Stabilization of the uniform rotations of a rigid body around a principal axis

Conclusion We sum up the results regarding the stabilization of the investigated motion. The system (1.2) is stabilizable in the linear approximation if the conditions (2.2) on the parameter are not satisfied, and also in the case 4 if a₂₂₃₀>0. The system (1.2) is stabilizable in the nonlinear approximation for ₀=0, ₁0. The system (1.2) is nonstabilizable in the case 4 if a₂₂₃₀<0. In the remaining cases the investigated motion is nonasymptotically stable.Comparing the results regarding stabilizability and controllability, we note that the relation between these properties is, possibly, more complex than for linear systems, since in the cases 2, 3, in which the necessary conditions for the nonlinear system (1.2) are satisfied, while the sufficient ones are not, we do have a nonasymptotic stability. The further investigation of these cases requires the determination of more general sufficient, possibly necessary and sufficient, conditions of controllability of nonlinear systems; this seems to be possible in any case for systems that are linear with respect to control.In conclusion we note that the critical cases of stabilization as well as the problem of the control of the motion of a rigid body by a reactive force have been investigated long ago (we mention 2, 5]) and, as shown by this paper, they have not been definitively solved and continue to present interest for both theory and practice.Institute of Applied Mathematics and Mechanics, Academy of Sciences of Ukraine, Kiev. University of Science, Sebha, Libya. Translated from Prikladnaya Mekhanika, Vol. 28, No. 9, pp. 73–79, September, 1992.