Fuel-optimal low thrust trajectory design with J2 perturbation

A multiple homotopy method is proposed to optimize the very low thrust fuel-optimal transfer trajectory under the effects of J2 perturbation. A simple problem of high thrust, energy-optimal transfers in the linear gravity without J2 perturbation is constructed as the homotopy initial problem. Three homotopic parameters are embedded in the kinetic equations, thrust magnitude and performance index, respectively, and the optimal control laws in the homotopy process are deduced according to the minimum principle. By solving the subproblems with iterative homotopic parameters, the high thrust energy-optimal transfers, low thrust energy-optimal transfers and fuel-optimal transfers are solved in turn. A numerical example about rendezvous mission of satellite and debris on sun-synchronous orbits is given to substantiate the effectiveness of the method in fuel-optimal low thrust trajectory design in the gravity with J2 perturbation. Using the proposed method, the difficulties of the highly nonlinearity of the dynamic system caused by J2 perturbation, the fuel optimal problem's discontinuous structure of Bang-Bang control and many revolution transfers can be solved.