Higher order derivatives of Liapunov functions

The problem considered is that of inferring asymptotic stability by examining higher order derivatives of a Liapunov function when examination of only the first derivative does not permit such an inference. It is shown that a seemingly plausible approach using only\dot{V}and\ddot{V}is vacuous. It is also shown that it is possible to infer asymptotic stability by examining\dot{V}and\ddot{V}, and perhaps\ddot{V}.