{
  "id": "1912.01906",
  "version": "v1",
  "published": "2019-12-04T11:37:43.000Z",
  "updated": "2019-12-04T11:37:43.000Z",
  "title": "Stability and phase transitions of dynamical flow networks with finite capacities",
  "authors": [
    "Leonardo Massai",
    "Giacomo Como",
    "Fabio Fagnani"
  ],
  "comment": "7 pages, 4 figures, submitted at IFAC 2020",
  "categories": [
    "math.DS"
  ],
  "abstract": "We study deterministic continuous-time lossy dynamical flow networks with constant exogenous demands, fixed routing, and finite flow and buffer capacities. In the considered model, when the total net flow in a cell ---consisting of the difference between the total flow directed towards it minus the outflow from it--- exceeds a certain capacity constraint, then the exceeding part of it leaks out of the system. The ensuing network flow dynamics is a linear saturated system with compact state space that we analyze using tools from monotone systems and contraction theory. Specifically, we prove that there exists a set of equilibria that is globally asymptotically stable. Such equilibrium set reduces to a single globally asymptotically stable equilibrium for generic exogenous demand vectors. Moreover, we show that the critical exogenous demand vectors giving rise to non-unique equilibria correspond to phase transitions in the asymptotic behavior of the dynamical flow network.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-04T11:37:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "phase transitions",
      "finite capacities",
      "demand vectors giving rise",
      "lossy dynamical flow networks",
      "continuous-time lossy dynamical flow"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}