  This paper presents a simple approach to low-thrust optimal-fuel and
optimal-time transfer problems between two elliptic orbits using the Cartesian
coordinates system. In this case, an orbit is described by its specific angular
momentum and Laplace vectors with a free injection point. Trajectory
optimization with the pseudospectral method and nonlinear programming are
supported by the initial guess generated from the Chang-Chichka-Marsden
Lyapunov-based transfer controller. This approach successfully solves several
low-thrust optimal problems. Numerical results show that the Lyapunov-based
initial guess overcomes the difficulty in optimization caused by the strong
oscillation of variables in the Cartesian coordinates system. Furthermore, a
comparison of the results shows that obtaining the optimal transfer solution
through the polynomial approximation by utilizing Cartesian coordinates is
easier than using orbital elements, which normally produce strongly nonlinear
equations of motion. In this paper, the Earth's oblateness and shadow effect
are not taken into account.
