  Motivated by recent applications of the Lyapunov's method in artificial
neural networks, which could be considered as dynamical systems for which the
convergence of the system trajectories to equilibrium states is a necessity. We
re-look at a well-known Krasovskii's stability criteria pertaining to a non
linear autonomous system. Instead, we consider the components of the same
autonomous system with the help of the elements of Jacobian matrix J(x), thus
proposing much simpler convergence criteria via the method of Lyapunov. We then
apply our results to artificial neural networks and discuss our results with
respect to recent ones in the field.
