%% formulation
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\begin{align*}
\min_{u_1(t),t_f} \quad & -x_2(t_f) x_4(t_f) \\
\text{subject to:} \quad & \dot{\bm{x}} = \begin{bmatrix}
h_1\left(x_1,x_3\right)x_1 - u_1\left(\frac{x_1}{500 x_4} \right) \\
h_2\left(x_3\right) x_1 - 0.01x_2 - u_1\left(\frac{x_2}{500 x_4}\right) \\
-h_1\left(x_1,x_3\right)\frac{x_1}{0.47} - h_2\left(x_3\right)\frac{x_1}{1.2} - x_1\left(\frac{0.029x_3}{0.0001+x_3}\right) + \frac{u_1}{x_4}\left(1 - \frac{x_3}{500}\right) \\
\frac{u_1}{500}
\end{bmatrix} \\
& \bm{x}(0) = (1.5,0,0,7) \\
& 0 \leq x_1(t) \leq 40 \\
& 0 \leq x_2(t) \leq 50 \\
& 0 \leq x_3(t) \leq 25 \\
& 0 \leq x_4(t) \leq 10 \\
& 0 \leq u_1(t) \leq 50 \\
\text{where:} \quad & h_1\left(x_1,x_3\right) = 0.11\left(\frac{x_3}{0.006x_1 + x_3}\right) \\
& h_2\left(x_3\right) = 0.0055\left(\frac{x_3}{0.0001 + x_3(1+10x_3)}\right)
\end{align*}
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%% solution