Energetically optimal control of the convergence of the spacecraft in non-central gravitational field of the Earth on the far guidance stage

The article is devoted to the development of algorithms for calculating energy-optimal programs for controlling the approach of a spacecraft to an orbital object at the long-range guidance stage using the maximum principle L.S. Pontryagin. It is assumed that the spacecraft is equipped with a longitudinal propulsion system running on chemical fuel. The programs  of optimal change in the second fuel consumption and the vector of cosine guides, which determine the orientation of the thrust force of the propulsion system, should be determined. As a criterion for optimal control, we consider a functional that determines the minimum consumption of the working fluid. The optimal control problem is solved in a limited region of the state space, which is determined by the range of variation of the angular range of the spacecraft within one revolution. The full equations of the boundary value problem of the maximum principle are given using the model of the object's motion in the normal gravitational field of the Earth, the boundary conditions, and also the analytical dependencies that determine the control structure in the optimal mode. The boundary optimization problem is solved with the Newton method. To determine the initial approximation of conjugate variables, an analytical solution is given to the problem of energy-optimal approach control in a uniform central field. The results of numerical studies of energy-optimal programs to control the interception in the normal gravitational field of the Earth with the final pitch at the stage of long-range guidance are presented. In general, the use of optimal approach control algorithms in the airborne and ground complex allows to reduce fuel costs when performing the maneuver, reduce time, and expand the reachability area of the target objects. In addition, the optimal solution can be considered as a reference, with which it is necessary to compare various variants of approximate control algorithms, evaluate their quality and make informed decisions on their practical use.