Definition:Lyapunov Function

Let $$x_0$$ be an equilibrium point of the system $$x' = f(x)$$.

Then a function $$V$$ is a Liapunov function of the system on an open set $$U$$ containing the equilibrium if:
 * $$V(x_0) = 0$$;
 * $$V(x) > 0 \,$$  if  $$\, x \in U - \{ x_0 \}$$;
 * $$\nabla V \cdot f \le 0$$  for  $$x \in U$$.

If the inequality is strict except at $$x_0$$, then $$V$$ is strict.