Geodesic
On the Lyapunov functionals method in the stability problem of Volterra integro-differential equations
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 20 (2018) no. 3, pp. 260-272.

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

In this paper, we consider the problem of applying the method of Lyapunov functionals to investigate the stability of non-linear integro-differential equations, the right-hand side of which is the sum of the components of the instantaneous action and also ones with a finite and infinite delay. The relevance of the problem is the widespread use of such complicated in structure equations in modeling the controllers using integral regulators for mechanical systems, as well as biological, physical and other processes. We develop the Lyapunov functionals method in the direction of revealing the limiting properties of solutions by means of Lyapunov functionals with a semi-definite derivative. We proved the theorems on the quasi-invariance of a positive limit set of bounded solution as well as ones on the asymptotic stability of the zero solution including a uniform one. The results are achieved by constructing a new structure of the topological dynamics of the equations under study. The theorems proved are applied in solving the stability problem of two model systems which are generalizations of a number of known models of natural science and technology.
Keywords: nonlinear systems of integro-differential equations, Lyapounov functional, stability, topological dynamics, limiting equation. 
@article{SVMO_2018_20_3_a0,
     author = {A. S. Andreev and O. A. Peregudova},
     title = {On the {Lyapunov} functionals method in the stability problem of {Volterra} integro-differential equations},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {260--272},
     publisher = {mathdoc},
     volume = {20},
     number = {3},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2018_20_3_a0/}
}
TY  - JOUR
AU  - A. S. Andreev
AU  - O. A. Peregudova
TI  - On the Lyapunov functionals method in the stability problem of Volterra integro-differential equations
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2018
SP  - 260
EP  - 272
VL  - 20
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SVMO_2018_20_3_a0/
LA  - ru
ID  - SVMO_2018_20_3_a0
ER  - 
%0 Journal Article
%A A. S. Andreev
%A O. A. Peregudova
%T On the Lyapunov functionals method in the stability problem of Volterra integro-differential equations
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2018
%P 260-272
%V 20
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SVMO_2018_20_3_a0/
%G ru
%F SVMO_2018_20_3_a0
A. S. Andreev; O. A. Peregudova. On the Lyapunov functionals method in the stability problem of Volterra integro-differential equations. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 20 (2018) no. 3, pp. 260-272. http://geodesic.mathdoc.fr/item/SVMO_2018_20_3_a0/

[1] V. Volterra, Theory of Functionals and Integral and Integro-Differential Equations, Dover, New York, 1959 (In Russ.)

[2] N. N. Krasovskii, Stability of Motion, Standford University Press, Standford, 1963 (In Russ.)

[3] L. Hatvani, “Marachkov type stability conditions for non-autonomous functional differential equations with unbounded right-hand sides”, Electronic Journal of Qualitive Theory of Differential Equations, 64 (2015), 1–11 | MR

[4] A. S. Andreev, “The Lyapunov functionals method in stability problems for functional differential equations”, Automation and Remote Control, 70 (2009), 1438–-1486 | Zbl

[5] T. A. Burton, Volterra Integral and Differential Equations, Mathematics in Science and Engineering, 202, 2nd ed., Elsevier B. V., Amsterdam, 2005 | MR | Zbl

[6] O. A. Peregudova, “Development of the Lyapunov function method in the stability problem for functional-differential equations”, Differential Equations, 44:12 (2008), 1701–-1710 | MR

[7] J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977 | MR | Zbl

[8] C. Corduneanu, and V. Lakshmikantham, “Equations with unbounded delay: a survey”, Nonlinear Analysis, Theory, Methods and Applications, 4 (1980), 831–877 | DOI | MR | Zbl

[9] J. Hale, and J. Kato, “Phase space for retarded equations with infinite delay”, Fukcialaj Ekvacioj, 21 (1978), 11–41 | MR | Zbl

[10] S. Murakami, “Perturbation theorems for functional differential equations with infinite delay via limiting equations”, J. Differential Eqns., 59 (1985), 314–335 | DOI | MR | Zbl

[11] G. Makay, “On the asymptotic stability of the solutions of functional differential equations with infinite delay”, J. Differential Eqns., 108 (1994), 139–151 | DOI | MR | Zbl

[12] V. S. Sergeev, “Resonance Oscillations in Some Systems with Aftereffect”, Journal of Applied Mathematics and Mechanics, 79:5 (2015), 432–439 (In Russ.)

[13] A. S. Andreev, and O. A. Peregudova, “Stabilization of the preset motions of a holonomic mechanical system without velocity measurement”, Journal of Applied Mathematics and Mechanics, 81:2 (2017), 95–105 (In Russ.)

[14] A. Andreev, O. Peregudova, “Non-linear PI regulators in control problems for holonomic mechanical systems”, Systems Science and Control Engineering, 6:1 (2018), 12–19 | DOI

[15] A. S. Andreev, O. A. Peregudova, “Nelineynyye regulyatory v zadache o stabilizatsii polozheniya golonomnoy mekhanicheskoy sistemy”, Journal of Applied Mathematics and Mechanics, 82:2 (2018), 156–176 (In Russ.)

[16] A. S. Andreev, O. A. Peregudova, and S. Y. Rakov, “On Modeling a nonlinear integral regulator on the base of the Volterra equations”, Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva, 18:3 (2016), 8–18 (In Russ.)

[17] Z. Artstein, “Topological dynamics of an ordinary differential equation”, J. Different. Equat., 23:2 (1977), 216–223 | DOI | MR | Zbl

[18] A. S. Andreyev, O. A. Peregudova, “The Comparison Method in Asymptotic Stability Problems,”, Journal of Applied Mathematics and Mechanics, 70:6 (2006), 865–875 | MR | Zbl

[19] N. Rouche, P. Habets, M. Laloy, Stability Theory by Lyapunov’s Direct Method, Springer, New York, 1977 (In Russ.) | MR

[20] F. Chen, “Global asymptotic stability in n-species non-autonomous Lotka-Volterra competitive systems with infinite delays and feedback control”, Applied Mathematics and Computation, 170 (2005), 1452–1468 | DOI | MR | Zbl