{
  "id": "2406.03110",
  "version": "v1",
  "published": "2024-06-05T09:58:15.000Z",
  "updated": "2024-06-05T09:58:15.000Z",
  "title": "Optimal Control of Semilinear Elliptic Partial Differential Equations with Non-Lipschitzian Nonlinearities",
  "authors": [
    "Constantin Christof"
  ],
  "categories": [
    "math.OC"
  ],
  "abstract": "We study optimal control problems that are governed by semilinear elliptic partial differential equations that involve non-Lipschitzian nonlinearities. It is shown that, for a certain class of such PDEs, the solution map is Fr\\'{e}chet differentiable even though the differential operator contains a nondifferentiable term. We exploit this effect to establish first-order necessary optimality conditions for minimizers of the considered control problems. The resulting KKT-conditions take the form of coupled PDE-systems that are posed in non-Muckenhoupt weighted Sobolev spaces and raise interesting questions regarding the regularity of optimal controls, the derivation of second-order optimality conditions, and the analysis of finite element discretizations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-05T09:58:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49J20",
      "49K20",
      "49K40"
    ],
    "keywords": [
      "semilinear elliptic partial differential equations",
      "optimal control",
      "non-lipschitzian nonlinearities",
      "first-order necessary optimality conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}