{
  "id": "1804.07565",
  "version": "v1",
  "published": "2018-04-20T11:57:20.000Z",
  "updated": "2018-04-20T11:57:20.000Z",
  "title": "Moments and convex optimization for analysis and control of nonlinear partial differential equations",
  "authors": [
    "Milan Korda",
    "Didier Henrion",
    "Jean-Bernard Lasserre"
  ],
  "categories": [
    "math.OC",
    "math.AP"
  ],
  "abstract": "This work presents a convex-optimization-based framework for analysis and control of nonlinear partial differential equations. The approach uses a particular weak embedding of the nonlinear PDE, resulting in a linear equation in the space of Borel measures. This equation is then used as a constraint of an infinite-dimensional linear programming problem (LP). This LP is then approximated by a hierarchy of convex, finite-dimensional, semidefinite programming problems (SDPs). In the case of analysis of uncontrolled PDEs, the solutions to these SDPs provide bounds on a specified, possibly nonlinear, functional of the solutions to the PDE; in the case of PDE control, the solutions to these SDPs provide bounds on the optimal value of a given optimal control problem as well as suboptimal feedback controllers. The entire approach is based purely on convex optimization and does not rely on spatio-temporal gridding, even though the PDE addressed can be fully nonlinear. The approach is applicable to a very broad class nonlinear PDEs with polynomial data. Computational complexity is analyzed and several complexity reduction procedures are described. Numerical examples demonstrate the approach.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-20T11:57:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlinear partial differential equations",
      "convex optimization",
      "broad class nonlinear pdes",
      "complexity reduction procedures",
      "infinite-dimensional linear programming problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}