%% formulation
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\begin{align*}
\min_{\bm{u}(t)} \quad & \int_0^{t_f} \left[ \bm{x}^T \bm{I} \bm{x} + \bm{u}^T \bm{I} \bm{u} \right] \mathrm{d}t \\
\text{subject to:} \quad & \dot{\bm{x}} =  \mathrm{tridiag}_n\left(\begin{bmatrix} 0.538998 & -1.173113 & 0.634115 \end{bmatrix}\right) \bm{x} + \cdots \\
& + \begin{bmatrix} 0.538998 & 0 & \cdots & 0 & 0 \\ 0 & 0 & \cdots & 0 & 0.634115  \end{bmatrix}^T \bm{u} \\
& \bm{x}(0) = -0.0307 - \frac{i - 1}{n - 1}(0.1273 - 0.0307)\qquad i = 1, \dots, n \\
& \bm{0} \leq \bm{u}
\end{align*}
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%% solution