Review of the works of V.\,N.~Shchennikova on the study of the convergence of nonlinear almost periodic systems by the comparison method

A. A. Kosov; A. V. Shchennikov; E. V. Shchennikova; R. V. Zhalnin; P. A. Shamanaev

Geodesic

Parcourir par

Review of the works of V.\,N.~Shchennikova on the study of the convergence of nonlinear almost periodic systems by the comparison method

A. A. Kosov ; A. V. Shchennikov ; E. V. Shchennikova ; R. V. Zhalnin ; P. A. Shamanaev

Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 2, pp. 175-186.

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

Résumé

The article provides an overview of the studies of V. N. Shchennikov on the problems of almost periodic convergence of nonlinear differential equations' systems. The problem of convergence established by linear or homogeneous approximation is considered. The conditions for convergence of complex systems are given, that are obtained by constructing Lyapunov vector functions and using the comparison method. It should be noted that in the course of the proof constructive estimates are made for the values of small parameters and interconnection functions. The dimensions of the region in which the limiting almost periodic mode is located are also specified. As an application, the problem of convergence in an electric circuit modeled by a second-order nonlinear differential equation with a small parameter is considered. In conclusion, possible applications and unsolved problems for new directions of research, on which V. N. Shchennikov worked in recent years, are discussed.

Mots-clés : convergence
Keywords: almost periodic solutions, Lyapunov vector function, nonlinear system of ordinary differential equations.

@article{SVMO_2019_21_2_a2,
     author = {A. A. Kosov and A. V. Shchennikov and E. V. Shchennikova and R. V. Zhalnin and P. A. Shamanaev},
     title = {Review of the works of {V.\,N.~Shchennikova} on the study of the convergence of nonlinear almost periodic systems by the comparison method},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {175--186},
     publisher = {mathdoc},
     volume = {21},
     number = {2},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2019_21_2_a2/}
}

TY  - JOUR
AU  - A. A. Kosov
AU  - A. V. Shchennikov
AU  - E. V. Shchennikova
AU  - R. V. Zhalnin
AU  - P. A. Shamanaev
TI  - Review of the works of V.\,N.~Shchennikova on the study of the convergence of nonlinear almost periodic systems by the comparison method
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2019
SP  - 175
EP  - 186
VL  - 21
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SVMO_2019_21_2_a2/
LA  - ru
ID  - SVMO_2019_21_2_a2
ER  -

%0 Journal Article
%A A. A. Kosov
%A A. V. Shchennikov
%A E. V. Shchennikova
%A R. V. Zhalnin
%A P. A. Shamanaev
%T Review of the works of V.\,N.~Shchennikova on the study of the convergence of nonlinear almost periodic systems by the comparison method
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2019
%P 175-186
%V 21
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SVMO_2019_21_2_a2/
%G ru
%F SVMO_2019_21_2_a2

A. A. Kosov; A. V. Shchennikov; E. V. Shchennikova; R. V. Zhalnin; P. A. Shamanaev. Review of the works of V.\,N.~Shchennikova on the study of the convergence of nonlinear almost periodic systems by the comparison method. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 2, pp. 175-186. http://geodesic.mathdoc.fr/item/SVMO_2019_21_2_a2/

Bibliographie
Cité par

[1] V. I. Zubov, Oscillations in nonlinear and controlled systems, Sudpromgiz, Leningrad, 1962, 631 pp. (In Russ.)

[2] V. M. Matrosov, “To the theory of motion stability”, Journal of Applied Mathematics and Mechanics, XXVI:6 (1962), 992–1000 (In Russ.)

[3] V. M. Matrosov, L. Yu. Anapol'skiy, S. N. Vasil'ev, Comparison method in mathematical systems theory, Nauka Publ., Novosibirsk, 1980, 480 pp. (In Russ.)

[4] V. M. Matrosov, “The comparison principle with a vector-valued Lyapunov function. IV”, Differentsialnye uravneniya, 5:12 (1969), 2129–2143 (In Russ.) | MR | Zbl

[5] V. M. Matrosov, “Method of Lyapunov vector functions in systems with feedback”, Autom. Remote Control, 33:9 (1972), 1458–1468 | Zbl

[6] V. N. Shchennikov, “The convergence phenomenon of a certain nonlinear system”, Differentsialnye uravneniya, 8:4 (1972), 737–739 (In Russ.) | MR

[7] V. N. Shchennikov, “On theorems on the existence of almost periodic solutions in nonlinear systems of differential equations”, Functional analysis and questions of the qualitative theory of differential equations, MRSU, Saransk, 1976, 158–161 (In Russ.)

[8] V. N. Shchennikov, “Investigation of convergence in a nonautonomous differential system by means Lyapunov vector-functions”, Differentsialnye uravneniya, 19:11 (1983), 1902–1907 (In Russ.) | MR

[9] V. N. Shchennikov, “The convergence phenomenon in complex systems of differential equations”, Differentsialnye uravneniya, 20:9 (1984), 1566–1571 (In Russ.) | MR

[10] V. N. Shchennikov, “Stability under continuous disturbances”, Autom. Remote Control, 46 (1985), 197–200

[11] V. N. Shchennikov, “Investigation of an almost periodic state of a nonlinear controllable system”, Differentsialnye uravneniya, 22:12 (1986), 2182–2183 (In Russ.) | MR

[12] V. N. Shchennikov, “On the theory of forced almost periodic oscillations in nonlinear controlled systems”, Matematicheskoe modelirovanie, 7:5 (1995), 29–30 (In Russ.)

[13] A. A. Kosov, V. N. Shchennikov, “On the convergence phenomen in complex almost periodic systems”, Differentsialnye uravneniya, 50:12 (2014), 1571–1581 (In Russ.) | DOI | MR | Zbl

[14] V. S. Elfimov, A. V. Shchennikov, V. N. Shchennikov, “Convergence of operated dynamic systems”, University Proceedings. Volga Region. Physical and Mathematical Sciences, 46:2 (2018), 86–94 (In Russ.)