Analytical Solution of Optimal Feedback Control for Radially Accelerated Orbits

The optimal feedback control problem for low-thrust trajectories with modulated inverse-square-distance radial thrust is studied in this paper. The problem is tackled by applying a generating-function method devised for linear systems. Instead of deriving open-loop solutions, arising from the two-point boundary-value problems in which the classical optimal control is stated, this technique allows us to obtain analytical closed-loop control laws. The idea behind this work consists of applying a globally diffeomorphic linearizing transformation that rearranges the original nonlinear dynamic system into a linear system of ordinary differential equations written in new variables. The generating-function technique is then applied to this new dynamic system, the optimal feedback control problem is solved, and the variables are transformed back into the original. Thus, we avoid the problem of expanding the vector field and truncating higher-order terms, because no remainders are lost in the approach undertaken. Practical examples are used to show the usefulness of the derived solution for modulated, inverse-square-distance, radially accelerated orbits.