  This paper is concerned with stability analysis of nonlinear time-varying
systems by using Lyapunov function based approach. The classical Lyapunov
stability theorems are generalized in the sense that the time-derivative of the
Lyapunov functions are allowed to be indefinite. The stability analysis is
accomplished with the help of the scalar stable functions introduced in our
previous study. Both asymptotic stability and input-to-state stability are
considered. Particularly, for asymptotic stability, several concepts such as
uniform and non-uniform asymptotic stability, and uniform and non-uniform
exponential stability are studied. The effectiveness of the proposed theorems
is illustrated by several numerical examples.
