{
  "id": "2302.07766",
  "version": "v1",
  "published": "2023-02-15T16:34:06.000Z",
  "updated": "2023-02-15T16:34:06.000Z",
  "title": "An optimal control problem subject to strong solutions of chemotaxis-consumption models",
  "authors": [
    "Francisco Guillén-González",
    "André Luiz Corrêa Vianna Filho"
  ],
  "categories": [
    "math.OC"
  ],
  "abstract": "In the present work we investigate a bilinear optimal control problem associated to the following chemotaxis-consumption model in a bounded domain $\\Omega \\subset \\mathbb{R}^3$ during a time interval $(0,T)$: $$\\partial_t u - \\Delta u = - \\nabla \\cdot (u \\nabla v), \\quad \\partial_t v - \\Delta v = - u^s v + f v 1_{\\Omega_c}$$ with $s \\geq 1$, endowed with isolated boundary conditions and initial conditions for $(u,v)$, $u$ being the cell density, $v$ the chemical concentration and $f$ the bilinear control acting in a subdomain $\\Omega_c \\subset \\Omega$. The existence of weak solutions $(u,v)$ to this model given $f \\in L^q((0,T) \\times \\Omega)$, for some $q > 5/2$, has been proved in \\cite{guillen2022optimal}. In this work the optimal control problem is studied in a strong solution setting. First we prove that the regularity criterion $u ^s,f \\in L^q((0,T) \\times \\Omega)$ allows us to get existence and uniqueness of global-in-time strong solutions. In the sequel, we show the existence of a global optimal solution. Finally, using a Lagrange multipliers theorem, we establish first order optimality conditions for any local optimal solution, proving existence, uniqueness and regularity of the associated Lagrange multipliers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-15T16:34:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49J20",
      "49K20",
      "49N60",
      "35Q92",
      "92C17"
    ],
    "keywords": [
      "optimal control problem subject",
      "strong solution",
      "chemotaxis-consumption model",
      "bilinear optimal control problem",
      "establish first order optimality conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}