{
  "abstract": "We provide a unified algebraic classification of approximate controllability, approximate null controllability, exact null controllability, and exact controllability for finite-dimensional linear stochastic control systems with constant coefficients and multidimensional Brownian noise. The controls may enter both the drift and the diffusion terms, and the joint diffusion-control matrix may have arbitrary rank. A finite decreasing sequence of subspaces defines a common geometric obstruction. Its vanishing characterizes the two approximate properties and exact null controllability. Exact controllability to arbitrary random targets additionally requires the joint diffusion-control matrix to be surjective. The codimension of the obstruction is given by a rank difference of explicit polynomial block matrices, yielding finite tests in the original coefficients. When the joint diffusion-control matrix is surjective, we obtain a finite Kalman-type full-rank condition for exact controllability. If the obstruction vanishes, a linear state feedback with a deterministic time-dependent gain drives every initial state to zero at any prescribed positive time with finite energy and polynomial small-time cost. An equivalent orthogonal elimination algorithm computes the rank differences with polynomial storage and cost, avoiding exponentially large block matrices. A two-noise example relates the input classification to terminal errors and control energy in penalized stochastic linear-quadratic control problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14975",
  "author_total": 2,
  "authors": [
    {
      "name": "Qi Lü"
    },
    {
      "name": "Yu Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2z4e50aq79xnr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14975",
  "fulltext_url": null,
  "id": "2z4e50aq79xnr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14975",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2207|Stability and Controllability of Differential Equations"
    ]
  ],
  "openalex": "W7221832826",
  "pdf_url": "https://arxiv.org/pdf/2610.14975",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2z4e50aq79xnr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14975",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14975"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "A Unified Kalman-Type Classification of Controllability for Linear Stochastic Systems",
  "title_text": "A Unified Kalman-Type Classification of Controllability for Linear Stochastic Systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/2z4e50aq79xnr",
  "venue": "arXiv",
  "versions": []
}