{
  "id": "1910.06198",
  "version": "v1",
  "published": "2019-10-14T15:15:50.000Z",
  "updated": "2019-10-14T15:15:50.000Z",
  "title": "Superexponential stabilizability of degenerate parabolic equations via bilinear control",
  "authors": [
    "Piermarco Cannarsa",
    "Cristina Urbani"
  ],
  "categories": [
    "math.OC",
    "math.AP"
  ],
  "abstract": "The aim of this paper is to prove the superexponential stabilizability to the ground state solution of a degenerate parabolic equation of the form \\begin{equation*} u_t(t,x)+(x^{\\alpha}u_x(t,x))_x+p(t)x^{2-\\alpha}u(t,x)=0,\\qquad t\\geq0,x\\in(0,1) \\end{equation*} via bilinear control $p\\in L_{loc}^2(0,+\\infty)$. More precisely, we provide a control function $p$ that steers the solution of the equation, $u$, to the ground state solution in small time with doubly-exponential rate of convergence.\\\\ The parameter $\\alpha$ describes the degeneracy magnitude. In particular, for $\\alpha\\in[0,1)$ the problem is called weakly degenerate, while for $\\alpha\\in[1,2)$ strong degeneracy occurs. We are able to prove the aforementioned stabilization property for $\\alpha\\in [0,3/2)$. The proof relies on the application of an abstract result on rapid stabilizability of parabolic evolution equations by the action of bilinear control. A crucial role is also played by Bessel's functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-14T15:15:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q93",
      "93C10",
      "35K10",
      "35K65"
    ],
    "keywords": [
      "degenerate parabolic equation",
      "bilinear control",
      "superexponential stabilizability",
      "ground state solution",
      "parabolic evolution equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}