%% formulation
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\begin{align*}
\min_{u(t)} \quad & \int_{34/15}^{4}\left[ x_1^2 + 10^{-3} u^2 \right] \mathrm{d}t \\
\text{subject to:} \quad & \dot{\bm{x}} = \begin{bmatrix}x_2 \\ u \end{bmatrix} \\
& x_1(34/15) = 302399/50625 \\
& x_2(34/15) = 70304/3375 \\
& -x_1(t) + 15 - (t-4)^4 \leq 0
\end{align*}
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\end{document}

%% solution