Optimization of low-thrust many-revolution transfers and Lyapunov-based guidance

Based on the dynamic evolution of mean equinoctial orbital elements, we developed a direct method to optimize low-thrust many-revolution orbit transfers and proposed a Lyapunov-based guidance, in which a Lyapunov control law with time-varying gains is employed. Within each transfer revolution, a parameterized control law, in the form of the optimal control derived from the calculus of variations, is formulated, and a periapsis- and apoapsis-centered burn structure is proposed in order to effectively solve fuel-saving orbit transfers. The parameters governing the control law and the burn structure within each transfer revolution are interpolated through a finite number of discrete nodes. The optimal orbit transfer problem is converted to the parameter optimization problem that is solved by nonlinear programming. Subsequently, a mapping between the parameterized control law and the Lyapunov control law is revealed, in terms of which the time-varying gains of the Lyapunov control law, called Lyapunov gains, can be obtained using trajectory optimization solutions. However, this mapping does not guarantee that all obtained Lyapunov gains are positive so that the Lyapunov-based guidance may not be strictly stable during an entire transfer period. Nevertheless, we showed that the Lyapunov-based guidance that is not strictly stable may still successfully guide the spacecraft for certain orbit transfer cases. Negative Lyapunov gains can be re-defined as appropriate positive values to warrant both stability and acceptable performance for the Lyapunov-based guidance.