On the partial asymptotic stability in nonautonomous differential equations

Abstract

A system of ordinary differential equations $dx/dt=X(t,x)$ which has a zero solution $x=0$ is considered. It is assumed that there exists a function $V(t,x)$, positive definite with respect to part of state variables such that its derivative $dV/dt$ is nonpositive. It is proved that if the function $\sum_{i=1}^jV_i^2$ is positive definite with respect to part of the studying variables, then the zero solution is asymptotically stable with respect to these variables. Here $V_1=dV/dt, V_{i}=dV_{i-1}/dt, \quad i=2, \dots,j;\quad j$ is some positive integer. The instability criterion is also obtained.