{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $x' = f(x, u)$ be a [[control system|SBQVGQPXT1]] on $\\R^n$ with control set $U \\subseteq \\R^m$.\nFix a time horizon $T \u003e 0$, an initial state $x_0 \\in \\R^n$ and a closed target set $S \\subseteq \\R^n$.\nAn *admissible control* for $(x_0, T, S)$ is a function $u \\colon [0, T] \\to \\R^m$ such that:\n\n1. $u$ is [[measurable|YYNCJSR0PJ]] and essentially bounded, and $u(t) \\in U$ for almost every $t \\in [0, T]$;\n2. there is an [[absolutely continuous|ZT6F4C2YHK]] trajectory $x \\colon [0, T] \\to \\R^n$ with $x(0) = x_0$ and $x'(t) = f(x(t), u(t))$ for almost every $t$;\n3. this trajectory satisfies $x(T) \\in S$.\n\n## Remarks\n\nBy the local Lipschitz condition on $f$, the trajectory in clause 2 is unique when it exists.\nClause 2 can fail because the solution blows up before time $T$.\nMany authors call any control satisfying clause 1 admissible, as in the [[control system|SBQVGQPXT1]] definition, and add clauses 2 and 3 as constraints of the problem.\nProblems with free final time, state constraints or other control classes, such as piecewise continuous controls, change the clauses accordingly.\n\nExample: $x' = u$ on $\\R$ with $U = [-1, 1]$, $x_0 = 0$, $T = 1$ and $S = \\set{\\tfrac12}$.\nThe constant control $u = \\tfrac12$ is admissible: it is bounded with values in $U$, and $x(t) = t/2$ gives $x(1) = \\tfrac12$.\n\nNon-example: the constant control $u = 1$ in the same setting satisfies clauses 1 and 2, with $x(t) = t$.\nIt fails clause 3, since $x(1) = 1 \\notin S$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "8M31QKB0GW",
    "kind": "definition",
    "msc": [
      "49J15",
      "93C15"
    ],
    "name": "Admissible control",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Optimal_control"
      }
    ],
    "tag": "8GETTS41P6",
    "uses": [
      "SBQVGQPXT1",
      "YYNCJSR0PJ",
      "ZT6F4C2YHK"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/8GETTS41P6/",
  "links": [
    {
      "field": "defines",
      "target": "8M31QKB0GW"
    },
    {
      "field": "uses",
      "target": "SBQVGQPXT1"
    },
    {
      "field": "uses",
      "target": "YYNCJSR0PJ"
    },
    {
      "field": "uses",
      "target": "ZT6F4C2YHK"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/8GETTS41P6.md",
  "msc": [
    "49J15",
    "93C15"
  ],
  "name": "Admissible control",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/8GE/8GETTS41P6.md",
  "status": null,
  "tag": "8GETTS41P6",
  "uri": "m2://8GETTS41P6",
  "url": "https://tamnd.github.io/math2/wiki/admissible-control-definition/",
  "writing": {
    "tag": "8GETTS41P6",
    "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
      }
    ]
  }
}