Goddard Rocket Problem

The Goddard rocket maximum altitude optimal control problem is stated as follows. Maximize the terminal altitude \[J=h(t_f)\] subject to the dynamic constraints \[\begin{array}{lcl}\dot{h} & = & v, \\ \dot{v} & = & \displaystyle \frac{T-D}{m}-g_0, \\ \dot{m} & = & \displaystyle -\frac{T}{c},\end{array}\] where the drag is \[D=k v^2\exp\!\left(-\frac{h}{H}\right).\] The problem is solved in three phases with linkage constraints between adjacent phases. The singular arc satisfies the path constraint \[T-D-mg_0-\left(\frac{c^2(1+v/c)}{Hg_0}-1-\frac{2c}{v}\right)\frac{mg_0}{1+4c/v+2c^2/v^2}=0.\] The initial conditions are \[h(0)=0,\qquad v(0)=0,\qquad m(0)=3,\] and the final mass condition is \[m(t_f)=1.\] The solution to the Goddard rocket maximum altitude problem using GPOPS-II is shown in the figures below.