Structure-Preserving Stabilization for Hamiltonian System and its Applications in Solar Sail

A structure-preserving controller is constructed to stabilize a hyperbolic Hamiltonian system. Bounded orbits for the planar solar sail three-body problem are generated by means of the controller. The invariant (stable, unstable, and center) manifolds of the equilibrium are used to stabilize a Hamiltonian system of 2 degrees of freedom using only position feedback. It is proved that the poles of the system can be assigned to any position on the imaginary axis by choosing the manifolds' gains properly. A new type of quasi-periodic orbit referred to as a stable Lissajous orbit is obtained. The orbit will degenerate to a periodic orbit in the case of resonance between modes and suitable initial values (Lyapunov orbit). Using the controller to solve the solar sail three-body problem yields a stable Lissajous orbit, which is quite different from the classical Lissajous orbit. We show that the sail equilibrium can be stabilized, and moreover that the orbit is bounded. The allocation law of the controller is also studied, which verifies that the controller is realizable.