On the asymptotic stability and instability of the zero solution of a non-autonomous system with respect to part of the variables

Abstract

A non-autonomous set of differential equations is considered which allows of the existence of a set of differential equations limiting it. Theorems of the asymptotic stability and instability of the zero solution of such systems with respect to part of the variables are proved in the presence of Liapunov function with derivatives of constant sign. Sufficient conditions are obtained for the partial asymptotic stability of non-autonomous holonomic mechanical system subjected to the action of dissipative forces with total or partial dissipation. The problem of the asymptotic stability of the equilibrium of a heavy solid with a fixed point in a homogeneous gravitational field of variable intensity, and of stabilizing the axis of symmetry of a symmetric satellite perpendicular to the orbital plane of the latter, whose centre of mass remains at the libration points of the limited circular three-body problem, are considered as examples.