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Lyapunov function

In mathematics, Lyapunov functions are functions which can be used to prove the stability of a certain fixed point in a dynamical system or autonomous differential equation. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions are important to stability theory and control theory.

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-candidate-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. Trial and error is the method to apply, when testing Lyapunov-candidate-functions on some equilibrium.

Definition of a Lyapunov candidate function

Let
$V:mathbb\left\{R\right\}^n to mathbb\left\{R\right\}$
be a scalar function.
$V$ is a Lyapunov-candidate-function if it is a locally positive-definite function, i.e.

$V\left(0\right) = 0 ,$
$V\left(x\right) > 0 quad forall x in Usetminus\left\{0\right\}$

With $U$ being a neighborhood region around $x = 0$

Definition of the equilibrium point of a system

Let

$g : mathbb\left\{R\right\}^n to mathbb\left\{R\right\}^n$
$dot\left\{y\right\} = g\left(y\right) ,$
be an arbitrary autonomous dynamical system with equilibrium point $y^* ,$:
$0 = g\left(y^*\right) ,$

There always exists a coordinate transformation $x = y - y^* ,$, such that:

$dot\left\{x\right\} = g\left(x + y^*\right) = f\left(x\right) ,$
$0 = f\left(x^*\right) quad Rightarrow quad x^* = 0 ,$

So the new system $f\left(x\right)$ has an equilibrium point at the origin.

Basic Lyapunov theorems for autonomous systems

Let

$x^* = 0 ,$
be an equilibrium of the autonomous system
$dot\left\{x\right\} = f\left(x\right) ,$

And let

$dot\left\{V\right\}\left(x\right) = frac\left\{partial V\right\}\left\{partial x\right\} frac\left\{dx\right\}\left\{dt\right\} = nabla V dot\left\{x\right\} = nabla V f\left(x\right)$
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:
$dot\left\{V\right\}\left(x\right) le 0 quad forall x in mathcal\left\{B\right\}$
for some neighborhood $mathcal\left\{B\right\}$, 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:
$dot\left\{V\right\}\left(x\right) < 0 quad forall x in mathcal\left\{B\right\}setminus\left\{0\right\}$
for some neighborhood $mathcal\left\{B\right\}$, then the equilibrium is proven to be locally asymptotically stable.

Globally asymptotically stable equilibrium

If the Lyapunov-candidate-function $V$ is globally positive definite, radially unbounded and the time derivative of the Lyapunov-candidate-function is globally negative definite:
$dot\left\{V\right\}\left(x\right) < 0 quad forall x in mathbb\left\{R\right\}^nsetminus\left\{0\right\},$
then the equilibrium is proven to be globally asymptotically stable.

The Lyapunov-candidate function $V\left(x\right)$ is radially unbounded if

$| x | to infty Rightarrow V\left(x\right) to infty$.