{
  "id": "2409.10169",
  "version": "v1",
  "published": "2024-09-16T10:58:52.000Z",
  "updated": "2024-09-16T10:58:52.000Z",
  "title": "Controllability Problems for the Heat Equation in a Half-Plane Controlled by the Neumann Boundary Condition with a Point-Wise Control",
  "authors": [
    "Larissa Fardigola",
    "Kateryna Khalina"
  ],
  "categories": [
    "math.AP",
    "math.OC"
  ],
  "abstract": "In the paper, the problems of controllability and approximate controllability are studied for the control system $w_t=\\Delta w$, $w_{x_1}(0,x_2,t)=u(t)\\delta(x_2)$, $x_1>0$, $x_2\\in\\mathbb R$, $t\\in(0,T)$, where $u\\in L^\\infty(0,T)$ is a control. To this aid, it is investigated the set $\\mathcal{R}_T(0)\\subset L^2((0,+\\infty)\\times\\mathbb R)$ of its end states which are reachable from $0$. It is established that a function $f\\in\\mathcal{R}_T(0)$ can be represented in the form $f(x)=g\\big(|x|^2\\big)$ a.e. in $(0,+\\infty)\\times\\mathbb R$ where $g\\in L^2(0,+\\infty)$. In fact, we reduce the problem dealing with functions from $L^2((0,+\\infty)\\times\\mathbb R)$ to a problem dealing with functions from $L^2(0,+\\infty)$. Both a necessary and sufficient condition for controllability and a sufficient condition for approximate controllability in a given time $T$ under a control $u$ bounded by a given constant are obtained in terms of solvability of a Markov power moment problem. Using the Laguerre functions (forming an orthonormal basis of $L^2(0,+\\infty)$), necessary and sufficient conditions for approximate controllability and numerical solutions to the approximate controllability problem are obtained. It is also shown that there is no initial state that is null-controllable in a given time $T$. The results are illustrated by an example.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-16T10:58:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "neumann boundary condition",
      "heat equation",
      "point-wise control",
      "sufficient condition",
      "half-plane"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}