  Low-thrust engines for interplanetary spacecraft transfers allow
cost-effective space missions with flexible launch and arrival dates. To find
fuel-optimal trajectories, an optimal control problem is to be solved.
Pontryagin's Maximum Principle shows that the structure of the optimal control
is bang-bang with the possibility of singular arcs. Even though the latter have
been heuristically shown to rarely appear in practical applications, a full
theoretical characterization does not exist in the literature. As a growing
number of missions are expected to adopt low-thrust engines in the near future,
such study is required to have a comprehensive understanding of the problem.
This work presents analytical necessary conditions for the existence of
singular arcs that only depend on three physical variables and not on the
costates. Moreover, it provides an analytical expression of the singular
control that depends on a limited set of physical variables. This is a
fundamental feature, as simple evaluation of the necessary condition and of the
singular control can be performed. Finally, it provides insightful information
on the reasons why singular arcs are rare and it quantifies the possibility of
their occurrence.

Optimization and Control