A finite-dimensional separation argument shows that this covariance radius equals the optimal common ellipsoidal norm of the resolvent family. Combined with the strict bounded real lemma, this yields a necessary and sufficient condition for Riccati stability involving only the Hurwitz property of the first matrix and a single scalar inequality. The invariant reduces to the ordinary spectral radius for one matrix, lies between the pointwise spectral-radius and unscaled small-gain levels of the resolvent family, and coincides with the operator-space spectral radius of Shalit and Shamovich for the natural resolvent function space. The main results have been formally verified in the Lean proof assistant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06275",
  "author_total": 1,
  "authors": [
    {
      "name": "Jeffrey Humpherys"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5j7x6ten71d2h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06275",
  "fulltext_url": null,
  "id": "5j7x6ten71d2h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06275",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.SY"
    ],
    [
      "arxiv",
      "eess.SY"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2207|Stability and Control of Uncertain Systems"
    ]
  ],
  "openalex": "W7211998703",
  "pdf_url": "https://arxiv.org/pdf/2609.06275",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5j7x6ten71d2h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2609.06275",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06275"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Riccati Stability Without Auxiliary Matrices",
  "title_text": "Riccati Stability Without Auxiliary Matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/5j7x6ten71d2h",
  "venue": "arXiv",
  "versions": []
}