Necessary condition on Lyapunov functions corresponding to the globally asymptotically stable equilibrium point

We give a necessary condition, independent of the system dynamics, for every Lyapunov function corresponding to the globally asymptotically stable equilibrium point of ẋ = f (x). This necessary condition is numerically easier to check than checking positive definiteness of a function. Therefore, it can be used as a first level test to check whether a given continuously differentiable function is a Lyapunov function candidate or not. We also propose a method, which we call a generalized steepest descent method, to check this condition numerically.