Talk:LaSalle's invariance principle

About Krasovsky spelling
Name Красовский can be translitirated in different ways: I think first variant is better. Mir76 14:27, 4 January 2007 (UTC)
 * Krasovskii (this spelling was used in his articles)
 * Krasovskiy or Krasovsky (this spelling is closer to Wikipedia, see Romanization of Russian )
 * Krasovskij (this spelling is closer to GOST)

Global asymptotic stability
Shouldn't V(x) also be unbounded for global asymptotic stability? -Roger 04:45, 3 October 2007 (UTC)

Possible Answer:

This equation is included in the Lypaunov function's list of properties:

$$ V(\mathbf x) \to \infty $$, if $$ \mathbf x \to \infty $$

Which I think is an attempt to claim radial un-boundedness, but according to [Khalil 1996: Nonlinear Systems Pg 110] where this theorem is stated there are two things wrong:


 * 1) Radially unbounded is defined as the norm(x) -> infinity, ie. $$ V (\mathbf x)\to \infty $$ as  $$ ||(\mathbf x)|| \to \infty $$
 * 2) Vdot(x) < 0, forall x not equal to zero, not Vdot(x) ≤ 0 as the page previously mentioned says... a corollary to this theorem that ties in LaSalle's invariance principle / theorem allows one to use a positive semidef lypaunov function --Brio50 (talk) 20:34, 29 January 2012 (UTC)

WikiProject class rating
This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 09:57, 10 November 2007 (UTC)

Proposed renaming
It is awfully strange to see this article under Krasovskii-LaSalle principle in the English language Wikipedia. This name for the result is far from standard in the literature (at least in papers/books published in English), where it is commonly known as LaSalle's invariance principle (or simply the invariance principle). Frankly, using Krasovskii-LaSalle is outright confusing; few people acquainted with the subject are likely to search for this (what's worse, Krasovskii is more commonly associated with Krasovskii's method for constructing Lyapunov functions, e.g. see ). Of course, naming conventions do vary, even among different authors working in the same field. I've also seen Barbashin-Krasovskii-LaSalle used in papers and books by Wassim Haddad. This would be more appropriate if historical considerations were really all that important, but is even more tedious and confusing. Hassan Khalil's book on Nonlinear Systems simply refers to the result as LaSalle's theorem, which is also not an ideal name, since it is more ambiguous. I strongly believe this article needs to be renamed to LaSalle's invariance principle or Invariance principle (or perhaps Invariance principle (stability theory)), which should bring it in line with terminology that is commonly accepted and understood in the (English language) applied mathematics community. The history of the result and the various naming conventions should really appear in a separate section in the article. Stablenode (talk) —Preceding undated comment added 18:09, 10 November 2015 (UTC)

To follow up, a quick search on Google Scholar yields: I believe this settles the naming issue. Stablenode (talk) 18:44, 10 November 2015 (UTC)
 * ″Krasovskii-LaSalle″ (About 420 results), ″Krasovskii-LaSalle principle″ (About 320 results), ″Krasovskii-LaSalle invariance principle″ (About 263 results)
 * ″Barbashin-Krasovskii-LaSalle″ (About 57 results), ″Barbashin-Krasovskii-LaSalle principle″ (About 47 results), ″Barbashin-Krasovskii-LaSalle invariance principle″ (About 46 results)
 * ″LaSalle's invariance principle″ (About 11,500 results)

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A "C1 function" should be replaced with "continuously differentiable function"
Please avoid usage of shorthands like the following: "If a {\displaystyle C^{1}}C^{1} function {\displaystyle V(\mathbf {x} )}V(\mathbf x) can be found such that"

Replace with If a continuously-differentiable function {\displaystyle V(\mathbf {x} )}V(\mathbf x) can be found such that

Because C1 depends on context which has not been provided nor linked. — Preceding unsigned comment added by Orangesherbet0 (talk • contribs) 08:29, 2 April 2020 (UTC)

Improve communication skills to a broader audience
Please translate into intuitive sentences and analogies. For inspiration, try reading Strogatz. We're here to help people, not intimidate them. — Preceding unsigned comment added by Orangesherbet0 (talk • contribs) 08:31, 2 April 2020 (UTC)

The global stability theory
The first general theorem on the global stability of nonlinear system with a single equilibrium via the global Lyapunov function was published by Barbashin and Krasovsky in 1952 for the first time. This was a further development of Krasovsky's work on solving the Aizerman problem on absolute stability and rigorous justification of Lurie-Postnikov's ideas on the Lyapunov function construction published in 1944. For the nonlinear systems with discontinuous nonlinearities corresponding global stability theorems were formulated by Gelig and Leonov in 1960-70-x, and for the cylindrical phase space by Leonov in 1970-x. See, e.g., a short survey: Kuznetsov N.V., Lobachev M.Y., Yuldashev M.V., Yuldashev R.V., Kudryashova E.V., Kuznetsova O.A., Rosenwasser E.N., Abramovich S.M., The birth of the global stability theory and the theory of hidden oscillations, 2020 European Control Conference (ECC), 2020, 769-774 (https://dx.doi.org/10.23919/ECC51009.2020.9143726) nk (talk) 05:15, 18 February 2021 (UTC)