{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $f \\colon [0, \\infty) \\to \\R$ be differentiable.\nAssume that the limit $\\lim_{t \\to \\infty} f(t)$ exists and is finite.\nAssume that $f'$ is [[uniformly continuous|0M4JJT780X]] on $[0, \\infty)$.\nThen $\\lim_{t \\to \\infty} f'(t) = 0$.\n\n## Remarks\n\nSuppose $f'(t) \\not\\to 0$.\nThen there are $\\varepsilon \u003e 0$ and times $t_n \\to \\infty$ with $\\abs{f'(t_n)} \\ge \\varepsilon$.\nUniform continuity gives $\\delta \u003e 0$, independent of $n$, with $\\abs{f'(t)} \\ge \\varepsilon / 2$ and constant sign on $[t_n, t_n + \\delta]$.\nThen $\\abs{f(t_n + \\delta) - f(t_n)} \\ge \\varepsilon \\delta / 2$ for all $n$, which contradicts the Cauchy criterion for the limit of $f$.\n\nThe hypothesis on $f'$ is necessary.\nFor $f(t) = \\sin(t^2) / (t + 1)$, the limit of $f$ is $0$, but $f'(t) = 2t \\cos(t^2) / (t + 1) - \\sin(t^2) / (t + 1)^2$ does not tend to $0$.\nHere $f'$ is not uniformly continuous.\n\nAn equivalent form: if $g \\colon [0, \\infty) \\to \\R$ is uniformly continuous and $\\int_0^\\infty g(t) \\, dt$ converges, then $g(t) \\to 0$.\nIn stability theory, it is applied to $f(t) = V(t, x(t))$ for a [[Lyapunov function|01REYCJ371]] $V$ along a solution $x$.\nIf $V$ is bounded below, $\\dot V \\le 0$ and $\\dot V$ is uniformly continuous in $t$, then $\\dot V(t, x(t)) \\to 0$.\n\n## Explanations\n\n### L1 human\n\nIf a quantity settles down to a limit and its rate of change cannot jump suddenly, then the rate of change must die out.\nWithout the second condition, the quantity could settle while still making ever faster tiny wiggles.\n\n### L2 human\n\nFor time-varying systems, LaSalle's invariance principle does not apply directly.\nBarbalat's lemma replaces it: from a Lyapunov function with $\\dot V \\le 0$ one gets $\\dot V \\to 0$ along solutions, which often forces some components of the state to tend to $0$.\nThis is a standard step in adaptive control.\nA common sufficient condition for uniform continuity of $\\dot V$ is that $\\ddot V$ is bounded.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "about": [
      "0F2SYCKDT2"
    ],
    "kind": "statement",
    "msc": [
      "34D05",
      "34D20"
    ],
    "name": "Barbalat's lemma",
    "role": "lemma",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lyapunov_stability"
      }
    ],
    "tag": "4KPNX1SZYR",
    "uses": [
      "0M4JJT780X",
      "01REYCJ371"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/4KPNX1SZYR/",
  "links": [
    {
      "field": "about",
      "target": "0F2SYCKDT2"
    },
    {
      "field": "uses",
      "target": "0M4JJT780X"
    },
    {
      "field": "uses",
      "target": "01REYCJ371"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/4KPNX1SZYR.md",
  "msc": [
    "34D05",
    "34D20"
  ],
  "name": "Barbalat's lemma",
  "prov": null,
  "role": "lemma",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/4KP/4KPNX1SZYR.md",
  "status": null,
  "tag": "4KPNX1SZYR",
  "uri": "m2://4KPNX1SZYR",
  "url": "https://math2.tamnd.com/wiki/barbalats-lemma/",
  "writing": {
    "tag": "4KPNX1SZYR",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}