A family of analytic extremals in problems of the optimal control of the rotation of a body

The problem of the optimal control of the rotation of an absolutely rigid body about the centre of mass is investigated. The main purpose of the control is to vary the angular velocity vector from its initial value to the required terminal value in a finite time so that the manoeuvre would require the smallest power consumption, which is characterized by an integral quadratic functional. The principal torque produced by the external forces applied to the body serves as the control. The change in orientation is not taken into account, i.e., the problem of the overspeed–braking control of the body, is studied. A new class of analytic extremals based on the use of space-time deformations of the solutions of the dynamical Euler equations for the free rotation of a rigid body is described. Sufficient conditions for the existence of such extremals for all types of symmetries are presented.