{
  "abstract": "We study the null controllability of the linear Kuramoto-Sivashinsky equation on a bounded interval using a single Dirichlet boundary control via the flatness method. The usual flatness representation for this fourth-order equation naturally involves two boundary traces, which serve as flat outputs. When two suitable boundary controls are available, these outputs can be chosen independently. With only one control, however, the remaining homogeneous boundary condition imposes a compatibility relation between the two flat outputs. The main novelty of this work is the construction of a one-output flat parametrization that automatically incorporates this relation. It is obtained through a weighted Bézout theorem for the entire functions associated with the boundary conditions. The required growth estimates for the Bézout coefficients are established using Hörmander's weighted ∂ estimate. Combining this parametrization with the regularizing effect of the free dynamics and a Gevrey cutoff, we construct the control and the corresponding trajectory as convergent flat series. We prove null controllability at any positive time by means of a boundary control whenever the anti-diffusion parameter lies outside a countable critical set determined by the length of the interval. In particular, the control has Gevrey regularity in time, while the state has Gevrey regularity in time and space away from the initial time.For critical parameter values, we exhibit an unobservable eigenmode and prove that null controllability under the considered boundary control fails. To the best of our knowledge, this is the first flatness-based analysis of the Kuramoto-Sivashinsky equation that successfully handles the critical-set phenomenon.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Subrata Majumdar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/31rr8cpkbk76s.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "31rr8cpkbk76s",
  "issns": [],
  "landing_url": "https://hal.science/hal-05765772v2",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05765772/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/31rr8cpkbk76s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:42Z",
      "native_id": "hal-05765772",
      "source": "hal",
      "url": "https://hal.science/hal-05765772v2"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "Flatness-based trajectory planning for the linear Kuramoto-Sivashinsky equation with a single boundary input",
  "title_text": "Flatness-based trajectory planning for the linear Kuramoto-Sivashinsky equation with a single boundary input",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:42Z",
  "url": "https://research.draconitic.com/paper/31rr8cpkbk76s",
  "venue": "HAL",
  "versions": []
}