%% formulation
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\begin{align*}
\min_{u_1(t),u_2(t)} \quad & \int_{0}^{20} \left[10^{2}(x_1^2 + x_2^2 + x_3^2 + x_4^2) + 10^{-2} (u_1^2 +u_2^2)\right] \mathrm{d}t \\
\text{subject to:} \quad & \dot{\bm{x}} = \begin{bmatrix}
-10 x_1 + u_1 + u_2 \\
-2 x_2 + u_1 + 2 u_2 \\
-3 x_3 + 5 x_4 + u_1 - u_2 \\
5 x_3 - 3 x_4 + u_1 + 3 u_2
\end{bmatrix} \\
& \bm{x}(0) = (2,1,2,1) \\
& \bm{x}(20) = (2,3,1,-2) \\
& x_1^2 + x_2^2 + x_3^2 + x_4^2 \geq 3 p(t,3,12) + 3 p(t,6,10) + 3 p(t,10,6) + 8 p(t,15,4) + 0.01 \\
\text{where:} \quad & p(t,a,b) = e^{-b(t-a)^2}
\end{align*}
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%% solution