Mayer and Lagrange problems

Problem

Let $U \subseteq \R^m$ be a nonempty set, and let $t_0 < t_1$ be fixed times. Let $f \colon [t_0, t_1] \times \R^n \times U \to \R^n$ be continuous, and let $S \subseteq \R^n \times \R^n$ be a closed set of allowed endpoint pairs. An admissible control is a measurable map $u \colon [t_0, t_1] \to U$ whose trajectory $x$ of the control system $x'(t) = f(t, x(t), u(t))$ is absolutely continuous on $[t_0, t_1]$ and satisfies $(x(t_0), x(t_1)) \in S$.

1. The Mayer problem: given a continuous function $\varphi \colon \R^n \times \R^n \to \R$, minimize $J(u) = \varphi(x(t_0), x(t_1))$ over all admissible controls $u$.
2. The Lagrange problem: given a continuous function $L \colon [t_0, t_1] \times \R^n \times U \to \R$, minimize the functional $J(u) = \int_{t_0}^{t_1} L(t, x(t), u(t)) \, dt$ over all admissible controls $u$.