The problem of the optimal stabilization of the spinning top motion.

The present work considers the optimal stabilization problem in motion of a Spinning Top when integrally small perturbations act during a finite interval of time. The optimal stabilization problem of considered motion is assumed and solved. In direction of the generalized coordinates introduced input controls, fully controllability of linear approximation of the obtained control system is checked up and the optimal stabilization problem of this system on classical sense is solved. Then, the problem will be limited to one input control, it is shown that the considered system is not fully controllable and for this case the optimal stabilization problem under integrally small perturbations of mentioned system is solved. For both cases optimal Lyapunov function is constructed, the optimal controls and the optimal value of performance index are obtained. The comparison between the optimal values of performance indexes proves that energy consumption at stabilization under integrally small perturbations is less than solving that issue in classical sense.