%% formulation
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\begin{align*}
\min_{u(t)} \quad & \int_{0}^{1}\left[ x_1^2 + x_2^2 + 0.005u^2 \right] \mathrm{d}t \\
\text{subject to:} \quad & \dot{\bm{x}} = \begin{bmatrix}0 & 1 \\ 0 & -1 \end{bmatrix} \bm{x} +\begin{bmatrix} 0 \\ 1 \end{bmatrix} u \\
& \bm{x}(0) = (0,-1) \\
& \begin{cases}
 & \text{Example 0} \\
x_2 - 8(t-0.5)^2 + 0.5 \leq 0 & \text{Example 1} \\
x_1 - 8(t-0.5)^2 + 0.5 \leq 0 & \text{Example 2} \\
x_1(0.5) = 0.5 & \text{Example 3}
\end{cases}
\end{align*}
}}
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%% solution