Minimality of Variable-Thrust Subarcs in Optimal Chemical Rocket Trajectories

摘要

The purpose of this Note is to review the necessary conditions for a relative minimum and to apply them to the variable-thrust subarc of a minimum-fuel rocket trajectory known as Lawden's spiral.rnThe necessary conditions are discussed for regular controls, singular controls, and mixed controls. Within each of these categories, the conditions are classified as strong or weak. The Weierstrass condition and the Legendre-Clebsch condition, which are valid for regular controls, are discussed and shown not to be applicable to singular controls. On a singular control, weak minimality can be tested by applying the generalized Legendre-Clebsch condition. There exists a strong condition, equivalent to the Weierstrass condition, but it has only been derived for a first-order singular control. This condition has the same formula as the generalized Legendre-Clebsch condition. Hence, the status of the generalized Legendre-Clebsch condition has been elevated from a weak condition to a strong condition in this case.