Hamilton–Jacobi–Isaacs equation

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 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.