Attractive Sets to Unstable Orbits Using Optimal Feedback Control

This paper investigates the combination of optimal feedback control with the dynamical structure of the three-body problem. The results provide new insights for the design of continuous low-thrust spacecraft trajectories. Specifically, the attracting set of an equilibrium point or a periodic orbit (represented as a fixed point) under optimal control with quadratic cost is obtained. The analysis reveals the relation between the attractive set and original dynamics. In particular, it is found that the largest dimensions of the set are found along the stable manifold and the least extent is along the left eigenvector of the unstable manifold. The asymptotic behavior of the structure of the attractive set when time tends to infinity is analytically revealed. The results generalize the use of manifolds for transfers to equilibrium points and periodic orbits in astrodynamic problems. The result is theoretical and developed for a linearized system, but it can be extended to nonlinear systems in the future.