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).

References

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