  This paper presents a novel neighboring extremal approach to establish the
neighboring optimal guidance (NOG) strategy for fixed-time low-thrust
multi-burn orbital transfer problems. Unlike the classical variational methods
which define and solve an accessory minimum problem (AMP) to design the NOG,
the core of the proposed method is to construct a parameterized family of
neighboring extremals around a nominal one. A geometric analysis on the
projection behavior of the parameterized neighboring extremals shows that it is
impossible to establish the NOG unless not only the typical Jacobi condition
(JC) between switching times but also a transversal condition (TC) at each
switching time is satisfied. According to the theory of field of extremals, the
JC and the TC, once satisfied, are also sufficient to ensure a multi-burn
extremal trajectory to be locally optimal. Then, through deriving the
first-order Taylor expansion of the parameterized neighboring extremals, the
neighboring optimal feedbacks on thrust direction and switching times are
obtained. Finally, to verify the development of this paper, a fixed-time
low-thrust fuel-optimal orbital transfer problem is calculated.
