{
  "id": "2411.18766",
  "version": "v1",
  "published": "2024-11-27T21:42:46.000Z",
  "updated": "2024-11-27T21:42:46.000Z",
  "title": "Collective steering in finite time: controllability on $\\text{GL}^+(n,\\mathbb{R})$",
  "authors": [
    "Mahmoud Abdelgalil",
    "Tryphon T. Georgiou"
  ],
  "comment": "15 pages, 2 figures",
  "categories": [
    "math.OC",
    "cs.SY",
    "eess.SY"
  ],
  "abstract": "We consider the problem of steering a collection of n particles that obey identical n-dimensional linear dynamics via a common state feedback law towards a rearrangement of their positions, cast as a controllability problem for a dynamical system evolving on the space of matrices with positive determinant. We show that such a task is always feasible and, moreover, that it can be achieved arbitrarily fast. We also show that an optimal feedback control policy to achieve a similar feat, may not exist. Furthermore, we show that there is no universal formula for a linear feedback control law to achieve a rearrangement, optimal or not, that is everywhere continuous with respect to the specifications. We conclude with partial results on the broader question of controllability of dynamics on orientation-preserving diffeomorphisms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-27T21:42:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "93-xx",
      "49-xx",
      "53-xx",
      "22-xx"
    ],
    "keywords": [
      "finite time",
      "controllability",
      "collective steering",
      "obey identical n-dimensional linear dynamics",
      "common state feedback law"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}