%% formulation
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\begin{align*}
\min_{\bm{u}(t)} \quad & \int_0^{1} \left[ \bm{x}^T \bm{I}_n \bm{x} + \bm{u}^T \bm{I}_n \bm{u} \right] \mathrm{d}t + 10 [x_1(1)]^2 \\
\text{subject to:} \quad & \dot{\bm{x}} = \begin{bmatrix}
0 & 1 & 0 & \cdots & 0 \\
0 & 0 & 1 & \ddots & \vdots \\
\vdots & \ddots & \ddots & \ddots & 0 \\
0 & \cdots & 0 & 0 & 1 \\
1 & -2 & 3 & \cdots & (-1)^{n+1}n
\end{bmatrix} \bm{x} + \bm{I}_n \bm{u} \\
& \bm{x}(0) = \begin{bmatrix} 1 & 2 & \cdots & n \end{bmatrix}
\end{align*}
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\end{document}

%% solution