Talk:Massera's lemma

Two questions

The article refers to "class-K funnction[s]". What are those?

The article says:

For any continuous function $u(t)$ that satisfies $0\leq u(t)\leq g(t)$ for all $t\geq 0$, there exist positive constants $k_{1}$ and $k_{2}$, independent of $u$, such that

etc. If you write

For any continuous function $u(t)$......, there exists.....

then the form of that sentence implies that whatever is asserted to exist does depend on u. But then it says "independent of u". That is unclear at best. Could it be that what was meant was something like the following?

There exist positive constants k₁, k₂, such that for any continuous function $u(t)$ that satisfies $0\leq u(t)\leq g(t)$, ....

Michael Hardy (talk) 15:44, 16 July 2008 (UTC)

Hi Michael - thanks for the clean up on this page.

Class-K functions are a generalization of the positive-real functions. A continuous function $[0,n)\rightarrow [0,\infty ]$ is said to belong to class K if it is strictly increasing and if $n(0)=0$. There is also a $K_{\infty }$ class - it's when $n=\infty$ and $n(a)\rightarrow \infty$ as $a\rightarrow \infty$. I'm sure there's a formal definition somewhere in group theory books.

As for your question on the confusing wording, it's because I wrote down things from memory - I'll get my nonlinear control book and get back to you on that. --Jiuguang (talk) 19:05, 16 July 2008 (UTC)

Jose Luis Massera

By the way - how sure are we that the lemma is named after Jose Luis Massera? He certainly looks right, but I can't find any sources for this statement. --Jiuguang (talk) 19:51, 16 July 2008 (UTC)

The book of Yoshizawa's review indicates a specific 1956 paper. The review of that paper has a very similar introduction as this article. The book review is MR0208086, and the paper is cited in the article now. JackSchmidt (talk) 18:25, 17 July 2008 (UTC)

This is not my area, but I think in the 1956 paper, Lemma 2 and its use, pp. 195–196, are fairly similar, but perhaps more general. The more verbatim copy appears as the first lemma of section 12 of Massera's 1949 paper. It appears the lemma gained popularity both from the fact that Massera's work was so revolutionary (according to some math reviews and obits), but also from his textbook which I also added to the references section. I don't have a copy handy, so don't have the exact page number. JackSchmidt (talk)

Thanks, Jack - this is very helpful. --Jiuguang (talk) 01:25, 19 July 2008 (UTC)