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While the Pontryagin Maximum Principle can be used to calculate candidateextremals for optimal orbital transfer problems, these candidates cannot beguaranteed to be at least locally optimal unless sufficient optimalityconditions are satisfied. In this paper, through constructing a parameterizedfamily of extremals around a reference extremal, some second-order necessaryand sufficient conditions for the strong-local optimality of the free-timemulti-burn fuel-optimal transfer are established under certain regularityassumptions. Moreover, the numerical procedure for computing these optimalityconditions is presented. Finally, two medium-thrust fuel-optimal trajectorieswith different number of burn arcs for a typical orbital transfer problem arecomputed and the local optimality of the two computed trajectories are testedthanks to the second-order optimality conditions established in this paper.

Optimality conditions applied to free-time multi-burn optimal orbital transfers