Energy-optimal low thrust optimization using finite-Fourier parametrization with nonstationary spectrum

Using low-thrust engines to change the orbit of a near-Earth spacecraft reduces the fuel required for the maneuver due to their high specific impulse. However, the smallness of the thrust relative to the Earth’s gravity leads to long flight times and spiral trajectories, the design of which requires advanced methods. The finite-Fourier method of approximating the periodic thrust acceleration allows one to find a control that implements a quasi-optimal low-thrust flight in terms of fuel consumption in the model of an ideally regulated thrust engine. In this paper, the finite-Fourier approach is generalized to a class of non-periodic controls by introducing the time dependence of the Fourier coefficients of the thrust acceleration. It is shown that the expansion of the control function class leads to a decrease in fuel costs. The trajectories obtained by this approach are useful in the design of flights in near-Earth space, since they can be used to estimate fuel costs in preliminary mission analysis. They can also serve as an initial guess for the problems of constructing the fuel-optimal control in the class of piecewise smooth functions.