Talk:Lyapunov function

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. — Preceding unsigned comment added by John Baez (talk • contribs) 00:40, 29 December 2013 (UTC)

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

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 $$-\nabla{V}\cdot g$$ is not strictly positive. In particular it is zero when g is zero. — Preceding unsigned comment added by 82.35.206.78 (talk) 05:26, 24 February 2021 (UTC)

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.


 * I agree the definition in this article is pretty abysmal - as are most mathematical definitions on Wikipedia. The concept of a Lyapunov Function could be explained with much more clarity, and without sacrificing precision. While the definition given here is precise and technically correct, the size of the audience capable of understanding and appreciating it - and furthermore the size of the audience it would actually benefit - is almost negligible. What then, is the point of having an article which benefits only a miserable fraction of the entire population, and an even more insignificant portion of the mathematical community? Doesn't that defeat the whole purpose of having a Wikipedia article in the first place?


 * Definitions such as these may be appropriate for Mathworld, but they don't belong in an open, global community such as Wikipedia. Is anyone aware that persons other than graduate students and elitist Mathematicians reference these articles? Most of the mathematical definitions I've read on here are written in the most advanced, arcane way possible... heaven forbid we ever write something accessible to the "laymen" out there who seek useful descriptions for the mathematical entities they encounter.

For me it is important, that an article is mathematically clear. I don't see the point of having easy to read articles which do not take into consideration some details which may become important and are of course not that easy to understand. I changed the following: - "Lypaunov functions can be used to prove the stability or instability of fixed points in dynamical systems and autonomous differential equations." This is wrong, a Lyapunov function by definition proves the stability of a certain equilibrium. Otherwise it is a Lyapunov candidate funciton. And there is no way to prove instability with a Lyapunov function.

Further on, I introduced a definition of the Lyapunov-candidate-function and a (as i hope) clear version of the "Basic Lyapunov theorems for autonomous systems", which can be used to prove stability of an equilibrium point of such a system. (FredTschanz 16:40, 4 July 2007 (UTC))

External link
What is the point of having an external link to a describtion of a book, which actually doesn't cover the matter. There are probably thousands of books which use Lyapunov theory. If every author would set a link to his book, informational links would get lost in link-spam. Stochastic theory of Markov-chains certainly uses Lyapunov's theorems, but Lyapunov's theory is not relatet to stochastic theory in particular.

If I'm looking for a book about something, I'm not going to search in Wikipedia. If I need further infos about some matter, I might be happy to find useful references to books which focus on the matter. Please, Mr S.P. Meyn, advertise somewhere else for your book. 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. 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"?. —Preceding unsigned comment added by 132.206.73.6 (talk) 01:29, 20 May 2010 (UTC)

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 Scholarpedia and 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, Answers.com.

Please, somebody who knows, compose precise definition for Lyapunov function. Arkadi kagan (talk) 21:13, 8 November 2010 (UTC)

Example
In the given example, in the development of the expression for $$\dot V(x)$$, wouldn't it be better to replace the $$f(x)$$ by a $$\dot x$$? I understand that $$f(x) = \dot x$$, but since $$\dot x$$ comes directly from the chain rule, I think the steps would be clearer and more natural that way. The complete equation would be: $$\dot V(x) = V'(x) \dot x = \mathrm{sign}(x)\cdot (-x) = -|x|<0.$$

Generalized Lyapunov equation
The article doesn't make any mention of the generalized lyapunov equation $$AXB' + BXA' +Q = 0$$

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) — Preceding unsigned comment added by 93.129.2.48 (talk) 18:54, 20 February 2017 (UTC)

Added text needs to be better formatted and integrated into article
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: "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 The-erinaceous-one (talk) 23:56, 18 April 2023 (UTC)
 * 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"