Module 4 extra - Publication data Advanced Topics in...

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6 Stability, Told by Its Developers Antonio Lor´ ıa and Elena Panteley C.N.R.S., Laboratoire des Signaux et Syst` emes, Sup´ elec, 3, Rue Joliot Curie, 91192 Gif-sur-Yvette, France. E-mail:, “The authors of the present manuscript would like to insist on the fact that only the attentive reading of the original documents can contribute to correct certain errors endlessly repeated by diFerent authors.” J. J. Samueli & J. C. Boudenot 1 6.1 Introduction 6.1.1 About the Chapter Lyapunov stability theory is probably the most useful qualitative method to study the behaviour of dynamical systems; it beneFts from at least 75 years of sustained development. It started with the memoir of A. M. Lyapunov [32], published in a Western 2 language in [33], and, starting with the 1930s, many reFnements to this stability theory have been established. The purpose of this chapter is to present basic deFnitions and theorems on stability, mostly on Lyapunov stability, through a concise and modest his- torical survey: it contains a short account of statements made by Lagrange, Lyapunov and other mathematicians: the developers of stability theory. With these notes, we intend to bring some clariFcations to important as- pects of stability theory which, otherwise, have been somewhat obscured due 1 Translated from H. Poincar´ e (1854-1912), physicien , Editions Ellipses: Paris, 2005. The citation is taken from the epilogue of the mentioned biography of the last universalist –as his biographers call H. Poincar´ e. The authors give interesting evidence of H. Poincar´ e’s shared discovery – with Lorentz – of restrained relativity cf. Comptes Rendus de l’Acad´ emie des Sciences, Paris 9th/June/1905. 2 The quali±er “Western” is used here to refer to European languages other than Russian as well as to non-Soviet nationals.
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200 Antonio Lor´ ıa and Elena Panteley to not always accurate translations from Russian – on occasions, double – into English and inexact “recursive” citations, i.e. citations made by an author A of the work of an author B, based on a text written by a tertiary author C 3 . In contrast, unless explicitly mentioned, our citations are in most cases from direct sources and, to avoid further ambiguity, we take special care in citing the exact formulations of concepts introduced in early literature and translations are made with a maximum of Fdelity, keeping at best the origi- nal words, the mathematical notations 4 , the numbering of equations, etc. We have taken special care in the accuracy of references; in particular, titles are original – Russian titles are phonetically transcripted from Cyrillic characters and translated. When considered necessary, comments are made to explain certain statements in “modern” language. We also emphasise that the chap- ter is written as a (historical) survey and not as a tutorial, i.e. we assume that the reader is familiar with the basic Lyapunov theory (main theorems, basic deFnitions, invariance principles etc. ). We hope with this brief histori-
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This note was uploaded on 05/06/2010 for the course ECE 514 taught by Professor Chaoukit.abdallah during the Spring '09 term at University of New Brunswick.

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Module 4 extra - Publication data Advanced Topics in...

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