{
  "id": "2106.10706",
  "version": "v1",
  "published": "2021-06-20T14:49:51.000Z",
  "updated": "2021-06-20T14:49:51.000Z",
  "title": "Feedback Nash Equilibria in Differential Games with Impulse Control",
  "authors": [
    "Utsav Sadana",
    "Puduru Viswanadha Reddy",
    "Georges Zaccour"
  ],
  "categories": [
    "math.OC",
    "cs.SY",
    "eess.SY"
  ],
  "abstract": "We study a class of deterministic finite-horizon two-player nonzero-sum differential games where players are endowed with different kinds of controls. We assume that Player 1 uses piecewise-continuous controls, while Player 2 uses impulse controls. For this class of games, we seek to derive conditions for the existence of feedback Nash equilibrium strategies for the players. More specifically, we provide a verification theorem for identifying such equilibrium strategies, using the Hamilton-Jacobi-Bellman (HJB) equations for Player 1 and the quasi-variational inequalities (QVIs) for Player 2. Further, we show that the equilibrium number of interventions by Player 2 is upper bounded. Furthermore, we specialize the obtained results to a scalar two-player linear-quadratic differential game. In this game, Player 1's objective is to drive the state variable towards a specific target value, and Player 2 has a similar objective with a different target value. We provide, for the first time, an analytical characterization of the feedback Nash equilibrium in a linear-quadratic differential game with impulse control. We illustrate our results using numerical experiments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-20T14:49:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "feedback nash equilibrium",
      "impulse control",
      "two-player linear-quadratic differential game",
      "two-player nonzero-sum differential games",
      "deterministic finite-horizon two-player nonzero-sum differential"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}