{
  "abstract": "This paper studies the time optimal control problem for the heat equation with singular inverse-square potentials under the Hardy critical condition. We first establish an observability inequality for the singular parabolic equation from general space-time measurable sets of positive Lebesgue measure. Rather than depending on the still-unknown Lebeau-Robbiano spectral inequality for the underlying singular operator, our proof combines Carleman-based observability results over open cylinders, real-analyticity estimates of solutions away from the singular origin, propagation of smallness estimates for real-analytic functions, and a telescoping-series technique. Using this observability result, we derive the null-controllability with controls supported on measurable subsets. Finally, we prove that the corresponding time optimal control is unique and obeys the bang-bang property almost everywhere over the control domain.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29059",
  "author_total": 3,
  "authors": [
    {
      "name": "Le Lu"
    },
    {
      "name": "Ke Wu"
    },
    {
      "name": "Can Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b1j15mvtzgkv3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29059",
  "fulltext_url": null,
  "id": "b1j15mvtzgkv3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29059",
  "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": "W7214330183",
  "pdf_url": "https://arxiv.org/pdf/2609.29059",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b1j15mvtzgkv3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29059",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29059"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Time optimal control for the heat equation with inverse-square potentials",
  "title_text": "Time optimal control for the heat equation with inverse-square potentials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/b1j15mvtzgkv3",
  "venue": "arXiv",
  "versions": []
}