Geodesic
On the stability of third order differential equations
Journal of the Belarusian State University. Mathematics and Informatics, Tome 2 (2018), pp. 25-33.

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

In this paper, we study the problem of stability of the equilibrium of nonlinear ordinary differential equations by the method of semi-definite Lyapunov’s functions. We have identified nonlinear third order differential equations of general form for which the choice of a semi-definite function does not present difficulties. For such equations, sufficient conditions of stability and asymptotic stability (local and global) are obtained. The results of asymptotic stability of the equilibrium coincide with necessary and sufficient conditions in the corresponding linear case. Consequently, they meet generally accepted requirements. The conducted studies show that the use of semi-defined positive functions can give advantages in comparison with the classical method of application of Lyapunov’s definite positive functions.
Keywords: differential equation, equilibrium, stability, semi-definite Lyapunov’s function. 
@article{BGUMI_2018_2_a3,
     author = {B. S. Kalitin},
     title = {On the stability of third order differential equations},
     journal = {Journal of the Belarusian State University. Mathematics and Informatics},
     pages = {25--33},
     publisher = {mathdoc},
     volume = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/BGUMI_2018_2_a3/}
}
TY  - JOUR
AU  - B. S. Kalitin
TI  - On the stability of third order differential equations
JO  - Journal of the Belarusian State University. Mathematics and Informatics
PY  - 2018
SP  - 25
EP  - 33
VL  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BGUMI_2018_2_a3/
LA  - ru
ID  - BGUMI_2018_2_a3
ER  - 
%0 Journal Article
%A B. S. Kalitin
%T On the stability of third order differential equations
%J Journal of the Belarusian State University. Mathematics and Informatics
%D 2018
%P 25-33
%V 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BGUMI_2018_2_a3/
%G ru
%F BGUMI_2018_2_a3
B. S. Kalitin. On the stability of third order differential equations. Journal of the Belarusian State University. Mathematics and Informatics, Tome 2 (2018), pp. 25-33. http://geodesic.mathdoc.fr/item/BGUMI_2018_2_a3/

[1] E. A. Barbashin, “Ob ustoichivosti resheniya odnogo uravneniya tretego poryadka”, Prikladnaya matematika i mekhanika, 16(3) (1952), 629–632 | Zbl

[2] S. N. Shimanov, “Ob ustoichivosti v tselom odnoi nelineinoi sistemy uravnenii”, Uspekhi matematicheskikh nauk, 8(6) (1953), 155–157 | Zbl

[3] V. A. Pliss, “Issledovanie odnogo nelineinogo differentsialnogo uravneniya tretego poryadka”, Doklady AN SSSR, 111(6) (1956), 1078–1180

[4] E. I. Zheleznov, “Ob ustoichivosti v tselom odnoi nelineinoi sistemy trekh uravnenii”, Trudy Uralskogo politekhnicheskogo instituta, 74 (1958), 41–45

[5] A. I. Ogurtsov, “Ob ustoichivosti v tselom reshenii nelineinykh differentsialnykh uravnenii tretego i chetvertogo poryadkov”, Izvestiya vuzov. Matematika, 1(2) (1958), 124–129 | MR | Zbl

[6] A. I. Ogurtsov, “Ob ustoichivosti reshenii dvukh nelineinykh differentsialnykh uravnenii tretego i chetvertogo poryadkov”, Prikladnaya matematika i mekhanika, 23(1) (1959), 179–181 | MR | Zbl

[7] A. I. Ogurtsov, “Ob ustoichivosti reshenii nekotorykh nelineinykh differentsialnykh uravnenii tretego i chetvertogo poryadkov”, Izvestiya vuzov. Matematika, 3 (1959), 200–209 | MR

[8] JOC. Ezeilo, “On the stability of the solutions of a certain differential equations of the third order”, The Quarterly Journal of Mathematics, 11(1) (1960), 64–69 | DOI | MR

