{
  "abstract": "We study an optimal control problem for the two-dimensional Navier-Stokes equations projected onto the tangent space of the $\\mathbb{H}$-unit sphere manifold $\\mathcal{M} := \\{\\boldsymbol{u} \\in \\mathbb{H} : \\|\\boldsymbol{u}\\|_{\\mathbb{H}}=1\\}$. This geometric projection yields a constrained evolution system of finite codimension, ensuring that the state remains on $\\mathcal{M}$ for all times. Using a Faedo-Galerkin approximation technique, we establish the global existence and uniqueness of strong solutions to the constrained Navier-Stokes equations with external controls $\\boldsymbol{U} \\in \\mathrm{L}^2(0,T;\\mathbb{H})$, together with uniform energy estimates. The associated optimal control problem, minimizing a quadratic cost functional subject to the manifold constraint, is shown to admit at least one optimal pair $(\\boldsymbol{u}^*,\\boldsymbol{U}^*)$. We derive the first-order necessary optimality conditions by formulating an optimal control system whose state-control pair $(\\boldsymbol{u}^*,\\boldsymbol{U}^*)$ evolves on the tangent bundle $T\\mathcal{M}$ of the constraint manifold $\\mathcal{M}$. In this setting, the control satisfies $\\boldsymbol{U}^*(t)\\in T_{\\boldsymbol{u}^*(t)}\\mathcal{M}$ for all $t\\in[0,T]$, ensuring that the dynamics remain confined to the tangent bundle $T\\mathcal{M}$. Furthermore, we establish the Fréchet differentiability of the control-to-state mapping $\\boldsymbol{U}\\mapsto \\boldsymbol{u}_{\\boldsymbol{U}}$ in $\\mathrm{C}([0,T];\\mathbb{H})\\cap\\mathrm{L}^2(0,T;\\mathbb{V})$, with the corresponding linearized state characterized as the unique weak solution of a linearized constrained Navier-Stokes system preserving the tangency condition. These results provide a rigorous analytical foundation for the optimal control of fluid flows constrained to nonlinear manifolds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00249",
  "author_total": 1,
  "authors": [
    {
      "name": "Manil T. Mohan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bks4mbe3a6fp3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00249",
  "fulltext_url": null,
  "id": "bks4mbe3a6fp3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00249",
  "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"
    ],
    [
      "openalex",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7215862458",
  "pdf_url": "https://arxiv.org/pdf/2610.00249",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bks4mbe3a6fp3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.00249",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00249"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Optimal Control of 2D Navier--Stokes Equations with Constraints of Finite Codimension",
  "title_text": "Optimal Control of 2D Navier--Stokes Equations with Constraints of Finite Codimension",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/bks4mbe3a6fp3",
  "venue": "arXiv",
  "versions": []
}