Stability investigation of invariant sets by means of semidefinite Lyapunov functions

A nonautonomous system of ordinary differential equations dx/dt = X (t, x),x = (y, z) admitting the invariant set y = 0 is considered. It is assumed that there exists a nonnegative Lyapunov function V(t, x) whose derivative is nonpositive. It is assumed that all solutions x(t) = (y(t), z(t)) of this system lying on the integral set V(t, x) = 0, have the property lim(t-infinity) parallel to y(t)parallel to = 0. The theorem on the uniform stability of the set y = 0 is proved. Crown Copyright (C) 2010 Published by Elsevier Ltd. All rights