Goddard problem

Problem

A rocket rises vertically from rest, with height $h(t)$, velocity $v(t)$ and mass $m(t)$. Its thrust $T(t)$ is the control, with $0 \le T(t) \le T_{\max}$. The motion follows the ordinary differential equations $$h' = v, \qquad v' = \frac{T - D(h, v)}{m} - g(h), \qquad m' = -\frac{T}{c},$$ where $D(h, v) \ge 0$ is the aerodynamic drag, $g(h) > 0$ is the gravitational acceleration and $c > 0$ is the exhaust speed. The initial mass $m(0) = m_0$ and the mass $m_f < m_0$ of the empty rocket are given. Choose the thrust program $T$ and the final time $t_f$, with $m(t_f) \ge m_f$, so that the final height $h(t_f)$ is as large as possible.