{
  "abstract": "We study the exact controllability of the cubic nonlinear Schrödinger equation in the exterior $Ω=\\mathbb{R}^n\\setminusΘ$ of a non-trapping obstacle $Θ$, with internal controls supported outside a large ball, and the corresponding boundary control problem on $Ω_0=B_{R_0}\\setminusΘ$ with Dirichlet controls acting only on the outer sphere $\\partial B_{R_0}$. Using the local smoothing effect of Burq, Gérard and Tzvetkov [9], we prove observability inequalities for the linear Schrödinger equation in the whole scale of Sobolev spaces $H^σ_D(Ω)$, $σ\\in[-2,2]$, associated with the Dirichlet Laplacian; they require only the non-trapping assumption (no star-shapedness of $Θ$), and their proofs use no normal boundary traces. Combined with Strichartz-type estimates and a perturbation argument, this yields local exact controllability in $H^1_0(Ω)$ and in $H^2(Ω)\\cap H^1_0(Ω)$, in dimensions two and three, around the zero solution and, for internal controls and under a unique continuation assumption, around nontrivial trajectories. The same method applies to the quintic equation, which is energy-critical in dimension three. Several open problems are discussed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11968",
  "author_total": 3,
  "authors": [
    {
      "name": "Roberto de A. Capistrano Filho"
    },
    {
      "name": "Lionel Rosier"
    },
    {
      "name": "Bingyu Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2afp3y87zwkts.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11968",
  "fulltext_url": null,
  "id": "2afp3y87zwkts",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11968",
  "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",
      "math.AP"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7221186029",
  "pdf_url": "https://arxiv.org/pdf/2610.11968",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2afp3y87zwkts.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.11968",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11968"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Nonlinear Schrödinger equation in an exterior domain",
  "title_text": "Nonlinear Schrödinger equation in an exterior domain",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/2afp3y87zwkts",
  "venue": "arXiv",
  "versions": []
}