Talk:Lyapunov stability

Application
You who wrote this page: are you sure that we cannot apply the Lyapunov stability to forced systems????
 * I did not write the page but know there have been attempts to apply it to forced systems. For example, I.G. Malkin in his book 'Stability of Motion' (Russian but it has been translated to English) proves that a asymptotically stable system remains stable under disturbances which are bounded if the bound is sufficiently small. But there is no estimate of how small.JFB80 (talk) 17:03, 21 December 2010 (UTC)

Overall coments
Hi,

I'm not in the mood to put it all correctly right now, but I'd like at least to comment some details.

First, I've already altered the definition here (I didn't have a login at that time, sorry... when I find out how, I promise to put my IP here), and basically that's the one present now, when it comes to &epsillon; and δ.

But the article is, actually, wrong in the following sense: we're dealing with Lyapunov Stability of Equilibrium Points. There's no need to specify if the system is unforced.

The concept of Lyapunov Stability is, actually, original from the study of trajectories from ODEs. Then, you should define when a solution of a ODE is Lyapunov Stable. It's a rather similar definition to the one on the page, but since the solution need not to be a equilibrium point anymore, you should compare the norm of the difference between y(t) (the solution you're studing) and x(t) as in the article.

As you can see (gotta find the reference to put it in the article too) in LaSalle's book on Stability, if you do make a change of variables based on a "translation by y(t)", then y(t) is the origin and a equilibrium point for the new system, and the definition just given in the article is applied.

Although we do not need to specify the kind of system for the definitions, it's very important to realize that a great part of classic results and theorems applies only to unforced systems (actually, I'd use "conservative" systems, since that's the context I do work with them everyday...)

Well, sorry for anything! This is my first writing here... so I'll try to learn it better how to contribute!

--Rfreire 22:40, May 5, 2005 (UTC)

Hi, in the definition don't we need to mention the system dynamics being considered are ordinary differential equations of the form xdot=f(x) (i.e. not time varying or other dynamics), and that they must be "well behaved" (e.g. f satisfies a Lipschitz condition within a ball of radius epsilon of the origin)? Also you might want to mention the norms involved. -

"entirely in the plane of the two primary bodies" Two points don't define a plane. I think what is meant is the plane of the orbits of the two bodies, which in most cases means the plane of the orbit of the smaller body around the larger. — Preceding unsigned comment added by 2601:644:8D7F:FD10:9C6C:1FEE:454F:3FE9 (talk) 19:18, 19 December 2021 (UTC)

forced systems can be stable ITSOL
Can't Lyapunov stability apply to forced systems, like closed-loop systems for example?

dx/dt - 4*x = f(t)

is unstable when f(t) = 0, but stable when f(t) = -5*x. The closed-loop system in this case is dx/dt + x = 0

It's true that the closed-loop system is unforced mathematically, but from the perspective of the original system, we have a system that was stabilized using the control force.

Equilibrium point
In the 9th line the 0 is not the EP.--Pokipsy76 18:57, 11 February 2006 (UTC)

The Van der Pol oscillator example is Incorrect.
In the Van der Pol oscillator example, we clearly have the origin as an equilibrium point to the system. The problem stems from the "claim" that $$\dot{V}(x)$$ is negative definite. Since $$\dot{V}(x)$$ has no $$x_1$$ contribution, this equation is negative semi-definite, i.e. $$\dot{V}(x) = 0$$ when $$x_2 = 0$$ and $$\forall x_1$$. To prove asymptotic stability, either a new Lyapunov candidate function needs to be considered, or LaSalle's invariance principal needs to be applied.

I agree. To use a Lyapunov function itself to prove asymptotic stability (without resorting to the invariance principle or Barbalat's Lemma), you have to choose a different function. I suggest changing the section to use a simpler first order system. S280Z28 06:06, 26 April 2007 (UTC)

lack of references
well a short bibliography would be useful, wouldn't it?

Lyapunov second theorem on stability
Isn't it also required that $$V(0) = 0$$? The definition given allows $$V(x) = 1/(1+|x|)$$ to prove that $$x(t) = t$$ is asymptotically stable.

