{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $I \\subseteq \\R$ be an interval and let $\\Omega \\subseteq \\R^n$ be an open set.\nLet $F$ be a [[multivalued function|03CPD1HGF3]] that assigns to each $(t, x) \\in I \\times \\Omega$ a nonempty subset $F(t, x)$ of $\\R^n$.\nThe *differential inclusion* given by $F$ is the relation\n$$x'(t) \\in F(t, x(t)).$$\nA *solution* on a subinterval $J \\subseteq I$ is an [[absolutely continuous|ZT6F4C2YHK]] function $x \\colon J \\to \\Omega$ such that $x'(t) \\in F(t, x(t))$ for almost every $t \\in J$.\n\n## Remarks\n\nIf $F(t, x) = \\set{f(t, x)}$ for a function $f$, the solutions are the solutions of the [[ordinary differential equation|0NT64JV96A]] $x' = f(t, x)$ in the sense of Carathéodory.\nExample: for $F(t, x) = [-1, 1]$ on $\\R \\times \\R$, the function $x(t) = \\abs{t}$ is a solution, since it is absolutely continuous and $x'(t) = \\pm 1$ for every $t \\ne 0$.\nNon-example: $x(t) = 2t$ is not a solution of the same inclusion, because $x'(t) = 2 \\notin [-1, 1]$ at every $t$.\nAn equation with a discontinuous right side, such as $x' = -\\sgn x$, is often replaced by the inclusion $x' \\in -\\operatorname{Sgn}(x)$, where $\\operatorname{Sgn}(0) = [-1, 1]$; then $x \\equiv 0$ is a solution.\nLocal existence of solutions holds when the values $F(t, x)$ are nonempty, compact and convex and $F$ satisfies suitable [[upper hemicontinuity|XNMKX7H4FD]] and measurability conditions.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "9ZSMF6NYKM",
    "kind": "definition",
    "msc": [
      "34A60",
      "49J21"
    ],
    "name": "Differential inclusion",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Differential_inclusion"
      }
    ],
    "tag": "1BKVNEHX96",
    "uses": [
      "03CPD1HGF3",
      "ZT6F4C2YHK",
      "0NT64JV96A",
      "XNMKX7H4FD"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/1BKVNEHX96/",
  "links": [
    {
      "field": "defines",
      "target": "9ZSMF6NYKM"
    },
    {
      "field": "uses",
      "target": "03CPD1HGF3"
    },
    {
      "field": "uses",
      "target": "ZT6F4C2YHK"
    },
    {
      "field": "uses",
      "target": "0NT64JV96A"
    },
    {
      "field": "uses",
      "target": "XNMKX7H4FD"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/1BKVNEHX96.md",
  "msc": [
    "34A60",
    "49J21"
  ],
  "name": "Differential inclusion",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/1BK/1BKVNEHX96.md",
  "status": null,
  "tag": "1BKVNEHX96",
  "uri": "m2://1BKVNEHX96",
  "url": "https://math2.tamnd.com/wiki/differential-inclusion-definition/",
  "writing": {
    "tag": "1BKVNEHX96",
    "kind": "definition",
    "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
      }
    ]
  }
}