DMOC approach to solve Goddard rocket problem in comparison with RPM

A new approach -DMOC (Discrete Mechanics and Optimal Control) is presented to solve the classic Goddard rocket problem, in comparison with RPM (Radau Pseudospectral Method). DMOC approach firstly discretizes the Lagrange-d´Alembert principle for the rocket system, the resulting forced discrete Euler-Lagrange equations then serve as constraints for the optimization of a given cost functional. Therefore the optimal control problem is converted into a nonlinear programming (NLP) problem. While RPM transcribes the optimal control problem to a NLP problem by parameterizing the state and control using global polynomials and collocating the differential-algebraic equations using nodes obtained from a Gaussian quadrature. Finally, DMOC approach and RPM both successfully solve the Goddard rocket problem, which is a terminal free, singular optimal control problem with path constrain; the results proved that DMOC approach and RPM each have its strong points.