{
  "abstract": "This paper addresses exact controllability and rapid stabilization issues for linear difference delay systems with finitely many pointwise delays. We first establish a criterion for exact controllability in $L^r$, $r \\in [1, +\\infty)$, characterized by a necessary and sufficient Hautus-type frequency-domain condition. The sufficiency of this condition in $L^r$ was already known [https://hal.science/hal-04830757], whereas its necessity had previously been established only in $L^1$ [https://hal.science/hal-03827918]. We answer [https://hal.science/hal-03827918, Conjecture 3.8] by proving that this condition is also necessary for $r \\in (1, +\\infty)$. The analysis is based on rewriting the system as an evolution equation in a suitable Banach space, which allows one to apply duality arguments in the spirit of [https://doi.org/10.1137/S036301299119795X]. Building on this controllability analysis and the abstract stabilization framework developed in [https://doi.org/10.1137/S0363012996301609, https://hal.science/hal-01272024], we prove that our exact controllability criterion in $L^r$ is also a sufficient condition for rapid stabilization in $L^2$, i.e., that the system of interest can be stabilized by a family of state feedback control laws ensuring that the closed-loop system is exponentially stable in $L^2$ with an arbitrarily large decay rate. As an application, we extend our results to linear one-dimensional hyperbolic partial differential equations, which can be reduced to linear difference delay systems via the method of characteristics, obtaining also a necessary and sufficient frequency-domain criterion for exact controllability in $L^r$, $r\\in [1, +\\infty)$, which, as before, is a sufficient condition for rapid stabilization in $L^2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31225",
  "author_total": 4,
  "authors": [
    {
      "name": "Yacine Chitour"
    },
    {
      "name": "Sébastien Fueyo"
    },
    {
      "name": "Pierre Lissy"
    },
    {
      "name": "Guilherme Mazanti"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cceft38jkxvan.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31225",
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  "id": "cceft38jkxvan",
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  "landing_url": "https://hal.science/hal-05835233v1",
  "language": "en",
  "licence": "CC-BY-NC-ND-4.0",
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  "pdf_url": "https://hal.science/hal-05835233/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cceft38jkxvan.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:01Z",
      "native_id": "hal-05835233",
      "source": "hal",
      "url": "https://hal.science/hal-05835233v1"
    },
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.31225",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31225"
    }
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  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Exact controllability and rapid stabilization of linear difference delay equations and one-dimensional hyperbolic systems",
  "title_text": "Exact controllability and rapid stabilization of linear difference delay equations and one-dimensional hyperbolic systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/cceft38jkxvan",
  "venue": "HAL",
  "versions": []
}