Also, my understanding was that such a $$V(x)$$ is called a Lyapunov function, and that the term 'candidate' is used to distinguish functions which have yet to be proven to be Lyapunov functions (candidate Lyapunov functions) from those which have been proven. LachlanA 19:00, 31 January 2007(UTC)

Nonsense. Counter-example given in this section is correct, but the conclusions are not. My point is that V must have global minima in origin. Functions of type W = V + const will also prove stability and they are also sometimes called 'Lyapunov functions'. It is vital to give either some references or full proofs. (or both) 95.67.191.95 (talk) 17:58, 25 November 2010 (UTC)

The course notes I have on this subject (not publicly accessible) state that, given equilibrium $$x_e$$, $$V(x) > V(x_e)$$ for all $$x$$ except $$x_e$$, is a condition for stability. So there must be global minimum in the equilibrium (which is not the case for $$V(x) = 1/(1+|x|)$$) Zeebrommer (talk) 13:24, 14 August 2014 (UTC)

Barbalat's lemma
This looks like a really cute lemma, but does it belong on this page? LachlanA 22:21, 24 February 2007 (UTC)


 * By the way, it is named after Romanian mathematician Ion Barbalat. -Heinrich ⅩⅦ von Bayern (talk) 08:40, 25 June 2015 (UTC)
 * It's commonly sub-chapter in the Lyapunov stability section of control textbooks like Slotine's. I rephrased a bit to make the connection more readily obvious to Lyapunov stability and explicitly cite Slotine who's example was already in the section. Given its probably too small to be its own page, unless other pages refer back to it, I think it is a good section. Treyra (talk) 19:45, 14 June 2024 (UTC)

'Iterated' systems
I never heard the term 'iterated' to refer to 'discrete-time' or just 'discrete' dynamics in the control literature. Though correct, I would suggest that anyone having a sounder background than me should consider replacing 'iterated' with 'discrete-time'. -- Biscay 12:10, 31 August 2007 (UTC).

'the origin is an equilibrium'
(Good article.) Forgive a naive question, but what kind of "equilibrium" is assumed, supposedly WLOG, at the origin? The term is not defined. Do you mean zero sum of forces? Stable orbits? Some other dynamical invariant? —Preceding unsigned comment added by Dratman (talk • contribs) 18:26, 17 August 2008 (UTC)

In the simplest case the origin is much rather 'in' equilibrium, and/or 'is' an equilibrium point. Also, in the simplest case, this is where the first derivative of position i.e. velocity is equal to 0. So, simply put, it is not moving in time and that is all that we mean here

Asymptotic Stability Implies Lyapunov Stability
It is redundant to define asymptotic stability as being Lyapunov stable, since it can be shown that asymptotic stability implies Lyapunov stability. —Preceding unsigned comment added by 130.15.101.152 (talk) 15:49, 18 November 2009 (UTC)


 * Asymptotic stability implies Lyapunov stability only since it is explicitly included in the definition. E.g. the system given in polar coordinates as $$\dot{r} = r(1-r), \dot{\theta} = \sin(\theta/2)$$ has an attractive equilibrium at (x,y)=(1,0) which is not Lyapunov stable. I think this case is impossible for hyperbolic equilibria, but I'm not sure. --132.68.52.136 (talk) 13:28, 8 December 2013 (UTC)

What is an equilibrium?
I can't find it here, nor in the dynamic system page. I'm assuming it's just another name for fixpoint, but some definition would be handy. 62.245.100.121 (talk) 20:39, 1 February 2014 (UTC)

Definition of exponential stability
Exponential stability doesn't require asymptotic stability. If the equilibrium is exponentially stable it is also stable (we can put δ=ε/α) and obviously the solutions from the δ-neighbourhood converge to it. — Preceding unsigned comment added by 91.122.175.225 (talk) 04:14, 25 April 2016 (UTC)

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Problem with the definition?
Point 1 of the definition sort of contradicts the lede, which discusses that this applies for a SMALL epsilon, rather than for ANY epsilon. This becomes a problem in the case of a Lyapunov Orbit which is a special example of a Lissajous Orbit. Using these terms, both have well defined and well bounded values of epsilon.

'Aleksandr Lyapunov' is not a link
Shouldn't the first mention of Aleksandr Lyapunov link to the wikipedia page about him? Like this: Aleksandr Lyapunov

— Preceding unsigned comment added by 121.212.147.109 (talk) 04:30, 3 March 2018 (UTC)

Lyapunov stability in the 3 body problem
The introduction talks about this but there is no description, illustration or reference in the further part of the article. As far as I know there is no such thing: the 3 body problem occurs in a conservative system whereas systems showing Lyapunov stability are non-conservative systems. I deleted the section but was promptly reverted without explanation or reference to literature which seems to me rather impolite and against Wikipedia policy. The heading to this page says 'Encycopedic content must be verifiable through citations to reliable sources' Is it possible to insist on this? JFB80 (talk) 14:33, 7 January 2022 (UTC)
 * Sorry, your edit looked like vandalism since you left your signature in place of a paragraph. I am not an expert on this so I can't comment on the content. Feel free to search for a reference or delete the unreferenced paragraph again (without signatures!). Thank you --Ita140188 (talk) 16:43, 7 January 2022 (UTC)
 * Yes it was a bit messy. But why edit on a subject you are not familiar with? JFB80 (talk) 16:21, 8 January 2022 (UTC)
 * I just deleted it again. The-erinaceous-one (talk) 08:20, 19 April 2023 (UTC)