Hamiltonian of an optimal control problem

Definition

Let $U \subseteq \R^m$ be a nonempty set and $I \subseteq \R$ an interval. Consider an optimal control problem with control system $x' = f(t, x, u)$ and running cost $\int L(t, x, u) \, dt$, where $f \colon I \times \R^n \times U \to \R^n$ and $L \colon I \times \R^n \times U \to \R$. The Hamiltonian of the problem is the function $H \colon I \times \R^n \times U \times \R^n \times \R \to \R$ given by $$H(t, x, u, p, \lambda_0) = p \cdot f(t, x, u) - \lambda_0 L(t, x, u).$$ The vector $p \in \R^n$ is the costate, and the number $\lambda_0 \ge 0$ is the cost multiplier. In the normal case $\lambda_0 = 1$ one writes $H(t, x, u, p) = p \cdot f(t, x, u) - L(t, x, u)$.