Trigonometric extremals in the optimal control problem of the reorientation of the axis of a dynamically symmetric rotating body

The problem of the optimal reorientation of an axisymmetric rigid body with a non-zero angular velocity is considered. An integral-quadratic functional, matched with the inertial symmetry of the body and characterizing the total energy consumption, is chosen as the criterion. The net torque of the applied external forces is the control. The control problem is investigated assuming that jet engines are used and the manœuvre leading to the least consumption of the working medium is therefore of interest. An explicit description of the family of analytical extremals, represented by trigonometric functions of time, is obtained. The construction of these extremals is based on a study of the space–time deformations of the solutions of the Euler–Poinsot equations for the free rotation of a body with similar symmetry characteristics.