LYAPUNOV FUNCTION

'Lyapunov functions' are of interest in mathematics, especially in stability theory.

'Lyapunov functions', named after Aleksandr Mikhailovich Lyapunov, are by definition functions which prove the stability of a certain fixed point in a dynamical system or autonomous differential equation.

Functions which might prove the stability of some equilibrium are called 'Lyapunov-candidate-functions'.
There is no general method to construct or find a Lyapunov-candidate-function which proves the stability of an equilibrium, and the inability to find a Lyapunov function is inconclusive with respect to stability, which means, that not finding a Lyapunov function doesn't mean that the system is unstable.
For dynamical systems (e.g. physical systems), conservation laws can often be used to construct a Lyapunov-candiadte-function.
The 'basic Lyapunov Theorems for autonomous systems' which are directly related to Lyapunov (candidate) functions are a useful tool to prove the stability of an equilibrium of an autonomous dynamical system.

One must be aware, that the basic Lyapunov Theorems for autonomous systems are a 'sufficient, but not necessary' tool to prove the stability of an equilibrium.

Finding a Lyapunov Function for a certain equilibrium might be a matter of luck. And trial and error is the method to apply, when testing Lyapunov-candidate-functions on some equilibrium.

Contents

Definition of a Lyapunov candidate function

Definition of the equilibrium point of a system

Basic Lyapunov theorems for autonomous systems

Stable equilibrium

Locally asymptotically stable equilibrium

Globally asymptotically stable equilibrium

See also

References

External Links

Definition of a Lyapunov candidate function

Let
:$V:mathbb\{R\}^n\; o\; mathbb\{R\}$
be a scalar function.

$V$ is a Lyapunov-candidate-function if it is a locally positive-definite function, i.e.
:$V(0)\; =\; 0\; ,$
:
With $U$ being a neighborhood region around $x\; =\; 0$

Definition of the equilibrium point of a system

Let
:$g\; :\; mathbb\{R\}^n\; o\; mathbb\{R\}^n$
:$dot\{y\}\; =\; g(y)\; ,$
be an arbitrary autonomous dynamical system with equilibrium point $y^$
★ ,:
:$0\; =\; g(y^$
★ ) ,
There always exists a coordinate transformation $x\; =\; y\; -\; y^$
★ ,, such that:
:$dot\{x\}\; =\; g(x\; +\; y^$
★ ) = f(x) ,
:$0\; =\; f(x^$
★ ) quad Rightarrow quad x^
★ = 0 ,
So the new system $f(x)$ has an equilibrium point at the origin.

Basic Lyapunov theorems for autonomous systems

:Main articles: Lyapunov stability

Let
:$x^$
★ = 0 ,
be an equilibrium of the autonomous system
:$dot\{x\}\; =\; f(x)\; ,$
And let
:
abla V dot{x} =
abla V f(x)
be the time derivative of the Lyapunov-candidate-function $V$.

Stable equilibrium

If the Lyapunov-candidate-function $V$ is locally positive definite and the time derivative of the Lyapunov-candidate-function is locally negative semidefinite:
:
for some neighborhood $mathcal\{B\}$, then the equilibrium is proven to be 'stable'.

Locally asymptotically stable equilibrium

If the Lyapunov-candidate-function $V$ is locally positive definite and the time derivative of the Lyapunov-candidate-function is locally negative definite:
:
for some neighborhood $mathcal\{B\}$, then the equilibrium is proven to be 'locally asymptotically stable'.

Globally asymptotically stable equilibrium

If the Lyapunov-candidate-function $V$ is globally positive definite and the time derivative of the Lyapunov-candidate-function is globally negative definite:
:
then the equilibrium is proven to be 'globally asymptotically stable'.

See also

★ Lyapunov stability

★ ordinary differential equations

References

★

★

External Links

★ Example of determining the stability of the equilibrium solution of a system of ODEs with a Lyapunov function