Residence Time of Microparticles on the Geostationary Orbit

The problem of rendezvous of a controllable spacecraft with a spacecraft moving along a Keplerian orbit is solved. The integral functional of quality that characterizes the energy expenditure and the time of transferring the controllable spacecraft from initial to final position or the characteristic spacecraft velocity are used as a minimized functional. The control (vector of acceleration or thrust of a jet engine) is assumed to be limited in magnitude. For solving the problem, the equations of motion of the spacecraft center of mass are written in a rotating coordinate system. Among these equations, there is a quaternion differential equation of instantaneous orientation of the spacecraft orbit with a compact symmetrical nondegenerate structure. Four variants of solutions to the problem are considered, differing by the type of minimized functional or by the type of control used. Using Pontryaginʹs maximum principle, we have obtained for all four variants the expressions for optimal control through the magnitude of spacecraft radius vector and conjugate variables; the transversality conditions that do not contain indeterminate Lagrange multipliers; and several first integrals of the problem equations, including the first quaternion integral common for all variants of the problem. This last integral implies that for any control (including optimal control), the instantaneous conjugate system of coordinates is rotated relative to a rotating coordinate system through one and the same angle about the Eulerian axis, which keeps its direction in the inertial coordinate system unchanged. Transformations are suggested that simplify nonlinear equations of the problem. Geometrical interpretations are given to these transformations and first integrals. A new form of the general analytical solution to the problem of spacecraft motion along a segment of optimum trajectory with zero thrust is obtained. This form differs from the known Lauden solution. Specific cases of the problem solution are considered. This work is a continuation of [1].