Analysis of optimal motion control for a material point in a central field with application of quaternions

This work is devoted to a survey and generalization of results obtained in the theory of optimal motion control for a material point in the central Newtonian gravitational field using the Pontryagin’s maximum principle and quaternion models of orbital motion. This theory is very important in space flight mechanics, being the background of the solution of optimal control problems of the motion of the center of mass of a space vehicle. In the first part of this work, a survey of quaternion models of the motion of a material point in a central Newtonian gravitational field is given, their advantages and disadvantages are analyzed. The formulation of the optimal control problem of the motion of a material point in the central Newtonian gravitational field and its correlation with the optimal control problem of the motion of the center of mass of a space vehicle is considered. The main problems arising in the solution of optimal control problems of the motion of a material point using the maximum principle, including instability in the Lyapunov’s sense of solutions to adjoint equations, are studied. It is shown that efficiency of analytical investigation and numerical solution of the corresponding boundary-value problems can be increased by the application of quaternion models of orbital motion.