The Goddard Problem with Constrained Time-of-Flight

Presently studied is the problem of the vertical flight of a rocket in a resisting medium, with a constraint of isoperimetric type. Only the case of maximum range for given flight-time has been analyzed, but the methodology applies to other types of constraints as well, e.g., the minimum-time problem, due to Mayer reciprocity. Analysis shows that a one-parameter family of singular extremals is generated according to the value of time-of-flight. Three cases of switching structures have been found, varying with the specified duration of flight, with the most interesting case featuring appearance of a second full-thrust subarc at the departure from the singular subarc owing to a low value of the upper bound on the thrust.