{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "T5BEKMG5W8"
    }
  ],
  "body": "\n## Statement\n\nLet $U \\subseteq \\R^m$ be a nonempty [[compact set|0R6EWFKRCF]] and $T \u003e 0$.\nLet $f \\colon [0, T] \\times \\R^n \\times U \\to \\R^n$ and $L \\colon [0, T] \\times \\R^n \\times U \\to \\R$ be bounded and continuous.\nAssume that both are Lipschitz in $x$, uniformly in $(t, u)$, and let $g \\colon \\R^n \\to \\R$ be bounded and Lipschitz.\nLet $V$ be the [[value function|K6G0DRQ3CP]] of the problem of minimizing $\\int_t^T L(s, y, u) \\, ds + g(y(T))$ subject to $y' = f(s, y, u)$, $y(t) = x$.\nLet $H(t, x, p) = \\max_{u \\in U} \\big( -p \\cdot f(t, x, u) - L(t, x, u) \\big)$.\nThen $V$ is bounded and Lipschitz on $[0, T] \\times \\R^n$, $V(T, \\cdot) = g$, and $V$ is a [[viscosity solution|H1ERJ3T75T]] of the [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]]\n$$-V_t + H(t, x, \\nabla_x V) = 0 \\quad \\text{in } (0, T) \\times \\R^n.$$\nThat is, for every $C^1$ function $\\varphi$ on $(0, T) \\times \\R^n$ and every point $(t_0, x_0) \\in (0, T) \\times \\R^n$:\n\n1. if $V - \\varphi$ has a local maximum at $(t_0, x_0)$, then $-\\varphi_t(t_0, x_0) + H(t_0, x_0, \\nabla_x \\varphi(t_0, x_0)) \\le 0$;\n2. if $V - \\varphi$ has a local minimum at $(t_0, x_0)$, then $-\\varphi_t(t_0, x_0) + H(t_0, x_0, \\nabla_x \\varphi(t_0, x_0)) \\ge 0$.\n\n## Remarks\n\nThe proof uses the dynamic programming principle.\nIf $V - \\varphi$ has a local maximum at $(t_0, x_0)$, use a constant control $u$ on $[t_0, t_0 + h]$: the principle gives $\\varphi(t_0, x_0) - \\varphi(t_0 + h, y(t_0 + h)) \\le \\int_{t_0}^{t_0 + h} L \\, ds$.\nDividing by $h$ and letting $h \\to 0$ gives $-\\varphi_t - \\nabla_x \\varphi \\cdot f(t_0, x_0, u) - L(t_0, x_0, u) \\le 0$ for every $u \\in U$, which is condition 1.\nCondition 2 uses nearly optimal controls in the same way, with the compactness of $U$.\nExample: for $f = u$, $U = [-1, 1]$, $L = 0$ and $g(x) = \\abs{x}$, the value function $\\max(\\abs{x} - (T - t), 0)$ is not differentiable, yet it is a viscosity solution of $-V_t + \\abs{V_x} = 0$.\nHere $g$ is unbounded, but one checks the two conditions directly at the corners.\nBy the comparison principle of Crandall and Lions, $V$ is the only bounded uniformly continuous viscosity solution with $V(T, \\cdot) = g$.\nSo the equation characterizes the value function even when no classical solution exists.\n\n## Explanations\n\n### L1 human\n\nThe optimal cost of a control problem should satisfy the Hamilton–Jacobi–Bellman equation, but it often has corners where derivatives do not exist.\nThis theorem says that it still satisfies the equation in the weak sense of viscosity solutions.\nTogether with uniqueness, it shows that the equation determines the optimal cost completely.\n\n### L2 human\n\nThe theorem is the bridge between optimal control and the theory of viscosity solutions.\nSmooth test functions $\\varphi$ touching $V$ from above or below replace the missing derivatives of $V$.\nCombined with the comparison principle, it gives convergence proofs for numerical schemes, which only need to be monotone, stable and consistent.\nIt also explains why the verification theorem for smooth solutions is not enough in general.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "K6G0DRQ3CP",
      "SJP9HW82VX",
      "H1ERJ3T75T"
    ],
    "kind": "statement",
    "msc": [
      "49L25",
      "49L12",
      "35F21"
    ],
    "name": "Viscosity solution of the Hamilton–Jacobi–Bellman equation",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Viscosity_solution"
      }
    ],
    "tag": "KREYT31SS1",
    "uses": [
      "K6G0DRQ3CP",
      "SJP9HW82VX",
      "H1ERJ3T75T",
      "0R6EWFKRCF"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/KREYT31SS1/",
  "links": [
    {
      "field": "about",
      "target": "K6G0DRQ3CP"
    },
    {
      "field": "about",
      "target": "SJP9HW82VX"
    },
    {
      "field": "about",
      "target": "H1ERJ3T75T"
    },
    {
      "field": "uses",
      "target": "K6G0DRQ3CP"
    },
    {
      "field": "uses",
      "target": "SJP9HW82VX"
    },
    {
      "field": "uses",
      "target": "H1ERJ3T75T"
    },
    {
      "field": "uses",
      "target": "0R6EWFKRCF"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/KREYT31SS1.md",
  "msc": [
    "49L25",
    "49L12",
    "35F21"
  ],
  "name": "Viscosity solution of the Hamilton–Jacobi–Bellman equation",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/KRE/KREYT31SS1.md",
  "status": null,
  "tag": "KREYT31SS1",
  "uri": "m2://KREYT31SS1",
  "url": "https://tamnd.github.io/math2/wiki/viscosity-solution-of-the-hamilton-jacobi-bellman-equation/",
  "writing": {
    "tag": "KREYT31SS1",
    "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
      }
    ]
  }
}