Moon Lander Problem

The moon lander optimal control problem is stated as follows. Minimize the cost functional $$J=\int_0^{t_f} u(t)dt$$ subject to the dynamic constraints $$\begin{array}{lcl}\dot{h}(t) & = & v(t), \\ \dot{v}(t) & = & -g + u(t)\end{array}$$ and the boundary conditions $$\begin{array}{lclclcl}h(0) & = & h_0 & , & h(t_f) & = & 0, \\ v(0) & = & v_0 &, & v(t_f) & = & 0. \end{array}$$ The solution to the moon lander optimal control problem using GPOPS-II is shown in the figures below.

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Image Courtesy of NASA.