{
  "aliases": [
    "invariance-principle",
    "krasovskii-lasalle-principle"
  ],
  "backlinks": null,
  "body": "\n## Statement\n\nLet $f : \\mathbb{R}^n \\to \\mathbb{R}^n$ be locally Lipschitz with $f(0) = 0$, and let $V : \\mathbb{R}^n \\to \\mathbb{R}$ be continuously differentiable.\nAssume that $\\dot{V}(x) = \\nabla V(x) \\cdot f(x) \\le 0$ for all $x \\in \\mathbb{R}^n$, that $V(0) = 0$ and $V(x) \u003e 0$ for $x \\neq 0$, and that $V(x) \\to \\infty$ as $\\|x\\| \\to \\infty$.\nLet $E = \\{x \\in \\mathbb{R}^n : \\dot{V}(x) = 0\\}$.\nAssume that the only solution $x(t)$ of $x' = f(x)$ with $x(t) \\in E$ for all $t \\ge 0$ is the solution $x(t) = 0$.\nThen the origin is globally asymptotically stable for $x' = f(x)$.\n\n## Remarks\n\nThe general principle: if $\\dot{V} \\le 0$, then each solution that is bounded for $t \\ge 0$ approaches the largest invariant subset of $E$.\nThe principle allows $\\dot{V} = 0$ outside the origin, where [[Lyapunov's stability theorem|0J6A752DTZ]] needs a strict Lyapunov function.\nFor the damped oscillator $x' = y$, $y' = -x - c y$ with $c \u003e 0$, take $V = \\frac{1}{2}(x^2 + y^2)$. Then $E = \\{y = 0\\}$, and a solution in $E$ has $y' = -x = 0$. So the origin is globally asymptotically stable.\nA local version holds when the conditions on $V$ hold only near the origin.\n\n## History\n\nThe principle is named after Joseph LaSalle. It is also named after Nikolay Krasovskii and Evgenii Barbashin.\n\n## Explanations\n\n### L1 human\n\nSometimes the energy only stops decreasing on a set of states, not only at the equilibrium.\nLaSalle's principle says that the motion must still end at the equilibrium, if no motion can stay forever in that set.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "aliases": [
      "invariance-principle",
      "krasovskii-lasalle-principle"
    ],
    "kind": "statement",
    "msc": [
      "34D20",
      "34D23"
    ],
    "name": "LaSalle's invariance principle",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q3922068",
      "wikipedia": "LaSalle's invariance principle"
    },
    "sources": [
      {
        "locator": "Global version",
        "url": "https://en.wikipedia.org/wiki/LaSalle%27s_invariance_principle"
      }
    ],
    "tag": "0659J5X7B5",
    "uses": [
      "0J6A752DTZ",
      "04P7B5XC22",
      "05Y1JAM3Y5"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0659J5X7B5/",
  "links": [
    {
      "field": "uses",
      "target": "0J6A752DTZ"
    },
    {
      "field": "uses",
      "target": "04P7B5XC22"
    },
    {
      "field": "uses",
      "target": "05Y1JAM3Y5"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0659J5X7B5.md",
  "msc": [
    "34D20",
    "34D23"
  ],
  "name": "LaSalle's invariance principle",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/065/0659J5X7B5.md",
  "status": null,
  "tag": "0659J5X7B5",
  "uri": "m2://0659J5X7B5",
  "url": "https://math2.tamnd.com/wiki/lasalles-invariance-principle/",
  "writing": {
    "tag": "0659J5X7B5",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}