  This brief gives a set of unified Lyapunov stability conditions to guarantee
the predefined-time/finite-time stability of a dynamical systems. The derived
Lyapunov theorem for autonomous systems establishes equivalence with existing
theorems on predefined-time/finite-time stability. The findings proposed herein
develop a nonsingular sliding mode control framework for an Euler-Lagrange
system to analyze its stability, and its upper bound for the settling time can
be arbitrarily determined a priori through predefined time constant.
