The comparison method in asymptotic stability problems

The problem of the stability of the unperturbed motion of a non-autonomous system is investigated on the basis of comparison equations. The principle of the quasi-invariance of the positive limit set of a perturbed motion is derived, which enables a new form of the necessary conditions for the stability of an unperturbed motion to be established using Lyapunov vector functions of fixed and constant sign. Problems concerning the stability conditions of unsteady motions and the stabilization of programme motions of mechanical systems are solved.