with $y(t) = (x(t),z(t))$, where $x$ and $u$ satisfy exactly the same admissibility, endpoint, path, and control constraints as in $\mathcal A$, and where
for $\mathcal{L}^1$-a.e. $t \in [t_0,t_1]$, together with $z(t_0)=0$. The augmented Mayer terminal cost is the map $\Psi: \mathbb{R}^{n+1} \to \mathbb{R}$ defined by
and let the augmented Mayer optimal value be the extended real number
paragraph
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\begin{align*}
\inf \{\Psi(x(t_1),z(t_1)) : (x,z,u) \text{ is admissible for the augmented Mayer problem}\},
\end{align*}
latex_env
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with the convention that the infimum of the empty set is $+\infty$. Then the Bolza problem and the augmented Mayer problem have the same optimal value. Moreover, a pair $(x,u) \in \mathcal A$ minimizes $J$ over $\mathcal A$ if and only if the triple $(x,z,u)$ minimizes the augmented Mayer problem, where