where $x_0 \in \mathbb{R}^n$, $M \subset \mathbb{R}^n$ is a fixed smooth terminal manifold independent of $t_1$, and the terminal cost has no explicit dependence on $t_1$. Define the normal Hamiltonian
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\begin{align*}
H: \mathbb{R}^n \times \mathbb{R}^n \times U &\to \mathbb{R}
\end{align*}
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by
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\begin{align*}
H(x,p,u) = p \cdot f(x,u) - L(x,u).
\end{align*}
such that on each smooth control arc $I_k=(s_{k-1},s_k)$ the restriction $u^*|_{I_k}$ has a $W^{1,1}(I_k;U)$ representative. Assume that $(x^*,p,u^*)$ is a normal Pontryagin extremal satisfying, for each $k \in \{1,\dots,N\}$ and for $\mathcal{L}^1$-a.e. $t \in I_k$, the state equation
has an absolutely continuous representative on each arc $I_k$, that the one-sided limits of these representatives agree across every switching time $s_k$ with $1 \le k \le N-1$, and that the free-final-time transversality condition is