%% formulation
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\begin{align*}
\min_{\bm{u}(t)} \quad & \int_0^1 \left[ u_1^2/10 + u_2^2/10 + u_1x_1 + u_1x_2 + 5\left(x_2-g(t)\right)^2 \right] \mathrm{d}t + \max_{0\leq t\leq 1} x_3(t) \\
\text{subject to:} \quad & \dot{\bm{x}} = \begin{bmatrix} -1 & 2 & 0 \\ 3 & -4 & 0 \\ 1 & 2 & -1 \end{bmatrix} \bm{x} +
\begin{bmatrix} 1 & 0 \\ -1 & 0 \\ 0 & 1/20 \end{bmatrix} \bm{u} \\
& x_1(0) = 2 \\
& x_3(0) = 1/2 \\
& x_2(0) - x_2(1) = 0 \\
& \int_0^1 x_1(t) \mathrm{d}t = 0 \\
& -x_1(t) + u_2(t)/12 \leq 0 \\
& x_2(t) \leq g(t) \\
& |u_2(t)| \leq 10
\end{align*}
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%% solution