{
  "id": "2009.11544",
  "version": "v1",
  "published": "2020-09-24T08:24:01.000Z",
  "updated": "2020-09-24T08:24:01.000Z",
  "title": "Koopman Resolvent: A Laplace-Domain Analysis of Nonlinear Autonomous Dynamical Systems",
  "authors": [
    "Yoshihiko Susuki",
    "Alexandre Mauroy",
    "Igor Mezic"
  ],
  "comment": "20 pages, 2 figures",
  "categories": [
    "math.DS",
    "cs.SY",
    "eess.SY",
    "math.OC"
  ],
  "abstract": "The motivation of our research is to establish a Laplace-domain theory that provides principles and methodology to analyze and synthesize systems with nonlinear dynamics. A semigroup of composition operators defined for nonlinear autonomous dynamical systems---the Koopman semigroup and its associated Koopman generator---plays a central role in this study. We introduce the resolvent of the Koopman generator, which we call the Koopman resolvent, and provide its spectral characterization for three types of nonlinear dynamics: ergodic evolution on an attractor, convergence to a stable equilibrium point, and convergence to a (quasi-)stable limit cycle. This shows that the Koopman resolvent provides the Laplace-domain representation of such nonlinear autonomous dynamics. A computational aspect of the Laplace-domain representation is also discussed with emphasis on non-stationary Koopman modes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-24T08:24:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlinear autonomous dynamical systems",
      "koopman resolvent",
      "laplace-domain analysis",
      "dynamical systems-the koopman semigroup",
      "autonomous dynamical systems-the koopman"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}