Optimal Rocket-Powered Ascent Study

The Goddard Problem is that of maximizing the final altitude for a vertically ascending, rocket-powered vehicle under the influence of an inverse square gravitational field and atmospheric drag. The present study is concerned with the effects of two additional constraints: a dynamic pressure limit [q
max
], and a specified final time [t
f
]. Nine different switching structures involving zero-thrust arcs, full-thrust arcs, singular-thrust arcs, and constrained arcs are obtained for prescribed values of q
max
and final time t
f
between t
f
,min and t
f
*. Here t
f
,min is the minimum possible time within which all the fuel can be burned, and t
f
* is the optimal final time. For all points in the above defined domain of the q
max
,t
f
-plane the associated optimal switching structure is clearly identified. Finally, a comparison between the optimal solutions and a simple intuitive feedback law is given.