Goddard problem in presence of a dynamic pressure limit

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 paper deals with the effects of two additional constraints, namely, a dynamic pressure limit and specified final time. Nine different switching structures involving zero-thrust arcs, full-thrust arcs, singular-thrust arcs, and state-constrained arcs are obtained when the value of the dynamic pressure limit is varied between zero and infinity and the final time is specified between the minimum possible time within which all of the fuel can be burned and the natural final time that emerges for the problem with final time unspecified. For all points in the aforementioned domain of dynamic pressure limit and prescribed final time, the associated optimal switching structure is clearly identified. Finally, a simple intuitive feedback law is presented for the free time problem. For all values of prescribed dynamic pressure limit, this strategy yields a loss in final altitude of less than 3 percent with respect to the associated optimal solution.