# Lyapunov function

A function defined as follows. Let be a fixed point of the system of differential equations

(that is, ), where the mapping is continuous and continuously differentiable with respect to (here is a neighbourhood of in ). In coordinates this system is written in the form

A differentiable function is called a Lyapunov function if it has the following properties:

1) for ;

2) ;

3)

The function was introduced by A.M. Lyapunov (see [1]).

Lyapunov's lemma holds: If a Lyapunov function exists, then the fixed point is Lyapunov stable (cf. Lyapunov stability). This lemma is the basis for one of the methods for investigating stability (the so-called second method of Lyapunov).

#### References

[1] | A.M. Lyapunov, "Stability of motion" , Acad. Press (1966) (Translated from Russian) |

[2] | E.A. Barbashin, "Lyapunov functions" , Moscow (1970) (In Russian) |

#### Comments

For additional references see Lyapunov stability.

**How to Cite This Entry:**

Lyapunov function.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Lyapunov_function&oldid=11336