{
  "id": "2409.00479",
  "version": "v1",
  "published": "2024-08-31T15:26:16.000Z",
  "updated": "2024-08-31T15:26:16.000Z",
  "title": "Optimal control of Newtonian fluids in a stochastic environment",
  "authors": [
    "Nikolai Chemetov",
    "Fernanda Cipriano"
  ],
  "categories": [
    "math.PR",
    "math.AP",
    "math.OC"
  ],
  "abstract": "We consider a velocity tracking problem for stochastic Navier-Stokes equations in a 2D-bounded domain. The control acts on the boundary through an injection-suction device with uncertainty, which acts in accordance with the non-homogeneous Navier-slip boundary conditions. After establishing a suitable stability result for the solution of the stochastic state equation, we prove the well-posedness of the stochastic linearized state equation and show that the G\\^ateaux derivative of the control-to-state mapping corresponds to the unique solution of the linearized equation. Next, we study the stochastic backward adjoint equation and establish a duality relation between the solutions of the forward linearized equation and the backward adjoint equation. Finally, we derive the first-order optimality conditions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-31T15:26:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "93E20",
      "49K45"
    ],
    "keywords": [
      "optimal control",
      "newtonian fluids",
      "stochastic environment",
      "stochastic backward adjoint equation",
      "linearized equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}