{
  "abstract": "For a controllable single-input linear system in dimension $n$, the controllability Gramian eigenvalues, ordered decreasingly, are known to have the successive small-time orders $T, T^3, \\ldots, T^{2n-1}$. We refine this leading-order hierarchy by explicitly computing the next-order coefficient of every eigenvalue and the first-order variation of its eigendirection and of the nested spectral subspaces. The resulting formulas have intrinsic expressions in the orthonormal Krylov basis: eigenvalue corrections describe the action of the dynamics along each Krylov direction, while variations of the spectral subspaces describe coupling to the next direction. We also establish local joint real-analytic dependence on $(T, A, b)$ near $T=0$ for the eigenvalues divided by their leading powers of $T$, consistently oriented eigenvectors, and spectral projectors. This provides convergent expansions with remainders uniform on compact families of controllable pairs. These results yield refined asymptotics for the worst-case minimum energy required to reach a unit target from the origin, the Gramian determinant and condition number, and the Ornstein--Uhlenbeck Gaussian profile in moving principal coordinates. The energy equals the reciprocal of the smallest Gramian eigenvalue, whose eigendirection identifies the most energy-demanding targets for sufficiently small times. We further derive an exact Gramian-weighted energy identity for the forward Fokker--Planck equation and sharp anisotropic short-time smoothing asymptotics using, respectively, the Lyapunov equation and an exact Fourier norm formula combined with the graded Gramian factorization. For symmetric dynamics, the spectral data reconstruct $A$ and determine $b$ up to its global sign.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37077",
  "author_total": 2,
  "authors": [
    {
      "name": "Emmanuel Trélat"
    },
    {
      "name": "Enrique Zuazua"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8e3h18yd6k63n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37077",
  "fulltext_url": null,
  "id": "8e3h18yd6k63n",
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  "landing_url": "https://hal.science/hal-05768599v1",
  "language": "en",
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  "openalex": "W7214954851",
  "pdf_url": "https://hal.science/hal-05768599/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8e3h18yd6k63n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:42Z",
      "native_id": "hal-05768599",
      "source": "hal",
      "url": "https://hal.science/hal-05768599v1"
    },
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37077",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37077"
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  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Two-term small-time spectral expansions for controllability Gramians",
  "title_text": "Two-term small-time spectral expansions for controllability Gramians",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/8e3h18yd6k63n",
  "venue": "HAL",
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}