{
  "id": "1703.10797",
  "version": "v1",
  "published": "2017-03-31T08:41:31.000Z",
  "updated": "2017-03-31T08:41:31.000Z",
  "title": "Tunneling estimates and approximate controllability for hypoelliptic equations",
  "authors": [
    "Camille Laurent",
    "Matthieu Léautaud"
  ],
  "categories": [
    "math.AP",
    "math.OC",
    "math.SP"
  ],
  "abstract": "This article is concerned with quantitative unique continuation estimates for equations involving a \"sum of squares\" operator $\\mathcal{L}$ on a compact manifold $\\mathcal{M}$ assuming: $(i)$ the Chow-Rashevski-H\\\"ormander condition ensuring the hypoellipticity of $\\mathcal{L}$, and $(ii)$ the analyticity of $\\mathcal{M}$ and the coefficients of $\\mathcal{L}$. The first result is the tunneling estimate $\\|\\varphi\\|_{L^2(\\omega)} \\geq Ce^{- \\lambda^{\\frac{k}{2}}}$ for normalized eigenfunctions $\\varphi$ of $\\mathcal{L}$ from a nonempty open set $\\omega\\subset \\mathcal{M}$, where $k$ is the hypoellipticity index of $\\mathcal{L}$ and $\\lambda$ the eigenvalue. The main result is a stability estimate for solutions to the hypoelliptic wave equation $(\\partial_t^2+\\mathcal{L})u=0$: for $T>2 \\sup_{x \\in \\mathcal{M}}(dist(x,\\omega))$ (here, $dist$ is the sub-Riemannian distance), the observation of the solution on $(0,T)\\times \\omega$ determines the data. The constant involved in the estimate is $Ce^{c\\Lambda^k}$ where $\\Lambda$ is the typical frequency of the data. We then prove the approximate controllability of the hypoelliptic heat equation $(\\partial_t+\\mathcal{L})v=1_\\omega f$ in any time, with appropriate (exponential) cost, depending on $k$. In case $k=2$ (Grushin, Heisenberg...), we further show approximate controllability to trajectories with polynomial cost in large time. We also explain how the analyticity assumption can be relaxed, and a boundary $\\partial \\mathcal{M}$ can be added in some situations. Most results turn out to be optimal on a family of Grushin-type operators. The main proof relies on the general strategy developed by the authors in arxiv:1506.04254.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-31T08:41:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "approximate controllability",
      "tunneling estimate",
      "hypoelliptic equations",
      "quantitative unique continuation estimates",
      "nonempty open set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}