{
  "abstract": "We study an optimal control problem (OCP) governed by the two-dimensional (2D) Boussinesq equations, where the right-hand sides of the velocity and temperature equations are projected onto the corresponding tangent spaces of the manifolds $\\mathcal{M}_1:=\\{y\\in\\mathbb{H}:\\|y\\|_{\\mathbb{H}}=1\\}$ and $\\mathcal{M}_2:=\\{θ\\in H:\\|θ\\|_H=1\\}$, respectively. The resulting evolution is subject to finite-codimensional constraints, under which the state $(y,θ)$ remains on $\\mathcal{M}:=\\mathcal{M}_1\\times\\mathcal{M}_2$ throughout the time interval. A Faedo--Galerkin construction provides a unique, global, strong solution to the constrained Boussinesq system for external controls $\\mathfrak{H}\\in L^2(0,T;\\mathbb{H})$, along with uniform energy bounds. In addition, we establish the existence of an optimal triplet $(y^*,θ^*,\\mathfrak{H}^*)$ that minimizes the quadratic cost subject to the constrained Boussinesq system. Necessary conditions of the first order are obtained through a coupled state--adjoint--control system. The optimal state $(y^*,θ^*)$ remains on $\\mathcal{M}$, while the optimal control lies in the tangent space: $\\mathfrak{H}^*(t)\\in T_{y^*(t)}\\mathcal{M}_1$, for all $t\\in[0,T]$. We also show that the map $\\mathfrak{H}\\mapsto(y_{\\mathfrak{H}},θ_{\\mathfrak{H}})$ in $(C([0,T];\\mathbb{H})\\cap L^2(0,T;\\mathbb{V}))\\times(C([0,T];H)\\cap L^2(0,T;V))$ is Fréchet differentiable. The linearized state is the unique weak solution to the constrained linearized Boussinesq system and satisfies the necessary tangency condition. Hence, the well-posedness and first-order optimality conditions are established for the Boussinesq system under nonlinear manifold constraints.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33828",
  "author_total": 3,
  "authors": [
    {
      "name": "Utpal Manna"
    },
    {
      "name": "Manil T. Mohan"
    },
    {
      "name": "Sangram Satpathi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ctyyg457gzz6s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33828",
  "fulltext_url": null,
  "id": "ctyyg457gzz6s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33828",
  "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",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7223944643",
  "pdf_url": "https://arxiv.org/pdf/2610.33828",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ctyyg457gzz6s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33828",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33828"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Optimal Control of the 2D Boussinesq Equations with Finite Codimension Constraints",
  "title_text": "Optimal Control of the 2D Boussinesq Equations with Finite Codimension Constraints",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/ctyyg457gzz6s",
  "venue": "arXiv",
  "versions": []
}