{
  "id": "1905.05422",
  "version": "v1",
  "published": "2019-05-14T07:14:27.000Z",
  "updated": "2019-05-14T07:14:27.000Z",
  "title": "Critical cones for sufficient second order conditions in PDE constrained optimization\\",
  "authors": [
    "Eduardo Casas",
    "Mariano Mateos"
  ],
  "comment": "21 pages",
  "categories": [
    "math.OC"
  ],
  "abstract": "In this paper, we analyze optimal control problems governed by semilinear parabolic equations. Box constraints for the controls are imposed and the cost functional involves the state and possibly a sparsity-promoting term, but not a Tikhonov regularization term. Unlike finite dimensional optimization or control problems involving Tikhonov regularization, second order sufficient optimality conditions for the control problems we deal with must be imposed in a cone larger than the one used to obtain necessary conditions. Different extensions of this cone have been proposed in the literature for different kinds of minima: strong or weak minimizers for optimal control problems. After a discussion on these extensions, we propose a new extended cone smaller than those considered until now. We prove that a second order condition based on this new cone is sufficient for a strong local minimum.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-14T07:14:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K59",
      "49K20"
    ],
    "keywords": [
      "sufficient second order conditions",
      "optimal control problems",
      "critical cones",
      "second order sufficient optimality conditions",
      "tikhonov regularization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}