Geodesic
Sufficient conditions for the existence of an asymptotic quiescent position in time-delay systems
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 20 (2018) no. 2, pp. 175-186.

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

In the present paper we study motions of time-delay systems. In particular, the case is studied, when the system has zero limit position which may not be an invariant set with respect to initial differential-difference equations. The concept of an asymptotic quiescent position for the trajectories of time-delay systems is introduced. Sufficient conditions for existence of an asymptotic quiescent position and an asymptotic quiescent position in the large are obtained. The method of proof is based on the modification of the second Lyapunov method, which was proposed by Razumikhin. Its idea is to use the classical Lyapunov functions, but to evaluate their derivatives along the solutions of the system not on the entire set of integral curves of the system, but on its certain subset. The article considers examples of non-linear time-delay equations that have an asymptotic quiescent position illustrating the theory being developed.
Keywords: Lyapunov stability, nonlinear time-delay systems, asymptotic stability of quiescent position, asymptotic quiescent position, Lyapunov function, Razumihin's approach. 
@article{SVMO_2018_20_2_a3,
     author = {U. P. Zaranik and S. E. Kuptsova and N. A. Stepenko},
     title = {Sufficient conditions for the existence of an asymptotic quiescent position in time-delay systems},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {175--186},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2018_20_2_a3/}
}
TY  - JOUR
AU  - U. P. Zaranik
AU  - S. E. Kuptsova
AU  - N. A. Stepenko
TI  - Sufficient conditions for the existence of an asymptotic quiescent position in time-delay systems
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2018
SP  - 175
EP  - 186
VL  - 20
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SVMO_2018_20_2_a3/
LA  - ru
ID  - SVMO_2018_20_2_a3
ER  - 
%0 Journal Article
%A U. P. Zaranik
%A S. E. Kuptsova
%A N. A. Stepenko
%T Sufficient conditions for the existence of an asymptotic quiescent position in time-delay systems
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2018
%P 175-186
%V 20
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SVMO_2018_20_2_a3/
%G ru
%F SVMO_2018_20_2_a3
U. P. Zaranik; S. E. Kuptsova; N. A. Stepenko. Sufficient conditions for the existence of an asymptotic quiescent position in time-delay systems. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 20 (2018) no. 2, pp. 175-186. http://geodesic.mathdoc.fr/item/SVMO_2018_20_2_a3/

[1] N. N. Krasovskiy, Some problems of the theory of stability of motions, State Publ.of phys. and math. literature, Moscow, 1959, 221 pp. (In Russ.)

[2] R. Bellman, K. L. Cooke, Differencial-Difference Equations, Mir Publ., Moscow, 1967, 548 pp. (In Russ.) | MR

[3] J. Hale, Theory of functional differential equations, Mir Publ., Moscow, 1980, 300 pp. (In Russ.)

[4] V. I. Zubov, Lectures on theory of control, Nauka Publ., Moscow, 1975, 496 pp. (In Russ.)

[5] V. I. Zubov, “To the theory of linear time-delay systems”, Proc. of Higher Educational Institutions. Mathematics, 6 (1958), 86-95 (In Russ.) | Zbl

[6] V. I. Zubov, Oscillations and waves, Leningrad. State University Publ., Leningrad, 1989, 416 pp. (In Russ.)

[7] S. E. Kuptsova, “An asymptotically invariant sets”, Control Processes and Stability, Proceedings of the 37th Intern. scientific conf. graduate and undergraduate students, eds. A. V. Platonov, N. V. Smirnov, SPbGU Publ., St. Petersburg, 2006, 50-56 (In Russ.)

[8] S. E. Kuptsova, “On the asymptotic behavior of solutions of systems of nonlinear nonstationary differential equations”, Trudy Srednevolzhskogo matematicheskogo obshchestva, 8:1 (2006), 235-243 (In Russ.) | MR

[9] O. G. Tikhomirov, E. V. Temkina, “Asymptotic quiescent position for systems of homogeneous non-autonomous differential equations”, Vestnik SPbGU. Series 10. Applied mathematics, computer science and control processes, 2014, no. 2, 58-65 (In Russ.)

[10] A. V. Ekimov, M. V. Svirkin, “Analysis of asymptotic equilibrium state of differential systems using Lyapunov function method”, International conference “Stability and control processes”, in memory of V. I. Zubov (2015), 45-47

[11] S. E. Kuptsova, “Asymptotic quiescent positions in systems of difference equations”, Sistemy upravleniia i informatsionnye tekhnologii, 56:2 (2014), 67-71 (In Russ.)

[12] S. Yu. Kuptsov, S. E. Kuptsova, U. P. Zaranik, “On asymptotic quiescent position of nonlinear difference systems with perturbations”, International conference “Stability and control processes”, in memory of V. I. Zubov (2015), 20-22

[13] B. S. Razumikhin, “On the stability of systems with a delay”, Prikladnaya matematika i mekhanika, 20:4 (1956), 500-512 (In Russ.)

[14] B. S. Razumikhin, “Application of Lyapunov’s method to problems in the stability of systems with a delay”, Automatika i telemekhanika, 21:6 (1960), 740-749 (In Russ.) | MR | Zbl

[15] V. L. Kharitonov, Time-Delay Systems: Lyapunov Functionals and Matrices, Birkhäuser, Basel, 2013, 311 pp. | MR