Symplectic multi-level method for solving nonlinear optimal control problem

By converting an optimal control problem for nonlinear systems to a Hamiltonian system, a symplecitc-preserving method is proposed. The state and costate variables are approximated by the Lagrange polynomial. The state variables at two ends of the time interval are taken as independent variables. Based on the dual variable principle, nonlinear optimal control problems are replaced with nonlinear equations. Furthermore, in the implementation of the symplectic algorithm, based on the 2^N algorithm, a multilevel method is proposed. When the time grid is refined from low level to high level, the initial state and costate variables of the nonlinear equations can be obtained from the Lagrange interpolation at the low level grid to improve e±ciency. Numerical simulations show the precision and the e±ciency of the proposed algorithm in this paper.