Let $T>0$, let $U$ be a nonempty control-value set, let $f:[0,T]\times\mathbb{R}^n\times U\to \mathbb{R}^n$ be a controlled vector field, let $\ell:[0,T]\times\mathbb{R}^n\times U\to \mathbb{R}$ be a running cost, and let $g:\mathbb{R}^n\to \mathbb{R}$ be a terminal cost.
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For each $0\leq a\leq b\leq T$ and $z\in\mathbb{R}^n$, let $\mathcal{A}_{a,z}[a,b]$ be a nonempty set of admissible controls on $[a,b]$ starting from $z$, where each $u\in\mathcal{A}_{a,z}[a,b]$ is a map $u:[a,b]\to U$. Assume that for every $u\in\mathcal{A}_{a,z}[a,b]$ the initial-value problem
Assume $V(t,x)\in\mathbb{R}$ for all $0\leq t\leq T$ and $x\in\mathbb{R}^n$.
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Assume the admissible control classes are stable under restriction and concatenation in the following sense.
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First, if $u\in\mathcal{A}_{t,x}[t,T]$ and $0\leq t\leq \tau\leq T$, then $u|_{[t,\tau]}\in\mathcal{A}_{t,x}[t,\tau]$ and $u|_{[\tau,T]}\in\mathcal{A}_{\tau,y}[\tau,T]$, where
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\begin{align*}
y=x^{t,x;u}(\tau).
\end{align*}
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The restricted trajectories agree with the full trajectory on their intervals:
Finally, assume that for every $0\leq t\leq \tau\leq T$, every $x\in\mathbb{R}^n$, every $u_1\in\mathcal{A}_{t,x}[t,\tau]$, every endpoint $y=x^{t,x;u_1}(\tau)$, and every $\varepsilon>0$, there exists $u_\varepsilon\in\mathcal{A}_{\tau,y}[\tau,T]$ such that