Editing Lyapunov stability (section)

==History==
Lyapunov stability is named after [[Aleksandr Mikhailovich Lyapunov]], a Russian mathematician who defended the thesis ''The General Problem of Stability of Motion'' at Kharkov University in 1892.<ref name=lyapunov>[[Aleksandr Lyapunov|Lyapunov, A. M.]] ''The General Problem of the Stability of Motion'' (In Russian), Doctoral dissertation, Univ. Kharkiv 1892 English translations: (1) ''Stability of Motion'', Academic Press, New-York & London, 1966 (2) ''The General Problem of the Stability of Motion'', (A. T. Fuller trans.) Taylor & Francis, London 1992. Included is a biography by Smirnov and an extensive bibliography of Lyapunov's work.</ref> A. M. Lyapunov was a pioneer in successful endeavors to develop a global approach to the analysis of the stability of nonlinear dynamical systems by comparison with the widely spread local method of linearizing them about points of equilibrium. His work, initially published in Russian and then translated to French, received little attention for many years. The mathematical theory of stability of motion, founded by A. M. Lyapunov, considerably anticipated the time for its implementation in science and technology. Moreover Lyapunov did not himself make application in this field, his own interest being in the stability of rotating fluid masses with astronomical application. He did not have doctoral students who followed the research in the field of stability and his own destiny was terribly tragic because of his suicide in 1918 {{Citation needed|reason=Cannot find a source. A different Lyapunov (Sergei Lyapunov) is affected by the Russian Revolution and could be a confusion here.|date=June 2019}}. For several decades the theory of stability sank into complete oblivion.  The Russian-Soviet mathematician and mechanician [[Nikolay Gur'yevich Chetaev]] working at the Kazan Aviation Institute in the 1930s was the first who realized the incredible magnitude of the discovery made by A. M. Lyapunov. The contribution to the theory made by N. G. Chetaev<ref>Chetaev, N. G. On stable trajectories of dynamics, Kazan Univ Sci Notes, vol.4 no.1 1936; The Stability of Motion, Originally published in Russian in 1946 by ОГИЗ. Гос. изд-во технико-теорет. лит., Москва-Ленинград.Translated by Morton Nadler, Oxford, 1961, 200 pages.</ref> was so significant that many mathematicians, physicists and engineers consider him Lyapunov's direct successor and the next-in-line scientific descendant in the creation and development of the mathematical theory of stability.

The interest in it suddenly skyrocketed during the [[Cold War (1953–62)|Cold War]] period when the so-called "Second Method of Lyapunov" (see below) was found to be applicable to the stability of aerospace [[guidance system]]s which typically contain strong nonlinearities not treatable by other methods. A large number of publications appeared then and since in the control and systems literature.<ref name=letov>{{cite book |last=Letov |first=A. M. |title=Устойчивость нелинейных регулируемых систем |trans-title=Stability of Nonlinear Control Systems |language=ru |location=Moscow |year=1955 |publisher=Gostekhizdat }} English tr. Princeton 1961</ref><ref name=rudolf1960>{{cite journal |author-link=Rudolf E. Kálmán |last1=Kalman |first1=R. E. |last2=Bertram |first2=J. F |title= Control System Analysis and Design Via the "Second Method" of Lyapunov: I—Continuous-Time Systems|journal= Journal of Basic Engineering|volume= 82|issue=2 |year=1960 |pages=371–393 |doi=10.1115/1.3662604 }}</ref><ref name=lasalle>{{cite book |last1=LaSalle |first1=J. P.|author1-link=Joseph P. LaSalle |author2-link=Solomon Lefschetz |last2=Lefschetz |first2=S. |title=Stability by Lyapunov's Second Method with Applications |location=New York |year=1961 |publisher=Academic Press }}</ref><ref name=parks1962>{{cite journal |last=Parks |first=P. C. |title=Liapunov's method in automatic control theory |journal=Control |volume=I Nov 1962 II Dec 1962 |year=1962 }}</ref><ref name=rudolf1963>{{cite journal |last=Kalman |first=R. E. |title=Lyapunov functions for the problem of Lur'e in automatic control |journal=[[Proceedings of the National Academy of Sciences of the United States of America|Proc Natl Acad Sci USA]] |year=1963 |volume=49 |issue=2 |pages=201–205 |doi= 10.1073/pnas.49.2.201|pmc=299777 |pmid=16591048|bibcode=1963PNAS...49..201K |doi-access=free }}</ref>
More recently the concept of the [[Lyapunov exponent]] (related to Lyapunov's First Method of discussing stability) has received wide interest in connection with [[chaos theory]]. Lyapunov stability methods have also been applied to finding equilibrium solutions in traffic assignment problems.<ref name=smith>{{cite journal |last1=Smith |first1=M. J. |last2=Wisten |first2=M. B. |title=A continuous day-to-day traffic assignment model and the existence of a continuous dynamic user equilibrium |journal=Annals of Operations Research |volume=60 |issue=1 |pages=59–79 |year=1995 |doi=10.1007/BF02031940 |s2cid=14034490 }}</ref>