Abstract
We consider periodic systems of delay equations. Exponential stability conditions for the zero solution are pointed out by using the Lyapunov-Krasovskiĭ functional. We give some estimates that characterize the decay rate of solutions at infinity.
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References
El’sgol’ts L. E. and Norkin S. B., Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Academic Press, New York and London 1973).
Hale J. K., Theory of Functional Differential Equations, Springer-Verlag, New York, Heidelberg, and Berlin 1977).
Korenevskiĭ D. G., Stability of Dynamical Systems Under Random Perturbations of Their Parameters. Algebraic Criteria [Russian], Naukova Dumka, Kiev 1989).
Azbelev N. V., Maksimov V. P., and Rakhmatullina L. F., Introduction to the Theory of Functional-Differential Equations [Russian], Nauka, Moscow 1991).
Dolgiĭ Yu. F., Stability of Periodic Differential-Difference Equations [Russian], Izdat. Ural Univ., Ekaterinburg 1996).
Kolmanovskii V. B. and Myshkis A. D., Introduction to the Theory and Applications of Functional-Differential Equations, Kluwer Acad. Publ., Dordrecht 1999) (Math. Appl.; vol. 463).
Gu K., Kharitonov V. L., and Chen J., Stability of Time-Delay Systems, Birkhäuser, Boston 2003).
Agarwal R. P., Berezansky L., Braverman E., and Domoshnitsky A., Nonoscillation Theory of Functional Differential Equations with Applications, Springer-Verlag, New York 2012).
Gil’ M. I., Stability of Neutral Functional Differential Equations, Atlantis Press, Paris 2014) (Atlantis Stud. Differ. Equ.; vol. 3).
MacDonald N., Biological Delay Systems: Linear Stability Theory, Camb. Univ. Press, Cambridge 1989) (Camb. Stud. Math. Biol.; vol. 8).
Kuang Y., Delay Differential Equations with Applications in Population Dynamics, Acad. Press, Boston 1993) (Math. Sci. Eng.; vol. 191).
Erneux T., Applied Delay Differential Equations, Springer-Verlag, New York 2009) (Surv. Tutorials Appl. Math. Sci.; vol. 3).
Demidenko G. V. and Matveeva I. I., “Stability of solutions to delay differential equations with periodic coefficients of linear terms,” Sib. Math. J., vol. 48, no. 5, 824–836 (2007).
Matveeva I. I., “Estimates of solutions to a class of systems of nonlinear delay differential equations,” J. Appl. Ind. Math., vol. 7, no. 4, 557–566 (2013).
Demidenko G. V. and Matveeva I. I., “On estimates of solutions to systems of differential equations of neutral type with periodic coefficients,” Sib. Math. J., vol. 55, no. 5, 866–881 (2014).
Demidenko G. V. and Matveeva I. I., “Estimates for solutions to a class of time-delay systems of neutral type with periodic coefficients and several delays,” Electron. J. Qual. Theory Differ. Equ., vol. 2015, no. 83, 1–22 (2015).
Demidenko G. V. and Matveeva I. I., “On stability of solutions to linear systems with periodic coefficients,” Sib. Math. J., vol. 42, no. 2, 282–296 (2001).
Daleckiĭ Ju. L. and Kreĭn M. G., Stability of Solutions to Differential Equations in Banach Space, Amer. Math. Soc., Providence 1974).
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The author was supported by the Russian Foundation for Basic Research (Grant 16–01–00592).
Novosibirsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 58, No. 2, pp. 344–352, March–April, 2017; DOI: 10.17377/smzh.2017.58.208.
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Matveeva, I.I. On exponential stability of solutions to periodic neutral-type systems. Sib Math J 58, 264–270 (2017). https://doi.org/10.1134/S0037446617020082
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DOI: https://doi.org/10.1134/S0037446617020082