Minimality of Variable-Thrust Subarcs in Optimal Chemical Rocket Trajectories

F ROM the beginning of rocket trajectory optimization, it has been known that the optimal trajectory of a rocket can be composed of three types of subarcs: maximum thrust, variable thrust, and zero thrust. To determine whether or not a particular subarc can be part of aminimum-fuel trajectory, it is possible to apply some tests (necessary conditions). Necessary conditions for a relative minimum are classified according to the types of the controls, that is, regular controls (nonsingular), singular controls, andmixed controls (regular and singular), and according to level, that is, weak or strong. Weak conditions are the Legendre–Clebsch condition [1] for regular controls, the generalized Legendre–Clebsch condition [2] for singular controls, andGoh’s condition [3] for mixed controls. Strong conditions are the Weierstrass condition [1] for regular controls, the equivalent of the Weierstrass conditon [4] for singular controls, and the equivalent of Goh’s condition for mixed controls (yet to be derived). A subarc is defined to be “minimizing” if the relevant necessary conditions for a minimum are satisfied. Although this use of the term minimizing might seem to be unconventional to some researchers, it is not. See for example Bliss [5] (p. 20), Bryson and Ho [6] (p. 247), Kopp and Moyer [7] (p. 1443), Goh [3] (p. 721), and Bell and Jacobson [2] (p. 3). For a chemical rocket, the particular variable-thrust subarc, known as Lawden’s spiral, was claimed by Lawden to be minimizing by applying the Weierstrass condition [8] (p. 112). Subsequently, Robbins [9], Kelley [10] , Kopp andMoyer [7], and Kelley et al. [11] developed the generalized Legendre–Clebsch condition and applied it to theLawden spiral. Their conclusionwas that theLawden spiral is not minimizing. The same conclusion was reached by Kelley [12] using state transformations. Later, Lawden [13] wrote that the variable-thrust subarc is not minimizing but did not say why the Weierstrass condition gave a conflicting result. This Note has two purposes. The first is to review the preceding necessary conditions and their use. The second is to explain why Lawden’s application of the Weierstrass condition did not eliminate the variable-thrust subarc.