A Lyapunov function is a scalar function defined on a region that is continuous, positive definite, for all ), and has continuous first-order partial
derivatives at every point of . The derivative of with respect to the system , written as is defined as the dot product

(1)

The existence of a Lyapunov function for which on some region containing the origin, guarantees the stability of the zero
solution of ,
while the existence of a Lyapunov function for which is negative definite on some region containing the origin guarantees the asymptotical stability
of the zero solution of .

For example, given the system

(2)

(3)

and the Lyapunov function , we obtain

(4)

which is nonincreasing on every region containing the origin, and thus the zero solution is stable.