{
  "id": "2210.13590",
  "version": "v1",
  "published": "2022-10-24T20:29:58.000Z",
  "updated": "2022-10-24T20:29:58.000Z",
  "title": "Hautus--Yamamoto criteria for approximate and exact controllability of linear difference delay equations",
  "authors": [
    "Yacine Chitour",
    "Sébastien Fueyo",
    "Guilherme Mazanti",
    "Mario Sigalotti"
  ],
  "categories": [
    "math.OC"
  ],
  "abstract": "The paper deals with the controllability of finite-dimensional linear difference delay equations, i.e., dynamics for which the state at a given time $t$ is obtained as a linear combination of the control evaluated at time $t$ and of the state evaluated at finitely many previous instants of time $t-\\Lambda_1,\\dots,t-\\Lambda_N$. Based on the realization theory developed by Y.Yamamoto for general infinite-dimensional dynamical systems, we obtain necessary and sufficient conditions, expressed in the frequency domain, for the approximate controllability in finite time in $L^q$ spaces, $q \\in [1, +\\infty)$. We also provide a necessary condition for $L^1$ exact controllability, which can be seen as the closure of the $L^1$ approximate controllability criterion. Furthermore, we provide an explicit upper bound on the minimal times of approximate and exact controllability, given by $d\\max\\{\\Lambda_1,\\dots,\\Lambda_N\\}$, where $d$ is the dimension of the state space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-24T20:29:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "exact controllability",
      "hautus-yamamoto criteria",
      "finite-dimensional linear difference delay equations",
      "explicit upper bound",
      "approximate controllability criterion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}