Neighboring optimal control for fixed-time multi-burn orbital transfers

Unlike the classical variational method to design the neighboring optimal control (NOC) by minimizing the second variation of cost functional, this paper, from a geometric point of view, presents a parametric approach to establish the NOC for the problem of fixed-time multi-burn orbital transfers. The key idea of the proposed approach is to construct a parameterized family of neighboring extremals around a reference one. By analyzing the canonical projection of the parameterized family, a geometric interpretation is given to show that, even though the gain matrix for the NOC is not explosive, it is impossible to construct the NOC for the multi-burn problem if conjugate points occur at switching times. Then, through deriving the Taylor series expansion of the parameterized extremals, the neighboring optimal feedback not only on thrust direction but also on switching times is readily devised and the procedure for computing the gain matrix is presented. Moreover, a scheme for performing the NOC is proposed such that the typical singularity when approaching the final time is avoided. Finally, the development of this paper is illustrated by two optimal multi-burn orbital transfer problems.