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 ;
The function was introduced by A.M. Lyapunov (see ).
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).
|||A.M. Lyapunov, "Stability of motion" , Acad. Press (1966) (Translated from Russian)|
|||E.A. Barbashin, "Lyapunov functions" , Moscow (1970) (In Russian)|
For additional references see Lyapunov stability.
Lyapunov function. V.M. Millionshchikov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Lyapunov_function&oldid=11336