{
  "id": "1301.2149",
  "version": "v1",
  "published": "2013-01-10T15:12:10.000Z",
  "updated": "2013-01-10T15:12:10.000Z",
  "title": "Numerical controllability of the wave equation through primal methods and Carleman estimates",
  "authors": [
    "Nicolae Cîndea",
    "Enrique Fernandez-Cara",
    "Arnaud Munch"
  ],
  "categories": [
    "math.OC"
  ],
  "abstract": "This paper deals with the numerical computation of boundary null controls for the 1D wave equation with a potential. The goal is to compute an approximation of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. We do not use in this work duality arguments but explore instead a direct approach in the framework of global Carleman estimates. More precisely, we consider the control that minimizes over the class of admissible null controls a functional involving weighted integrals of the state and of the control. The optimality conditions show that both the optimal control and the associated state are expressed in terms of a new variable, the solution of a fourth-order elliptic problem defined in the space-time domain. We first prove that, for some specific weights determined by the global Carleman inequalities for the wave equation, this problem is well-posed. Then, in the framework of the finite element method, we introduce a family of finite-dimensional approximate control problems and we prove a strong convergence result. Numerical experiments confirm the analysis. We complete our study with several comments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-01-10T15:12:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "carleman estimates",
      "primal methods",
      "numerical controllability",
      "finite-dimensional approximate control problems",
      "1d wave equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}