{
  "abstract": "We study the parameter dependence and transversality properties of trajectories of the controlled ODE $$ U_x = P, \\qquad P_x = - f(U) - βP + \\tilde α(x)U, $$ arising from traveling-wave solutions of a bistable reaction-diffusion equation with control. Optimal profiles minimizing the $L^1$-norm of the control $\\tilde α$ are determined by the unstable and stable manifolds $Γ_u$, $Γ_s$, the critical curve $$ P^*(U) = \\sqrt{Uf(U)}, $$ and the associated function $$ F(U) = \\frac{dP^*}{dU} + \\frac{f(U)}{P^*(U)}. $$ We establish generic transversality properties for these objects with respect to perturbations of the speed $β$ and the nonlinear source $f$, together with quantitative estimates on the first- and second-order dependence of the invariant manifolds on these parameters. We also analyze the loss of regularity at the critical speed $β^{**}$, where the unstable manifold approaches the strongly stable manifold of the equilibrium $(U^*,0)$. These results provide the ODE ingredients used in the analysis of the structure and regularity of optimal profiles in paper [S. Bianchini and C. Trifone. Traveling profiles and control cost for a PDE describing the evolution of invasive species. 2026. arXiv: 2606.23979 [math.OC]].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12105",
  "author_total": 2,
  "authors": [
    {
      "name": "Stefano Bianchini"
    },
    {
      "name": "Chiara Trifone"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0x98xe0eppwya.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12105",
  "fulltext_url": null,
  "id": "0x98xe0eppwya",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12105",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2207|Stability and Controllability of Differential Equations"
    ]
  ],
  "openalex": "W7221109413",
  "pdf_url": "https://arxiv.org/pdf/2610.12105",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0x98xe0eppwya.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.12105",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12105"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Invariant manifolds and transversality for a traveling-wave ODE in a controlled reaction-diffusion model",
  "title_text": "Invariant manifolds and transversality for a traveling-wave ODE in a controlled reaction-diffusion model",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/0x98xe0eppwya",
  "venue": "arXiv",
  "versions": []
}