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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 spa...cecraft trajectories. Specifically we solve for the attracting set of an equilibrium point or a periodic orbit (represented as a fixed point) under optimal control with quadratic cost. The analysis reveals the relation between the attractive set and original dynamics. In particular we find 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 problem is worked out in detail analytically and we develop several proofs regarding the structure of the attractive set for an optimal transfer. Our result is theoretical and developed for a linearized system, but can be extended to nonlinear and more realistic situations.show more
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