Lyapunov function: Difference between revisions

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{{Short description|Concept in the analysis of dynamical systems}}{{No footnotes|date=October 2023}}

In the theory of [[ordinary differential equations]] (ODEs), '''Lyapunov functions''', named after [[Aleksandr Lyapunov]], are scalar functions that may be used to prove the stability of an [[equilibrium point|equilibrium]] of an ODE. Lyapunov functions (also called Lyapunov’s second method for stability) are important to [[stability theory]] of [[dynamical system]]s and [[control theory]]. A similar concept appears in the theory of general state -space [[Markov chain]]s, usually under the name Foster–Lyapunov functions.

For certain classes of ODEs, the existence of Lyapunov functions is a necessary and sufficient condition for stability. ThereWhereas there is no general technique for constructing Lyapunov functions for ODEs, however, depending on formulation type, a systematic method to construct Lyapunov functions for ordinary differential equations using their most general form in autonomous cases was given by Prof. Cem Civelek. Though, in many specific cases the construction of Lyapunov functions is known. For instance, according to a lot of applied mathematicians{{Citation needed|date=July 2023|reason=who?}}, for a dissipative gyroscopic system a Lyapunov function could not be constructed. However, using the method expressed in the publication above, even for such a system a Lyapunov function could be constructed as per related article by C. Civelek and Ö. Cihanbegendi. In addition, [[quadratic function|quadratic]] functions suffice for systems with one state;, the solution of a particular [[linear matrix inequality]] provides Lyapunov functions for linear systems, and [[Conservation law (physics)|conservation law]]s can often be used to construct Lyapunov functions for [[physical system]]s.

== Definition ==

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{{main article|Lyapunov stability}}

Let <math>x^* = 0</math> be an equilibrium point of the autonomous system

:<math>\dot{x} = f(x).</math>

and use the notation <math>\dot{V}(x)</math> to denote the time derivative of the Lyapunov-candidate-function <math>V</math>:

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===Locally asymptotically stable equilibrium===

If the equilibrium point is isolated, the Lyapunov-candidate-function <math>V</math> is locally positive definite, and the time derivative of the Lyapunov-candidate-function is locally negative definite:

:<math>\dot{V}(x) < 0 \quad \forall x \in \mathcal{B}(0)\setminus\{0\},</math>

for some neighborhood <math>\mathcal{B}(0)</math> of origin, then the equilibrium is proven to be locally asymptotically stable.

===Stable equilibrium===

If <math>V</math> is a Lyapunov function, then the equilibrium is [[Lyapunov stable]]. The converse is also true, and was proved by [[José Luis Massera]].

===Globally asymptotically stable equilibrium===

If the Lyapunov-candidate-function <math>V</math> is globally positive definite, [[Radially unbounded function|radially unbounded]], the equilibrium isolated and the time derivative of the Lyapunov-candidate-function is globally negative definite:

:<math>\dot{V}(x) < 0 \quad \forall x \in \R ^n\setminus\{0\},</math>

then the equilibrium is proven to be [[Stability theory|globally asymptotically stable]].

The Lyapunov-candidate function <math>V(x)</math> is radially unbounded if

:<math>\| x \| \to \infty \Rightarrow V(x) \to \infty. </math>

(This is also referred to as norm-coercivity.)

The converse is also true,<ref name=Massera1949>{{Citation

| author = Massera, José Luis

| year = 1949

| title = On Liapounoff's conditions of stability

| journal = Annals of Mathematics |series=Second Series

| volume = 50

| issue = 3

| pages = 705–721

| doi = 10.2307/1969558

| mr = 0035354

| jstor = 1969558

}}</ref> and was proved by [[José Luis Massera]] (see also [[Massera's lemma]]).

==Example==

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* {{cite book |first=Joseph |last=La Salle |first2=Solomon |last2=Lefschetz |title=Stability by Liapunov's Direct Method: With Applications |___location=New York |publisher=Academic Press |year=1961 }}

*{{PlanetMath attribution|id=4386|title=Lyapunov function}}

* Civelek, C. (2018). Archives of Control Sciences, volume 28 (LXIV), No. 2, pages 201–222 Doi:10.24425/123456

* Civelek, C.; Cihanbeğendi, Ö. (2020). Frontiers of Information Technology & Electronic Engineering, volume 21, pages 629–634 Doi: 10.1631/FITEE.1900014

==External links==

* [https://web.archive.org/web/20110926230621/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