Lyapunov function

A function defined as follows. Let $ x _ {0} $ be a fixed point of the system of differential equations

$$ \dot{x} = f ( x , t ) $$

(that is, $ f ( x _ {0} , t ) \equiv 0 $), where the mapping $ f ( x , t ) : U \times \mathbf R ^ {+} \rightarrow \mathbf R ^ {n} $ is continuous and continuously differentiable with respect to $ x $( here $ U $ is a neighbourhood of $ x _ {0} $ in $ \mathbf R ^ {n} $). In coordinates this system is written in the form

$$ \dot{x} ^ {i} = f ^ { i } ( x ^ {1} \dots x ^ {n} , t ) ,\ \ i = 1 \dots n . $$

A differentiable function $ V ( x) : U \rightarrow \mathbf R $ is called a Lyapunov function if it has the following properties:

1) $V(x) > 0$ for $x \neq x_{0}$;

2) $V(x_{0}) = 0$;

3)

$$ 0 \geq \frac{d V(x)}{dx} f ( x , t ) = \sum_{i=1 }^ { n } \frac{\partial V ( x ^ {1} \dots x ^ {n} ) }{\partial x ^ {i} } f ^ { i } ( x ^ {1} \dots x ^ {n} , t ) . $$

The function $V(x)$ 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).

Comments

For additional references see Lyapunov stability.

References

[1]	A.M. Lyapunov, "Stability of motion" , Acad. Press (1966) (Translated from Russian)
[2]	E.A. Barbashin, "Lyapunov functions" , Moscow (1970) (In Russian)