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=0y=0 is considered. It is assumed that there exists a nonnegative Lyapunov function V(t,x)V(t,x) whose derivative is nonpositive. It is assumed that all solutions x(t)=(y(t),z(t))x(t)=(y(t),z(t)) of this system lying on the integral set V(t,x)=0V(t,x)=0, have the property limt→∞‖y(t)‖=0limt→∞‖y(t)‖=0. The theorem on the uniform stability of the set y=0y=0 is proved.