Geodesic
On stability of the inverted pendulum motion with a~vibrating suspension point
Sibirskij žurnal industrialʹnoj matematiki, Tome 21 (2018) no. 4, pp. 39-50.

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

Under study is the stability of the inverted pendulum motion whose suspension point vibrates according to a sinusoidal law along a straight line having a small angle with the vertical. Formulating and using the contracting mapping principle and the criterion of asymptotic stability in terms of solvability of a special boundary value problem for the Lyapunov differential equation, we prove that the pendulum performs stable periodic movements under sufficiently small amplitude of oscillations of the suspension point and sufficiently high frequency of oscillations.
Keywords: inverted pendulum, asymptotic stability, Lyapunov differential equation, contracting mapping principle. 
@article{SJIM_2018_21_4_a3,
     author = {G. V. Demidenko and A. V. Dulepova},
     title = {On stability of the inverted pendulum motion with a~vibrating suspension point},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {39--50},
     publisher = {mathdoc},
     volume = {21},
     number = {4},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2018_21_4_a3/}
}
TY  - JOUR
AU  - G. V. Demidenko
AU  - A. V. Dulepova
TI  - On stability of the inverted pendulum motion with a~vibrating suspension point
JO  - Sibirskij žurnal industrialʹnoj matematiki
PY  - 2018
SP  - 39
EP  - 50
VL  - 21
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJIM_2018_21_4_a3/
LA  - ru
ID  - SJIM_2018_21_4_a3
ER  - 
%0 Journal Article
%A G. V. Demidenko
%A A. V. Dulepova
%T On stability of the inverted pendulum motion with a~vibrating suspension point
%J Sibirskij žurnal industrialʹnoj matematiki
%D 2018
%P 39-50
%V 21
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJIM_2018_21_4_a3/
%G ru
%F SJIM_2018_21_4_a3
G. V. Demidenko; A. V. Dulepova. On stability of the inverted pendulum motion with a~vibrating suspension point. Sibirskij žurnal industrialʹnoj matematiki, Tome 21 (2018) no. 4, pp. 39-50. http://geodesic.mathdoc.fr/item/SJIM_2018_21_4_a3/

[1] Bogolyubov N. N., “Teoriya vozmuschenii v nelineinoi mekhanike”, Sb. tr. In-ta stroitelnoi mekhaniki AN USSR, 14, 1950, 9–34

[2] Kapitsa P. L., “Dinamicheskaya ustoichivost mayatnika pri koleblyuscheisya tochke podvesa”, Zhurn. eksperiment. i teor. fiziki, 21:5 (1951), 588–597

[3] Bogolyubov N. N., Mitropolskii Yu. A., Asimptoticheskie metody teorii nelineinykh kolebanii, Fizmatlit, M., 1963

[4] Mitropolskii Yu. A., Metod usredneniya v nelineinoi mekhanike, Nauk. dumka, Kiev, 1971

[5] Demidenko G. V., Matveeva I. I., “Ob ustoichivosti reshenii kvazilineinykh periodicheskikh sistem differentsialnykh uravnenii”, Sib. mat. zhurn., 45:6 (2004), 1271–1284 | MR | Zbl

[6] Demidenko G. V., Dulina K. M., Matveeva I. I., “Asimptoticheskaya ustoichivost reshenii odnogo klassa nelineinykh differentsialnykh uravnenii vtorogo poryadka s parametrami”, Vestn. YuUrGU. Ser. Mat. modelirovanie i programmirovanie, 2012, no. 14, 39–52 | Zbl

[7] Trenogin V. A., Funktsionalnyi analiz, Fizmatlit, M., 2007

[8] Demidenko G. V., Matveeva I. I., “Ob ustoichivosti reshenii lineinykh sistem s periodicheskimi koeffitsientami”, Sib. mat. zhurn., 42:2 (2001), 332–348 | MR | Zbl