A ranking alternative of the extremals in the problem of the optimal reorientation of an axisymmetric rigid body

The problem of the optimal control of the spatial orientation of a rotating rigid body with an axis of symmetry is considered. New geometric properties of the extremals of this variational problem are established in the non-degenerate case. The property of a “collapse” of the extremal field, a ranking alternative and the connection with the family of trigonometric extremals constructed earlier in a similar problem are described in detail. The results obtained are based on an analysis of the system of equations obtained as the result of using the formalism of Pontryagin's maximum principle.The intrinsic non-linearity of the equations of motion in problems of optimizing the control of the reorientation of a rotating body1, 2, 3, 4 and 5 has led to the development of new approaches assuming additional geometric optimal turning properties. For a non-zero initial rotational velocity, the initial problem reduces to the optimization of successive manoeuvres, that is, the body is first completely decelerated and then reorientated from a position of rest to a position of rest.A new property of the extremals has been described6 in the problem of the optimal control of the reorientation of a rotating spherically-symmetric body. This is associated with the possibility of an abrupt change in the dimension of a certain linear subspace generated by the extremals. It is shown below that similar properties can also be established in the case of the more general problem of the optimal reorientation of an axisymmetric rigid body.