Symplectic adaptive algorithm for solving nonlinear two-point boundary value problems in Astrodynamics

In this paper, from a Hamiltonian point of view, the nonlinear optimal control problems are transformed into nonlinear two-point boundary value problems, and a symplectic adaptive algorithm based on the dual variational principle is proposed for solving the nonlinear two-point boundary value problem. The state and the costate variables within a time interval are approximated by using the Lagrange polynomial and the costate variables at two ends of the time interval are taken as independent variables. Then, based on the dual variational principle, the nonlinear two-point boundary value problems are replaced by a system of nonlinear equations which can preserve the symplectic structure of the nonlinear optimal control problem. Furthermore, the computational efficiency of the proposed symplectic algorithm is improved by using the adaptive multi-level iteration idea. The performance of the proposed algorithm is tested by the problems of Astrodynamics, such as the optimal orbital rendezvous problem and the optimal orbit transfer between halo orbits.