[9] E. A. Barbashin, “Funktsii Lyapunova”, Moskva: Nauka, 1970 | MR | Zbl

[10] A. M. Lyapunov, “Obschaya zadacha ob ustoichivosti dvizheniya”, Moskva, Leningrad: Gostekhizdat, 1950 | MR | Zbl

[11] N. G. Bulgakov, B. S. Kalitin, “Obobschenie teorem vtorogo metoda Lyapunova. Teoriya”, Izvestiya AN BSSR. Seriya fizikomatematicheskikh nauk, 3(3) (1978), 32–36 | MR | Zbl

[12] N. G. Bulgakov, B. S. Kalitin, “Obobschenie teorem vtorogo metoda Lyapunova. Primery”, Izvestiya AN BSSR. Seriya fiziko-matematicheskikh nauk, 1(1) (1979), 70–74 | Zbl

[13] B. Kalitine, “Sur la stabilite des ensembles compacts positivement invariants des systemes dynamiques”, RAIRO Automatique/Systems Analysis and Control, 16(3) (1982), 275–286 | MR | Zbl

[14] B. S. Kalitin, “K ustoichivosti kompaktnykh mnozhestv”, Vestnik BGU. Fizika. Matematika. Mekhanika, 3 (1984), 61–62 | Zbl

[15] B. S. Kalitin, “V-ustoichivost i problema Florio – Seiberta”, Differentsialnye uravneniya, 35(4) (1999), 453–463 | Zbl

[16] B. Kalitine, “Sur le theoreme de la stabilite non asymptotique dans la methode directe de Lyapunov”, Comptes Rendus Mathematique, 338(2) (2004), 163–166 | DOI | MR | Zbl

[17] A. S. Andreev, “Ustoichivost neavtonomnykh funktsionalno-differentsialnykh uravnenii”, Ulyanovsk: Izdatelstvo UlGU, 2005

[18] S. V. Pavlikov, “Metod funktsionalov Lyapunova v zadachakh ustoichivosti i stabilizatsii”, Naberezhnye Chelny: Izdatelstvo Instituta upravleniya, 2010

[19] B. S. Kalitin, “Ustoichivost differentsialnykh uravnenii (Metod znakopostoyannykh funktsii Lyapunova)”, Saarbryukken: LAP, 2012

[20] B. S. Kalitin, “Ustoichivost neavtonomnykh differentsialnykh uravnenii”, Minsk: BGU, 2013

[21] B. S. Kalitin, “O problemakh ustoichivosti neavtonomnykh differentsialnykh uravnenii (Metod znakopostoyannykh funktsii Lyapunova)”, Saarbryukken: LAP, 2015

[22] B. S. Kalitin, “O reshenii zadach ustoichivosti pryamym metodom Lyapunova”, Differentsialnye uravneniya, 52(6) (2016), 839–840 | MR

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

[24] B. S. Kalitin, “O reshenii zadach ustoichivosti pryamym metodom Lyapunova”, Izvestiya vuzov. Matematika, 6 (2017), 33–43 | MR | Zbl

[25] N. O. Sedova, “Globalnaya asimptoticheskaya ustoichivost i stabilizatsiya v nelineinoi kaskadnoi sisteme s zapazdyvaniem”, Izvestiya vuzov. Matematika, 11 (2008), 68–79 | Zbl

[26] N. Rouche, P. Habets, M. Laloy, “Stability Theory Liapunov’s Direct Method”, 22, New York: Springer, 1977 | DOI | MR | Zbl

[27] L. Amerio, “Studio asintotico del moto di un punto su una linea chinsa per azione di forze indipendenti dal tempo”, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3e serie, 3(1–4) (1950), 19–57 | MR | Zbl

[28] N. P. Bhatia, G. Szego, “Stability theory of Dynamical systems”, Berlin, Heidelberg: Springer, 1970 | MR | Zbl