---
tag: Q5N7E7P6F6
kind: definition
name: Hamilton–Jacobi–Isaacs equation
msc: [49L12, 49N70, 91A23]
defines: WHTBZ3K4FB
uses: [THSHGNJNPS, 5TRXQKP7DY, SJP9HW82VX, H1ERJ3T75T]
sources:
  - url: https://en.wikipedia.org/wiki/Differential_game
---

## Definition

Let $A \subseteq \R^{k}$ and $B \subseteq \R^{l}$ be nonempty compact sets and let $T > 0$.
Let $f \colon \R^n \times A \times B \to \R^n$, $h \colon \R^n \times A \times B \to \R$ and $g \colon \R^n \to \R$ be continuous.
Consider the two-player zero-sum [[differential game|THSHGNJNPS]] with dynamics $x' = f(x, a, b)$ and payoff $\int_t^T h(x(s), a(s), b(s)) \, ds + g(x(T))$.
The player with control $a(s) \in A$ minimises the payoff, and the player with control $b(s) \in B$ maximises it.
The *upper* and *lower Hamiltonians* are, for $x, p \in \R^n$,
$$H^+(x, p) = \min_{a \in A} \max_{b \in B} \left\{ f(x, a, b) \cdot p + h(x, a, b) \right\}, \qquad H^-(x, p) = \max_{b \in B} \min_{a \in A} \left\{ f(x, a, b) \cdot p + h(x, a, b) \right\}.$$
The *Hamilton–Jacobi–Isaacs equations* are the terminal value problems
$$V_t + H^\pm(x, \nabla_x V) = 0 \quad \text{in } (0, T) \times \R^n, \qquad V(T, x) = g(x),$$
for an unknown function $V \colon [0, T] \times \R^n \to \R$, one with $H^+$ and one with $H^-$.
The game satisfies *Isaacs' condition* if $H^+ = H^-$, and then the two equations coincide.

## Remarks

Always $H^- \le H^+$.
If $B$ has one point, both equal $\min_{a} \{ f \cdot p + h \}$ and the equation is the [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]], a special [[Hamilton–Jacobi equation|5TRXQKP7DY]].
Assume also that $f$, $h$ and $g$ are bounded and Lipschitz in $x$.
Then the upper and lower values of the game, defined with nonanticipating strategies, are the [[viscosity solutions|H1ERJ3T75T]] of these equations.
Under Isaacs' condition the two values are equal, so the game has a value.

Example: if $f(x, a, b) = f_1(x, a) + f_2(x, b)$ and $h(x, a, b) = h_1(x, a) + h_2(x, b)$, then the expression splits into a part that depends only on $a$ and a part that depends only on $b$.
So $\min_a \max_b$ and $\max_b \min_a$ are both the sum of $\min_a$ of the first part and $\max_b$ of the second, and Isaacs' condition holds.

Non-example: $n = 1$, $A = B = [-1, 1]$, $f(x, a, b) = (a - b)^2$ and $h = 0$.
For $p = 1$, $\max_b (a - b)^2 = (1 + \abs{a})^2$, so $H^+(x, 1) = 1$, while $\min_a (a - b)^2 = 0$, so $H^-(x, 1) = 0$.
Isaacs' condition fails, and the two equations are different.

## Explanations

### L1 human

The equation says how the value of the game changes in time when both players play their best against each other at each instant.