{
  "id": "1708.02464",
  "version": "v1",
  "published": "2017-08-08T12:36:41.000Z",
  "updated": "2017-08-08T12:36:41.000Z",
  "title": "Optimal control of a Vlasov-Poisson plasma by an external magnetic field - The basics for variational calculus",
  "authors": [
    "Patrik Knopf"
  ],
  "categories": [
    "math.AP",
    "math.OC"
  ],
  "abstract": "We consider the three dimensional Vlasov-Poisson system that is equipped with an external magnetic field to describe a plasma. The aim of various concrete applications is to control a plasma in a desired fashion. This can be modeled by an optimal control problem. For that reason the basics for calculus of variations will be introduced in this paper. We have to find a suitable class of fields that are admissible for this procedure as they provide unique global solutions of the Vlasov-Poisson system. Then we can define a field-state operator that maps any admissible field onto its corresponding distribution function. We will show that this field-state operator is Lipschitz continuous and (weakly) compact. Last we will consider a model problem with a tracking type cost functional and we will show that this optimal control problem has at least one globally optimal solution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-08T12:36:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49J20",
      "35Q83"
    ],
    "keywords": [
      "external magnetic field",
      "vlasov-poisson plasma",
      "variational calculus",
      "optimal control problem",
      "field-state operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}