Geodesic
Stability of Li\'enard equation
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2018), pp. 17-28.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider stability problem of balance on nonlinear differential equations of the second type of the Liénard type. Investigations are carried out by means of constant-sign Lyapunov's functions for problems of stability, asymptotical stability (local and global) and instability. Implicitly we formulate a method of construction of constant-sign functions suitable for solving problems of stability of motion. Special attention is paid to a problem of non-asymptotical stablity, where we demonstrate possibilities of new assertions that rely upon a usage of constant-sign Lyapunov functions.
Keywords: differential equation, balance, stability, Lyapunov's constant-sign function. 
@article{IVM_2018_10_a1,
     author = {B. S. Kalitine},
     title = {Stability of {Li\'enard} equation},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {17--28},
     publisher = {mathdoc},
     number = {10},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2018_10_a1/}
}
TY  - JOUR
AU  - B. S. Kalitine
TI  - Stability of Li\'enard equation
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2018
SP  - 17
EP  - 28
IS  - 10
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2018_10_a1/
LA  - ru
ID  - IVM_2018_10_a1
ER  - 
%0 Journal Article
%A B. S. Kalitine
%T Stability of Li\'enard equation
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2018
%P 17-28
%N 10
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2018_10_a1/
%G ru
%F IVM_2018_10_a1
B. S. Kalitine. Stability of Li\'enard equation. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2018), pp. 17-28. http://geodesic.mathdoc.fr/item/IVM_2018_10_a1/

[1] Liénard A., “Etüde des oscillations autoentretienes”, Rev. Gen. Elec., 23, 901–902; (1928), 946–954

[2] Barbashin E. A., Krasovskii N. N., “Ob ustoichivosti dvizheniya v tselom”, DAN SSSR, 86:3 (1952), 453–456 | Zbl

[3] Lya-Sall Zh., Lefshets S., Issledovanie ustoichivosti pryamym metodom Lyapunova, Mir, M., 1964

[4] Barbashin E. A., Funktsii Lyapunova, Nauka, M., 1970

[5] Amerio L., “Studio asintotico del moto di un punto su una linea chinsa per azione di forze indipendenti dal tempo”, Ann. scuola Norm. Sup. Pisa, 3:3 (1950), 19–57 | MR | Zbl

[6] Amelkin V. V., Lukashevich N. A., Sadovskii A. P., Nelineinye kolebaniya v sistemakh vtorogo poryadka, BGU, Minsk, 1982 | MR

[7] Lyapunov A. M., Obschaya zadacha ob ustoichivosti dvizheniya, Gostekhizdat, M.–L., 1950

[8] Bulgakov N. G., Kalitin B. S., “Obobschenie teorem vtorogo metoda Lyapunova. 1. Teoriya”, Izv. AN BSSR. Ser. fiz.-matem. nauk, 3:3 (1978), 32–36

[9] Kalitine B., “Sur la stabilité des ensembles compacts positivement invariants des systèmes dynamiques”, RAIRO, Automat. Syst. Anal. Control, 16:3 (1982), 275–286 | MR | Zbl

[10] Kalitin B. S., “$B$-ustoichivost i zadacha Florio–Seiberta”, Differents. uravneniya, 35:4 (1999), 453–463 | Zbl

[11] Kalitine B., “Sur le théorème de la stabilité non asymptotique dans la méthode directe de Lyapunov”, C. R. Acad. Sci. Paris, Ser. I, 338:2 (2004), 163–166 | DOI | MR | Zbl

[12] Andreev A. S., Ustoichivost neavtonomnykh funktsionalno-differentsialnykh uravnenii, UlGU, Ulyanovsk, 2005

[13] Pavlikov S. V., Metod funktsionalov Lyapunova v zadachakh ustoichivosti i stabilizatsii, Naberezhnye Chelny, 2010

[14] Kalitin B. S., Ustoichivost differentsialnykh uravnenii (Metod znakopostoyannykh funktsii Lyapunova), LAP Lambert Academic Publishing, Saarbrucken, 2012

[15] Kalitin B. S., O problemakh ustoichivosti neavtonomnykh differentsialnykh uravnenii (Metod znakopostoyannykh funktsii Lyapunova), LAP Lambert Academic Publishing, Saarbrucken, 2015

[16] Kalitin B. S., “K pryamomu metodu Lyapunova dlya poludinamicheskikh sistem”, Matem. zametki, 100:4 (2016), 531–543 | DOI | Zbl

[17] Kalitin B. S., “O reshenii zadach ustoichivosti pryamym metodom Lyapunova”, Izv. vuzov. Matem., 2017, no. 6, 33–43

[18] Sedova N. O., “Globalnaya asimptoticheskaya ustoichivost i stabilizatsiya v nelineinoi kaskadnoi sisteme s zapazdyvaniem”, Izv. vuzov. Matem., 2008, no. 11, 68–79 | MR | Zbl

[19] Rush N., Abets P., Lalua M., Pryamoi metod Lyapunova v teorii ustoichivosti, Mir, M., 1980

[20] Bautin N. N., Leontovich E. A., Metody i priemy kachestvennogo issledovaniya dinamicheskikh sistem na ploskosti, Nauka, M., 1990