{
  "id": "2001.07057",
  "version": "v1",
  "published": "2020-01-20T11:14:57.000Z",
  "updated": "2020-01-20T11:14:57.000Z",
  "title": "Approximation of linear controlled dynamical systems with small random noise and fast periodic sampling",
  "authors": [
    "Shivam Dhama",
    "Chetan D. Pahlajani"
  ],
  "categories": [
    "math.PR",
    "math.DS"
  ],
  "abstract": "In this paper, we study the dynamics of a linear control system with given state feedback control law in the presence of fast periodic sampling at temporal frequency $1/\\delta$ ($0 < \\delta \\ll 1$), together with small white noise perturbations of size $\\varepsilon$ ($0<\\varepsilon \\ll 1$) in the state dynamics. For the ensuing continuous-time stochastic process indexed by two small parameters $\\varepsilon,\\delta$, we obtain effective ordinary and stochastic differential equations describing the mean behavior and the typical fluctuations about the mean in the limit as $\\varepsilon,\\delta \\searrow 0$. The effective fluctuation process is found to vary, depending on whether $\\delta \\searrow 0$ faster than/at the same rate as/slower than $\\varepsilon \\searrow 0$. The most interesting case is found to be the one where $\\delta,\\varepsilon$ are comparable in size; here, the limiting stochastic differential equation for the fluctuations has both a diffusive term due to the small noise and an effective drift term which captures the cumulative effect of the fast sampling. In this regime, our results yield a Markov process which provides a strong (pathwise) approximation of the original non-Markovian process, together with estimates on the ensuing error.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-20T11:14:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear controlled dynamical systems",
      "fast periodic sampling",
      "small random noise",
      "stochastic differential equation",
      "continuous-time stochastic process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}