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.

This entry contributed by Martin Keller-Ressel

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Keller-Ressel, Martin. "Lyapunov Function." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/LyapunovFunction.html