Low-thrust displaced orbits by weak Hamiltonian-Structure-Preserving control

For the requirements from some missions, the spacecraft is placed near the unstable region around the hyperbolic equilibrium. To stabilize the motion, a novel concept of the weak Hamiltonian-Structure-Preserving (HSP) control parameterized by the Coriolis acceleration is introduced. Based on the dynamical model of low-thrust displaced orbits, this paper searches for the mechanism in orbital stability change by the weak HSP control and provides specific strategies on how the control is implemented. Since the weak control has no effect on the topology of the original system, two invariant equilibria, a hyperbolic one and an elliptic one are solved. Dynamical system techniques are employed to investigate the controlled motions near the two equilibria, illustrating the Lyapunov orbit in different Coriolis acceleration cases and presenting two basic periodic modes measuring the bounded trajectories in the interior region. Controlled trajectories in the neck region are analyzed for their characteristics, classification and transition to make preparation for the stability change. One of the important contributions of this paper is to numerically demonstrate that the weak HSP control preserves the homoclinic orbits to the hyperbolic equilibrium as well as to the Lyapunov orbit, and another is to achieve the orbital transfer within, or even beyond KAM tori based on two methods in determining whether the controlled trajectory integrated from an initial point can exhibit transit.