Conjugate points on extremal rocket paths

Abstract

Of the methods in use for optimizing rocket trajectories, many do not test whether or not the Jacobi necessary condition is satisifed or otherwise guarantee the minimizing character of the extremal obtained. Thus the possibility of the occurrence of Jacobi conjugate points on rocket extremals motivates the present study of the simplified case of planar vacuum flight in an inverse square law gravity field under constant thrust. The results indicate that conjugate points do indeed occur and shed some light upon the circumstances incident to their appearance.

The first order necessary conditions and the conjugate point condition are reviewed for the rocket trajectory problem. The conjugate point test for Mayer problems is reduced to the examination of a single minor of the usual test matrix plus a Lagrange multiplier.

Initially rocket paths are studied with the central angle ignored. The three state variables are the radius and the radial and circumferential velocity components. Vertical launch trajectories are shown to be extremals. The state variation defined by an extremal adjacent to this trajectory obeys a single second order differential equation which may exhibit an oscillatory solution whose zeros specify conjugate points. When the rocket is launched nonvertically with zero initial velocity, an analytic proof shows that the sign of the circumferential velocity component must change before a conjugate point occurs. It is conjectured that this is also true for trajectories with arbitrary initial velocity. A plausibility argument and numerical results are offered in substantiation.

Rocket paths are also examined with the central angle included with the three state variables mentioned above. Conjugate points are found numerically but no qualitative rules governing them have been deduced.

This research was partially sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force under AFOSR Contracts Nos. AF 49(638)-1207 and AF 49(638)-1512 and Contract NAS 12-114 with NASA Electronics Research Center, Cambridge, Massachusetts. The complete paper may be found in the Proceedings of the 19th Congress of the International Astronautical Federation published by Pergamon Press.