%% formulation
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\begin{align*}
\min_{u(t),m(t)} \quad & -ax(t_f)e^{-\rho t_f} - bp(t_f)e^{-\rho t_f} - \int_{0}^{t_f} e^{-\rho t} \left[ ws(t) - hx(t) - ru^2(t) - qu(t) - d - cm(t) \right] \mathrm{d}t \\
\text{subject to:} \quad & \dot{x}(t) = p(t) u(t) - s(t) \\
& \dot{p}(t) = - \left[ \alpha(t) + m(t) \right] p(t) + m(t)  \\
& (x,p)(0) = (x_0,p_0) \\
& 0 \leq u(t) \\
& 0 \leq m(t) \leq M \\
& 0 \leq x(t)
\end{align*}
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%% solution