- Lyapunov function
In
mathematics , Lyapunov functions are functions which can be used to prove the stability of a certain fixed point in adynamical system orautonomous differential equation . Named after the Russianmathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions are important tostability theory andcontrol 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 system s (e.g.physical system s),conservation law s 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{R}^n o mathbb{R}be a
scalar function .
V is a Lyapunov-candidate-function if it is a locallypositive-definite function , i.e.:V(0) = 0 ,:V(x) > 0 quad forall x in Usetminus{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|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} frac{dx}{dt} = abla V dot{x} = abla V f(x)be the time derivative of the Lyapunov-candidate-function V.
table 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}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::dot{V}(x) < 0 quad forall x in mathcal{B}setminus{0}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,
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}^nsetminus{0},then the equilibrium is proven to be globally asymptotically stable.The Lyapunov-candidate function V(x) is radially unbounded if:x | o infty Rightarrow V(x) o infty .
ee also
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Ordinary differential equation sReferences
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* cite book
author = Khalil, H.K.
year = 1996
title = Nonlinear systems
publisher = Prentice Hall Upper Saddle River, NJ
isbn =
*External links
* [http://www.exampleproblems.com/wiki/index.php/ODELF1 Example] of determining the stability of the equilibrium solution of a system of ODEs with a Lyapunov function
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