%% formulation
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\begin{align*}
\min_{u_1(t),u_2(t)} \quad & \ \int_{0}^{t_f} [u_1x_2 + \rho(u_1^2 +u_2^2)] \mathrm{d}t \\
\text{subject to:} \quad & \dot{\bm{x}} = \begin{bmatrix}
x_2\\
h(x_1) - (a + bx_2 + cx_2^2) + u_1 -u_2\\
\end{bmatrix} \\
& \bm{x}(0) = \bm{x}_0 \\
& \bm{x}(t_f) = \bm{x}_f\\
& u_{1,\min} \leq u_1(t) \leq u_{1,\max} \\
& u_{2,\min} \leq u_2(t) \leq u_{2,\max} \\
\text{where:} \quad & h(x_1) = \sum_{j=1}^{2}\left[ \frac{s_{j+1} - s_j}{\pi} \tan^{-1}\frac{x_1 - z_j}{\epsilon }\right]
\end{align*}
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%% solution