Let $n,m \in \mathbb{N}$, and let $t_0,t_1 \in \mathbb{R}$ with $t_0 < t_1$. Let $U_c \subset \mathbb{R}^m$ be nonempty and compact, and let $K \subset \mathbb{R}^n$ be nonempty and compact. Let $f: [t_0,t_1] \times \mathbb{R}^n \times U_c \to \mathbb{R}^n$ and $L: [t_0,t_1] \times \mathbb{R}^n \times U_c \to \mathbb{R}$ be continuous maps. Assume that there exist constants $a,b \geq 0$ such that

\begin{align*} |f(t,x,u)| \leq a + b|x| \end{align*}

for every $(t,x,u) \in [t_0,t_1] \times \mathbb{R}^n \times U_c$.

Let $\Phi: K \to \mathbb{R} \cup \{\infty\}$ be lower semicontinuous, and let $E \subset \mathbb{R}^n \times \mathbb{R}^n$ be closed. A pair $(x,u)$ is called admissible if $u: [t_0,t_1] \to U_c$ is Lebesgue measurable, $x: [t_0,t_1] \to K$ is absolutely continuous, $\dot{x}(t) = f(t,x(t),u(t))$ for $\mathcal{L}^1$-a.e. $t \in [t_0,t_1]$, and $(x(t_0),x(t_1)) \in E$.

For each $(t,x) \in [t_0,t_1] \times K$, define the velocity-cost epigraph

\begin{align*} Q(t,x) := \{(v,r) \in \mathbb{R}^n \times \mathbb{R} : \text{there exists } u \in U_c \text{ such that } v = f(t,x,u) \text{ and } r \geq L(t,x,u)\}. \end{align*}

Assume that $Q(t,x)$ is convex for every $(t,x) \in [t_0,t_1] \times K$.

Define the Bolza functional on admissible pairs by

\begin{align*} J[x,u] := \Phi(x(t_1)) + \int_{[t_0,t_1]} L(t,x(t),u(t))\,d\mathcal{L}^1(t). \end{align*}

If the admissible set is nonempty and $J$ is bounded below on admissible pairs, then there exists an admissible pair $(x_*,u_*)$ such that

\begin{align*} J[x_*,u_*] = \inf\{J[x,u] : (x,u) \text{ is admissible}\}. \end{align*}