{
  "abstract": "We show that the geometric control condition (GCC) does not guarantee internal observability for a wave equation with a smooth time-dependent potential. On a bounded connected domain with analytic boundary, we construct a nonempty open observation set such that the pair satisfies GCC and the free Dirichlet wave is observable. For this same domain and observation set, we construct a smooth $2π$-periodic potential for which there exists a nonzero smooth Dirichlet wave that vanishes on the observed cylinder for all time. The wave is the unique mild solution in the space $X_Ω=L^2(Ω)\\times H^{-1}(Ω)$ on every finite time interval, so no finite observation horizon yields an observability inequality for this fixed potential. The construction starts from a real-valued local counterexample to unique continuation across a noncharacteristic hyperplane. This local statement is of independent interest, since earlier constructions for the wave operator use complex-valued waves and potentials. The global counterexample is then obtained by joining flat oscillatory solutions along the edges of a polygonal observation set, repairing their sum at the vertices without changing its zero set, and using a spatial transition to a common time-independent pair near the boundary. The planar construction also yields counterexamples in every space dimension $n\\ge3$. For comparison, and to identify a geometry that supplies both propagation and continuation, we show on the unit disk that an observation region containing a central disk and a neighborhood of an embedded cross-cut yields observability for every weak wave at sufficiently large observation time, with constants uniform over bounded measurable potentials. The proof uses a cut, logarithmic polar coordinates, and a Carleman weight whose favorable flux is confined to the observed artificial boundary.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25327",
  "author_total": 3,
  "authors": [
    {
      "name": "Zhonghua Liao"
    },
    {
      "name": "Qi Lü"
    },
    {
      "name": "Xu Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3pe3f6ae0mvf5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25327",
  "fulltext_url": null,
  "id": "3pe3f6ae0mvf5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25327",
  "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.AP"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2207|Stability and Controllability of Differential Equations"
    ]
  ],
  "openalex": "W7223041919",
  "pdf_url": "https://arxiv.org/pdf/2610.25327",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3pe3f6ae0mvf5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.25327",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25327"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Observability of wave equations with time- and space-dependent potentials: a counterexample under GCC and a positive criterion on the disk",
  "title_text": "Observability of wave equations with time- and space-dependent potentials: a counterexample under GCC and a positive criterion on the disk",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/3pe3f6ae0mvf5",
  "venue": "arXiv",
  "versions": []
}