The problem of joint optimization for the main design parameters of the electric propulsion system and the trajectory of the spacecraft is considered. We consider two problems: to minimize thrust and to maximize the useful mass of the spacecraft with the optimum characteristics of the propulsion system (for example, thrust, exhaust velocity and power). The first problem is related to the existence of optimal trajectories with finite thrust. If there is no a priori information on the existence of solutions, then it is difficult to construct stable and efficient methods of numerical optimization: if the numerical method does not converge to the desired solution, then it can not be determined whether the method fails or the optimal control problem does not have solutions at all. Therefore, it is important to find the region for the existence of a solution for the numerical solution. The latter problem is directly related to the optimization of the characteristics of the propulsion system within the domain of existence of the solution.

A method has been developed for solving problems of minimum propulsion thrust and maximum payload of the spacecraft for a combined scheme of launching into geostationary orbit. A scheme is considered of a flight from a low Earth orbit (reference orbit) to the geostationary orbit of a spacecraft with a marching propulsion system, in which the booster block takes the spacecraft into some intermediate orbit (the parameters of which must be optimized), after which the spacecraft is brought into a final orbit using a cruise propulsion system. The necessary optimality conditions are obtained in both problems under consideration, including the necessary conditions for the optimal connection of the flight sections with large and small thrust and the necessary optimality conditions for the thrust of the electric propulsion system. In addition, for the problem of the maximum payload mass of the spacecraft, the necessary optimality conditions for the specific impulse of the electric propulsion system are obtained. The method for solving the problems of the minimum thrust of the electric propulsion system and the maximum useful mass of the spacecraft for the combined launch scheme is to use the maximum principle to reduce the problem of end-to-end optimization of the flight path with sections of large and small thrust to the boundary problem for a system of ordinary differential equations. To ensure the computational stability and convergence of the method, we used the numerical averaging of the differential equations of the optimal motion, the continuation method with respect to the parameter, and the sequential solution of the problems with minimum reactive acceleration, minimum thrust, and maximum net mass. To solve the latter problem, we used a simplified mass model of a spacecraft with a marching electric propulsion system, which allows us to estimate the useful mass from the known values ​​of the initial mass of the spacecraft, the required mass of the working body of the electric propulsion system, thrust and specific impulse of the electric propulsion system.

This study was supported by the grant in form of subsidies from the federal budget, allocated for state support of scientific research under supervision of leading scientists in Russian institutions of higher education, scientific foundations and state research centers of the Russian Federation (7th stage,  Decree of the Government of the Russian Federation No. 220 of 09 April 2010), project No. 075-15-2019-1894.