Lyapunov Control for the Circular Restricted Three-Body Problem Using Normal Forms

Lyapunov-based control laws have been widely studied for transfers between Keplerian orbits in the two-body problem. However, in the context of orbit transfers near the equilibrium points of the circular restricted three-body problem (CR3BP), there is a lack of significant literature. The present study introduces a Lyapunov-based controller for low-thrust orbital transfers in the CR3BP. The approach uses action-angle coordinates based on the Birkhoff normal form, which can be used as local orbital elements around the equilibrium points of a Hamiltonian system. For the unstable equilibrium points of the CR3BP, a strategy using a combination of Lyapunov-based and linear quadratic regulator controllers is presented to perform a stabilized orbital transfer. The performance of the proposed controller is then demonstrated by application to numerical examples of low-thrust orbital transfers around different equilibrium points of the CR3BP.