An adaptive stabilization scheme for autonomous system oscillations

A smooth autonomous system of general form is considered. A global family of non-degenerate periodic solutions by the parameter h is constructed; the period varies monotonically on this family. The problem of stabilizing the oscillations of the reduced controlled system is solved. A smooth autonomous control law with a parameter depending on h is applied, and an attracting cycle is constructed. The results are concretized for an nth-order differential equation. The relation of these results with the conclusions obtained for the reversible mechanical system is established. An adaptive control scheme for the reduced conservative system is proposed to stabilize any oscillation of the family. Some applications are presented.