{
  "abstract": "We study one-dimensional cascades of heat equations controlled through their last component. For a fixed time $T>0$, we consider the Hilbert space $\\mathcal{C}_T$ of all initial data that can be steered exactly to zero at time $T$, endowed with the norm of the minimal $L^2$ control. This space can be strongly anisotropic: in explicit examples, it is incomparable with every isotropic spectral Sobolev or sub-Gaussian Gevrey scale associated with an adapted basis. For two equations, under the block Fattorini--Hautus condition, we give an almost sharp spectral description of $\\mathcal{C}_T$. More precisely, in an adapted Riesz basis of generalized eigenvectors we define an explicit increasing scale $(\\mathcal{V}_τ)_{τ\\in\\R}$ of Hilbert spaces such that $\\mathcal{V}_{T-\\varepsilon}\\hookrightarrow\\mathcal{C}_T\\hookrightarrow\\mathcal{V}_T$ for every $\\varepsilon>0$. The weights of $\\mathcal{V}_T$ combine heat dissipation with the loss of observation inside each double generalized eigenspace. When the coupling and control regions are disjoint, this recovers the known minimal null-control time $T^\\star$ and identifies it as a transition time for controllable data: for $TT^\\star$, the natural state space is properly contained in $\\mathcal{C}_T$. We extend the analysis to arbitrary finite upper triangular heat cascades. The observation losses of the different directions in each spectral block may have distinct exponential orders, leading to a hierarchy of critical times rather than a single threshold. This hierarchy describes successive gains of null controllability on adapted modal subspaces and accounts for the interactions and cancellations between different coupling paths.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14134",
  "author_total": 3,
  "authors": [
    {
      "name": "Hugo Lhachemi"
    },
    {
      "name": "Christophe Prieur"
    },
    {
      "name": "Emmanuel Trélat"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ct20r9yenv9xn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14134",
  "fulltext_url": null,
  "id": "ct20r9yenv9xn",
  "issns": [],
  "landing_url": "https://hal.science/hal-05788850v1",
  "language": "en",
  "licence": "LicenseRef-publisher-terms",
  "licence_label": "Publisher terms",
  "licence_source": "article",
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  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
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      "2207|Stability and Controllability of Differential Equations"
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  "openalex": "W7221742652",
  "pdf_url": "https://hal.science/hal-05788850/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ct20r9yenv9xn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:52Z",
      "native_id": "hal-05788850",
      "source": "hal",
      "url": "https://hal.science/hal-05788850v1"
    },
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14134",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14134"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Null-controllable data for cascade heat equations: almost sharp spectral spaces and modal critical times",
  "title_text": "Null-controllable data for cascade heat equations: almost sharp spectral spaces and modal critical times",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/ct20r9yenv9xn",
  "venue": "HAL",
  "versions": []
}