Analytical solutions in the problem of the optimal control of the rotation of an axisymmetric body

The problem of the optimal control of the rotation of an axisymmetric rigid body is investigated. An integral functional, characterizing the power consumption to carry out a manoeuvre is chosen as the criterion, and the boundary conditions for the angular velocity vector are arbitrary. The principal moment of the applied external forces serves as the control. The necessary conditions of the maximum principle are used to solve the problem in the case of a fixed completion time. New non-trivial first integrals are established for the canonical system of direct and conjugate differential equations obtained, which enable the set of all extremals to be parametrized. Hence, the optimal-control problem is reduced to a problem of non-linear mathematical programming. It is shown that there cannot be more than two different solutions in the latter, and a family of boundary conditions is established when the optimum rotation is determined in a uniquely explicit form.