%% formulation
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\begin{align*}
\min_{u_r(t), u_t(t)} \quad & \int_0^{t_f} \| \bm{u}(t) \|_p \mathrm{d}t \\
\text{subject to:} \quad
& \dot{r} = v_r \\
& \dot{\theta} = \frac{v_t}{r} \\
& \dot{v}_r = \frac{v_t^2}{r} - \frac{1}{r^2} + u_r \\
& \dot{v}_t = -\frac{v_r v_t}{r} + u_t \\
& (r,\theta,v_r,v_t)(0) = (1, 0, 0, 1) \\
& (r,v_r,v_t)(t_f) = (4,0,0.5) \\
& \| \bm{u}(t) \|_q \leq u_{\max} \\
\text{where:} \quad & \frac{1}{p} + \frac{1}{q} = 1
\end{align*}
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%% solution