{
  "id": "2304.01977",
  "version": "v1",
  "published": "2023-04-04T17:33:01.000Z",
  "updated": "2023-04-04T17:33:01.000Z",
  "title": "Frequency domain approach for the stability analysis of a fast hyperbolic PDE coupled with a slow ODE",
  "authors": [
    "Gonzalo Arias",
    "Swann Marx",
    "Guilherme Mazanti"
  ],
  "comment": "6 pages, 3 figures, double column format",
  "categories": [
    "math.AP",
    "math.OC"
  ],
  "abstract": "This paper deals with the exponential stability of systems made of a hyperbolic PDE coupled with an ODE with different time scales, the dynamics of the PDE being much faster than that of the ODE. Such a difference of time scales is modeled though a small parameter $\\varepsilon$ multiplying the time derivative in the PDE, and our stability analysis relies on the singular perturbation method. More precisely, we define two subsystems: a reduced order system, representing the dynamics of the full system in the limit $\\varepsilon = 0$, and a boundary-layer system, which represents the dynamics of the PDE in the fast time scale. Our main result shows that, if both the reduced order and the boundary-layer systems are exponentially stable, then the full system is also exponentially stable for $\\varepsilon$ small enough, and our strategy is based on a spectral analysis of the systems under consideration. Our main result improves a previous result in the literature, which was proved using a Lyapunov approach and required a stronger assumption on the boundary-layer system to obtain the same conclusion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-04T17:33:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "frequency domain approach",
      "fast hyperbolic pde",
      "stability analysis",
      "slow ode",
      "time scale"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}