On the asymptotic stability of equilibrium ✩

The classical criterion of asymptotic stability of the zero solution of equations x = f (t,x) is that there exists a positive definite function V which has infinitesimal upper bound such that dV dt is negative definite. In this paper we prove that if dm+1V dtm+1 is bounded then the condition that dV dt is negative definite can be weakened and replaced by that dV dt 0 and − dV dt + d2V dt2 + · · · + dmV dtm + dm+pV dtm+p is negative definite.  2004 Elsevier Inc. All rights reserved