{
  "id": "math/0307158",
  "version": "v2",
  "published": "2003-07-11T11:16:35.000Z",
  "updated": "2003-11-05T18:07:19.000Z",
  "title": "Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time",
  "authors": [
    "Luc Miller"
  ],
  "comment": "26 pages, uses elsart.sty, typos and section 5.3 corrected",
  "categories": [
    "math.AP",
    "math.OC"
  ],
  "abstract": "Given a control region $\\Omega$ on a compact Riemannian manifold $M$, we consider the heat equation with a source term $g$ localized in $Omega$. It is known that any initial data in $L^2(M)$ can be stirred to 0 in an arbitrarily small time $T$ by applying a suitable control $g$ in $L^2([0,T]xOmega)$, and, as $T$ tends to 0, the norm of $g$ grows like $e^(C/T)$ times the norm of the data. We investigate how $C$ depends on the geometry of $Omega$. We prove $C\\geq d^{2}/4$ where $d$ is the largest distance of a point in $M$ from $\\Omega$. When $M$ is a segment of length $L$ controlled at one end, we prove $C\\leq alpha L^{2}$ for some $alpha < 2$. Moreover, this bound implies $C\\leq alpha L_{Omega}^2$ where $L_{Omega}$ is the length of the longest generalized geodesic in $M$ which does not intersect $\\Omega$. The control transmutation method used in proving this last result is of a broader interest.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2003-11-05T18:07:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B37",
      "35K05"
    ],
    "keywords": [
      "heat equation",
      "geometric bounds",
      "growth rate",
      "null-controllability cost",
      "compact riemannian manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}