{
  "id": "1605.09750",
  "version": "v1",
  "published": "2016-05-31T18:12:07.000Z",
  "updated": "2016-05-31T18:12:07.000Z",
  "title": "Nonconvex penalization of switching control of partial differential equations",
  "authors": [
    "Christian Clason",
    "Armin Rund",
    "Karl Kunisch"
  ],
  "categories": [
    "math.OC"
  ],
  "abstract": "A standard approach to treat constraints in nonlinear optimization is penalization, in particular using $L^1$-type norms. Applying this approach to the pointwise switching constraint $u_1(t)u_2(t)=0$ leads to a nonsmooth and nonconvex infinite-dimensional minimization problem which is challenging both analytically and numerically. Adding $H^1$ regularization or restricting to a finite-dimensional control space allows showing existence of optimal controls. First order necessary optimality conditions are then derived using tools of nonsmooth analysis. Their solution can be computed using a combination of Moreau--Yosida regularization and a semismooth Newton method. Numerical examples illustrate the properties of this approach.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-31T18:12:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "partial differential equations",
      "nonconvex penalization",
      "switching control",
      "first order necessary optimality conditions",
      "nonconvex infinite-dimensional minimization problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}