Talk:Lyapunov function: Difference between revisions

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== What's a Lyapunov function? ==

The bad thing about this page is that it doesn't really say. It gives a very precise definition of a "Lyapunov candidate function" and says a lot about Lyapunov candidate functions, but it doesn't bother to define a Lyapunov function, except in a sketchy way near the start. <small><span class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:John Baez|John Baez]] ([[User talk:John Baez|talk]] • [[Special:Contributions/John Baez|contribs]]) 00:40, 29 December 2013 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->

Well it looks like it's defined now, but I'm not sure I trust it... -Sam Winnick

[[Special:Contributions/2607:9880:1A18:10A:3830:F357:B9F5:6B3D|2607:9880:1A18:10A:3830:F357:B9F5:6B3D]] ([[User talk:2607:9880:1A18:10A:3830:F357:B9F5:6B3D|talk]]) 05:38, 26 March 2021 (UTC)

== Definition ==

I’d say this definition is wrong: strictly positive and locally positive definite are completely different. The function <math>-\nabla{V}\cdot g</math> is not strictly positive. In particular it is zero when g is zero. <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/82.35.206.78|82.35.206.78]] ([[User talk:82.35.206.78#top|talk]]) 05:26, 24 February 2021 (UTC)</small> <!--Autosigned by SineBot-->

I think that http://mathworld.wolfram.com/LyapunovFunction.html gives a much better definition than the one provided by http://planetmath.org/?op=getobj&from=objects&id=4386 . The planetmath definition does not specifiy that V is a ''scalar'' function, and it uses an example in only two independent variables (x,y) when in general V is a function in n variables. Also it is proving instability when the much more common utility of a Lyapunov function is to prove stability. I would say that the planetmath "definition" is really more of an example than a definition.

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Please, Mr S.P. Meyn, advertise somewhere else for your book.

<br>[[User:FredTschanz|FredTschanz]] ([[User talk:FredTschanz|talk]]) 10:49, 3 February 2008 (UTC)

== Backslash? ==

What does the backslash mean in the positive-definite requirements? It should be specified, if it's intentional. [[User:LokiClock|LokiClock]] ([[User talk:LokiClock|talk]]) 10:22, 26 September 2009 (UTC)

== Asymptotically stable or globally asymptotically stable? ==

The last phrase in the example:

"This correctly shows that the origin is asymptotically stable",

Shouldn't it say "globally asymptotically stable"?. <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/132.206.73.6|132.206.73.6]] ([[User talk:132.206.73.6|talk]]) 01:29, 20 May 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

== Real definition is missing ==

Please, note that from this article we can't cite precise definition of what is Lyapunov function.

Unfortunately, the same is true for [http://www.answers.com/topic/lyapunov-function Scholarpedia] and [http://mathworld.wolfram.com/LyapunovFunction.html Wolfram].

In this respect Scholarpedia cites partially some unreferenced textbook. Wolfram do not provide definition similarly to this article.

There are various definitions for this functions in the Internet. For example, [http://www.answers.com/topic/lyapunov-function Answers.com].

Please, somebody who knows, compose precise definition for Lyapunov function.

[[User:Arkadi kagan|Arkadi kagan]] ([[User talk:Arkadi kagan|talk]]) 21:13, 8 November 2010 (UTC)

== Example ==

In the given example, in the development of the expression for <math>\dot V(x)</math>, wouldn't it be better to replace the <math>f(x)</math> by a <math>\dot x</math>? I understand that <math>f(x) = \dot x</math>, but since <math>\dot x</math> comes directly from the [[chain rule]], I think the steps would be clearer and more natural that way.

The complete equation would be: <math>\dot V(x) = V'(x) \dot x = \mathrm{sign}(x)\cdot (-x) = -|x|<0.</math>

== Generalized Lyapunov equation ==

The article doesn't make any mention of the generalized lyapunov equation

<math>AXB' + BXA' +Q = 0</math>

[[User:Jeroendv|Jeroendv]] ([[User talk:Jeroendv|talk]]) 15:57, 20 November 2012 (UTC)

== Article, How to think of Lyapunov functions ==

This article is a mess, it holds almost everything, but looks bad.

It should be short and comprehensible, when you look for (mathematical) a definition.

One can think of these functions as energy description of a system to understand what the reasoning behind it is.

1. Lyapunov function exists which holds V'(x)<= 0 -> Lyapunov stability given(it just says that system is bounded by some value[of energy],

it does not need to converge (to 0)

2. V'(x)<0 means that systems energy converges [around some area defined by x] ->(locally) asymptotically converges to stable equilibrium point

(if x=0 is such one) <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/93.129.2.48|93.129.2.48]] ([[User talk:93.129.2.48#top|talk]]) 18:54, 20 February 2017 (UTC)</small> <!--Autosigned by SineBot-->

== Added text needs to be better formatted and integrated into article ==

{{ping|Salih Ertan}} you added the following section to this article but didn't format it very well. Adding a "CORRECTION" in the middle of an article does not fit with the encyclopedic nature of Wikipedia. Please also look at [[WP:CITE]] to learn how to format citations.

The reverted text is copied here:

{{quote|CORRECTION: 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 in 'Civelek, C. (2018). Archives of Control Sciences, volume 28 (LXIV), No. 2, pages 201–222 Doi:10.24425/1234562' though ... According to a lot of applied mathematicians, 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 given in 'Civelek, C.; Cihanbegendi, Ö. (2020). Frontiers of Information Technology & Electronic Engineering, volume 21, pages 629–634, https://doi.org/10.1631/FITEE.1900014'. In addition,...

References

* 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

}} [[User:The-erinaceous-one|The-erinaceous-one]] ([[User talk:The-erinaceous-one|talk]]) 23:56, 18 April 2023 (UTC)