{
  "id": "1602.05066",
  "version": "v1",
  "published": "2016-02-15T15:12:32.000Z",
  "updated": "2016-02-15T15:12:32.000Z",
  "title": "On Characterization of Inverse Data in the Boundary Control Method",
  "authors": [
    "Mikhail Belishev",
    "Aleksei Vakulenko"
  ],
  "comment": "33 pages, 1 figure",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We deal with a dynamical system \\begin{align*} & u_{tt}-\\Delta u+qu=0 && {\\rm in}\\,\\,\\,\\Omega \\times (0,T)\\\\ & u\\big|_{t=0}=u_t\\big|_{t=0}=0 && {\\rm in}\\,\\,\\,\\overline \\Omega\\\\ & \\partial_\\nu u = f && {\\rm in}\\,\\,\\,\\partial\\Omega \\times [0,T]\\,, \\end{align*} where $\\Omega \\subset {\\mathbb R}^n$ is a bounded domain, $q \\in L_\\infty(\\Omega)$ a real-valued function, $\\nu$ the outward normal to $\\partial \\Omega$, $u=u^f(x,t)$ a solution. The input/output correspondence is realized by a response operator $R^T: f \\mapsto u^f\\big|_{\\partial\\Omega \\times [0,T]}$ and its relevant extension by hyperbolicity $R^{2T}$. Ope\\-rator $R^{2T}$ is determined by $q\\big|_{\\Omega^T}$, where $\\Omega^T:=\\{x \\in \\Omega\\,|\\,\\,{\\rm dist\\,}(x,\\partial \\Omega)<T\\}$. The inverse problem is: Given $R^{2T}$ to recover $q$ in $\\Omega^T$. We solve this problem by the boundary control method and describe the {\\it ne\\-ces\\-sary and sufficient} conditions on $R^{2T}$, which provide its solvability.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-15T15:12:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R30",
      "35Qxx",
      "35L51",
      "F.2.2",
      "I.2.7"
    ],
    "keywords": [
      "boundary control method",
      "inverse data",
      "characterization",
      "input/output correspondence",
      "response operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160205066B"
    }
  }
}