{
  "id": "2105.07678",
  "version": "v1",
  "published": "2021-05-17T08:55:03.000Z",
  "updated": "2021-05-17T08:55:03.000Z",
  "title": "A sufficient condition for $k$-contraction of the series connection of two systems",
  "authors": [
    "Ron Ofir",
    "Michael Margaliot",
    "Yoash Levron",
    "Jean-Jacques Slotine"
  ],
  "journal": "IEEE TAC vol 67, issue 9 (2022), 4994 - 5001",
  "doi": "10.1109/TAC.2022.3177715",
  "categories": [
    "math.DS"
  ],
  "abstract": "The flow of contracting systems contracts 1-dimensional parallelotopes, i.e., line segments, at an exponential rate. One reason for the usefulness of contracting systems is that many interconnections of contracting sub-systems yield an overall contracting system. A generalization of contracting systems is $k$-contracting systems, where $k\\in\\{1,\\dots,n\\}$. The flow of such systems contracts the volume of $k$-dimensional parallelotopes at an exponential rate, and in particular they reduce to contracting systems when $k=1$. It was shown by Muldowney and Li that time-invariant $2$-contracting systems have a well-ordered asymptotic behaviour: all bounded trajectories converge to the set of equilibria. Here, we derive a sufficient condition guaranteeing that the system obtained from the series interconnection of two sub-systems is $k$-contracting. This is based on a new formula for the $k$th multiplicative and additive compounds of a block-diagonal matrix, which may be of independent interest. As an application, we find conditions guaranteeing that $2$-contracting systems with an exponentially decaying input retain the well-ordered behaviour of time-invariant 2-contracting systems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-17T08:55:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sufficient condition",
      "series connection",
      "exponential rate",
      "contraction",
      "line segments"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "IEEE"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}