Geodesic
Stability of equilibrium with respect to a white noise
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 1, pp. 112-121 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

A system of ordinary differential equations with a local asymptotically stable equilibrium is considered. The problem of stability with respect to a persistent perturbation of the white noise type is discussed. The stability with given estimates is proved on a large time interval with a length of the order of the squared reciprocal magnitude of the perturbation. The proof is based on the construction of a barrier function for the Kolmogorov parabolic equation associated with the perturbed dynamical system.
Keywords: dynamical system; random perturbation; stability; parabolic equation; barrier function. 
@article{TIMM_2015_21_1_a10,
     author = {L. A. Kalyakin},
     title = {Stability of equilibrium with respect to a white noise},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {112--121},
     year = {2015},
     volume = {21},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2015_21_1_a10/}
}
TY  - JOUR
AU  - L. A. Kalyakin
TI  - Stability of equilibrium with respect to a white noise
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2015
SP  - 112
EP  - 121
VL  - 21
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TIMM_2015_21_1_a10/
LA  - ru
ID  - TIMM_2015_21_1_a10
ER  - 
%0 Journal Article
%A L. A. Kalyakin
%T Stability of equilibrium with respect to a white noise
%J Trudy Instituta matematiki i mehaniki
%D 2015
%P 112-121
%V 21
%N 1
%U http://geodesic.mathdoc.fr/item/TIMM_2015_21_1_a10/
%G ru
%F TIMM_2015_21_1_a10
L. A. Kalyakin. Stability of equilibrium with respect to a white noise. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 1, pp. 112-121. http://geodesic.mathdoc.fr/item/TIMM_2015_21_1_a10/

[1] Gikhman I.I., Skorokhod A.V., Vvedenie v teoriyu sluchainykh protsessov, Fizmatgiz, M., 1977, 568 pp.

[2] Schuss Z., Theory and applications of stochastic processes, Appl. Math. Sci., 170, Springer, New York, 2010, 468 pp. | DOI | MR | Zbl

[3] Lyapunov M.A., Obschaya zadacha ob ustoichivosti dvizheniya, Gostekhizdat, M.; L., 1950, 472 pp.

[4] Kats I.Ya., Krasovskii N.N., “Ob ustoichivosti sistem so sluchainymi parametrami”, Prikl. matematika i mekhanika, 24:5 (1960), 809–823 | Zbl

[5] Khasminskii R.Z., Ustoichivost sistem differentsialnykh uravnenii pri sluchainykh vozmuscheniyakh ikh parametrov, Nauka, M., 1969, 367 pp. | MR

[6] Khapaev M. M., Asimptoticheskie metody i ustoichivost v teorii nelineinykh kolebanii, Vysshaya shk., M., 1988, 184 pp. | MR

[7] Malkin I.G., Teoriya ustoichivosti dvizheniya, GITTL, M.; L., 1952, 432 pp. | MR

[8] Krasovskii N.N., Nekotorye zadachi teorii ustoichivosti dvizheniya, Fizmatgiz, M., 1959, 211 pp. | MR

[9] Germaidze V.E., Krasovskii N.N., “Ob ustoichivosti pri postoyanno deistvuyuschikh vozmuscheniyakh”, Prikl. matematika i mekhanika, 21:6 (1957), 769–774 | MR

[10] Bogolyubov N.N., Mitropolskii Yu.A., Asimptoticheskie metody v teorii nelineinykh kolebanii, Nauka, M., 1974, 504 pp. | MR

[11] Arnold V.I., Kozlov V.V., Neishtadt A.I., “Matematicheskie aspekty klassicheskoi i nebesnoi mekhaniki”, Sovremennye problemy matematiki. Fundamentalnye napravleniya, v. 3, Itogi nauki i tekhniki, VINITI, M., 1985, 5–290 | MR

[12] Kalyakin L.A., “Asimptoticheskii analiz modelei avtorezonansa”, Uspekhi mat. nauk, 63:5 (2008), 3–72 | DOI | MR | Zbl

[13] Khasminskii R.Z., “Ob ustoichivosti pri postoyanno deistvuyuschikh sluchainykh vozmuscheniyakh”, Teoriya peredachi informatsii. Opoznanie obrazov, cb., Nauka, M., 1965, 74–87

[14] Kats I.Ya., Metod funktsii Lyapunova v zadachakh ustoichivosti i stabilizatsii sistem sluchainoi struktury, Izd-vo UrGAPS, Ekaterinburg, 1998, 222 pp.

[15] Venttsel A.D., Freidlin M.I., Fluktuatsii v dinamicheskikh sistemakh pod deistviem malykh sluchainykh vozmuschenii, Nauka, M., 1979, 424 pp. | MR

[16] Ishii H., Souganidis P.E., “Metastability for parabolic equation with drift: Part 1”, 2013, 32 pp., arXiv: math.1312.5504

[17] Kalyakin L., “Stability under persistent perturbation by white noise”, J. Phys: Conf. Ser, 482 (2014), 1–8

[18] Kalyakin L.A., “Lyapunov functions in barriers for parabolic equations and in stability problems with respect to “white noise””, Russ. J. Math. Phys, 21:4 (2014), 472–483 | DOI | MR

[19] Ladyzhenskaya O.A., Solonnikov V.A., Uraltseva N.N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967, 736 pp. | MR

[20] Oksendal B., Stokhasticheskie differentsialnye uravneniya, Mir, M., 2003, 407 pp.

[21] Krylov N.V., Upravlyaemye protsessy diffuzionnogo tipa, Nauka, M., 1977, 400 pp. | MR

[22] A. Artemyev, D. Vainchtein, A. Neishtadt, L. Zelenyi, “Resonant acceleration of charged particles in the presence of random fluctuations”, Phys. Rev. E, 84 (2011), 046213, 1–5 | DOI

[23] I. Bart, L. Friedland, E. Sarid, A.G. Shagalov, “Autoresonance transition in the presence of noise and self-fields”, Phys. Rev. Letters, 103 (2009), 155001, 1–4

[24] Kalyakin L.A., Sultanov O.A., “Ustoichivost modelei avtorezonansa”, Differents. uravneniya, 49:3 (2013), 279–293 | MR | Zbl