On the possibility of dissipative stabilization of periodic motion of a system with one degree of freedom

A conservative system with one degree of freedom admitting a periodic motion is considered. The system is located on a translationally moving base. Linear viscous friction forces are added to the forces acting on the points of the system. We determine the law of motion of the base that allows one to preserve the periodic motion of the initial system relative to this base. The conditions when the periodic motion becomes Lyapunov asymptotically stable have been obtained by using the Vazhevsky inequality.