Lyapunov function: Definition from Answers.com

Lyapunov fractal with the sequence AAAABBB

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. A similar concept appears in the theory of general state space Markov Chains, usually under the name Lyapunov-Foster functions.

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. As the areas of equal stability often follow lines in 2D, the computer generated images of Lyapunov exponents look nice and are very popular.

1 Definition of a Lyapunov candidate function
2 Definition of the equilibrium point of a system
3 Basic Lyapunov theorems for autonomous systems
4 Example
5 See also
6 References
7 External links

Definition of a Lyapunov candidate function

Let

$V:\mathbb{R}^n \to \mathbb{R}$

be a scalar function.
$V$ is a Lyapunov-candidate-function if it is a locally positive-definite function, i.e.

$V(0) = 0 \,$

$V(x) > 0 \quad \forall x \in U\setminus\{0\}$

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

Definition of the equilibrium point of a system

Let

$g : \mathbb{R}^n \to \mathbb{R}^n$

$\dot{y} = g(y) \,$

be an arbitrary autonomous dynamical system with equilibrium point $y^* \,$ :

$0 = g(y^*) \,$

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

$\dot{x} = g(x + y^*) = f(x) \,$

$f(0) = 0 \,$

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

Basic Lyapunov theorems for autonomous systems

Main article: Lyapunov stability

Let

$x^* = 0 \,$

be an equilibrium of the autonomous system

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

And let

$\dot{V}(x) = \frac{\partial V}{\partial x}\cdot \frac{dx}{dt} = \nabla V \cdot \dot{x} = \nabla V\cdot f(x)$

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:

$\dot{V}(x) \le 0 \quad \forall x \in \mathcal{B}\setminus\{0\}$

for some neighborhood $\mathcal{B}$ of $0$ , 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:

$\dot{V}(x) < 0 \quad \forall x \in \mathcal{B}\setminus\{0\}$

for some neighborhood $\mathcal{B}$ of $0$ , 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:

$\dot{V}(x) < 0 \quad \forall x \in \mathbb{R}^n\setminus\{0\},$

then the equilibrium is proven to be globally asymptotically stable.

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

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

Example

Consider the following differential equation on $\mathbb{R}$ :

$\dot x = -x.$

The velocity vector points always towards the origin, so the distance from it decreases with time and is a natural candidate for a Lyapunov function. Take $V (x) = | x |$ on $\mathbb{R}\setminus\{0\}$ . Then

$\dot V(x) = V'(x) f(x) = \mathrm{sign}(x)\cdot (-x) = -|x|<0.$

This correctly shows that the origin is asymptotically stable.

References

Weisstein, Eric W., "Lyapunov Function" from MathWorld.
Khalil, H.K. (1996). Nonlinear systems. Prentice Hall Upper Saddle River, NJ.
This article incorporates material from Liapunov function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

External links

Example of determining the stability of the equilibrium solution of a system of ODEs with a Lyapunov function
Some Lyaponov diagrams

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Lyapunov function

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