Asymptotic solution to the problem of optimal low-thrust energy increase.

Abstract

HE problem to be considered is that of optimal ascent from an initial circular planetary orbit to some specified final energy level by a spacecraft equipped with a low-thrust engine. Optimal will be defined as minimum time; and since it will be assumed that the engine produces continuous thrust with constant thrust-acceleration, fuel expenditure is minimized. Analytical solutions to this problem have previously been found by Lawden, 1 and by Breakwell and Rauch.2 Until now, Lawden's was the only analysis which used a small parameter perturbation approach; and his results failed to predict the oscillatory nature of the optimal control program. Breakwell and Rauch's work was directed primarily toward the analytical representation of a nominal trajectory and guidance coefficients for a neighboring optimal guidance scheme. Their solution is basically a series solution in the radial distance, but also contains some additional correction terms which were found by developing a set of defining differential equations, assuming a periodic solution with variable coefficients, and employing the method of averaging to determine those coefficients. The solution correctly, represents the characteristics of the control program and trajectory and, for at least eight revolutions, matches a numerically generated solution to within 1%. The use of the radial distance as independent variable and the series form of the solution, however, make an analysis of the motion and a comparison with other spiral trajectories difficult. The purpose of this Note is to present an accurate small parameter perturbation solution to the problem. In addition, the optimal trajectory is analyzed and compared with a tangential thrust trajectory. Analysis The problem is formulated using the equations of motion