Stability of solutions to class of nonlinear systems of integro-differential delay equations

I. I. Matveeva

Čelâbinskij fiziko-matematičeskij žurnal, Tome 9 (2024) no. 4, pp. 609-621.

Résumé

We consider a class of nonlinear systems of nonautonomous differential equations with time-varying concentrated and distributed delays that can be unbounded. Using a special Lyapunov — Krasovskii functional, conditions for exponential stability of the zero solution are obtained. We establish estimates for attraction sets and estimates characterizing stabilization rates of solutions at infinity.

Détail

Keywords: time-varying delay systems, estimates for solutions, exponential stability, attraction sets, Lyapunov — Krasovskii functional.

Bibliographie

[1] El'sgol'ts L.E., Norkin S.B., Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Academic Press, New York, 1973

[2] Hale J., Theory of Functional Differential Equations, Springer, New York, 1977

[3] Korenevskiĭ D.G., Stability of Dynamical Systems under Random Perturbations of Parameters. Algebraic Criteria, Naukova Dumka, Kiev, 1989 (In Russ.)

[4] Dolgiĭ Yu.F., Stability of Periodic Differential-Difference Equations, Ural State University, Ekaterinburg, 1996 (In Russ.)

[5] Kolmanovskii V. B., Myshkis A. D., Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Acad. Publ., Dordrecht, 1999

[6] Azbelev N.V., Selected Works, Institute of Computer Science, Moscow and Izhevsk, 2012

[7] Agarwal R. P., Berezansky L., Braverman E., Domoshnitsky A., Nonoscillation Theory of Functional Differential Equations with Applications, Springer, New York, 2012

[8] Kharitonov V. L., Time-Delay Systems. Lyapunov Functionals and Matrices. Control Engineering, Birkhäuser, New York, 2013

[9] Gil' M. I., Stability of neutral functional differential equations, v. 3, Atlantis Studies in Differential Equations, Atlantis Press, Paris, 2014

[10] Michiels W., Niculescu S. I., Stability, Control, and Computation for Time-Delay Systems. An Eigenvalue-Based Approach, SIAM, Philadelphia, 2014

[11] Park J. H., Lee T. H., Liu Y., Chen J., Dynamic Systems with Time Delays: Stability and Control, Springer, Singapore, 2019

[12] Demidenko G.V., Matveeva I.I., “Stability of solutions to delay differential equations with periodic coefficients of linear terms”, Siberian Mathematical Journal, 48:5 (2007), 824–836

[13] Matveeva I.I., “Estimates for solutions to one class of nonlinear delay differential equations”, Journal of Applied and Industrial Mathematics, 7:4 (2013), 557–566

[14] Demidenko G.V., Matveeva I.I., Skvortsova M.A., “Estimates for solutions to neutral differential equations with periodic coefficients of linear terms”, Siberian Mathematical Journal, 60:5 (2019), 828–841

[15] Matveeva I.I., “Estimates of exponential decay of solutions to one class of nonlinear systems of neutral type with periodic coefficients”, Computational Mathematics and Mathematical Physics, 60:4 (2020), 601–609

[16] Matveeva I.I., “Estimates for solutions to a class of nonautonomous systems of neutral type with unbounded delay”, Siberian Mathematical Journal, 62:3 (2021), 468–481

[17] Matveeva I. I., “Estimates for solutions to one class of nonlinear nonautonomous systems with time-varying concentrated and distributed delays”, Siberian Electronic Mathematical Reports, 18:2 (2021), 1689–1697

[18] Matveeva I.I., “Estimates of the exponential decay of solutions to linear systems of neutral type with periodic coefficients”, Journal of Applied and Industrial Mathematics, 13:3 (2019), 511–518

[19] Hartman Ph., Ordinary Differential Equations, Wiley, New York, 1964

[20] Yskak T., “Estimates for solutions to one class of systems of nonlinear differential equations with distributed delay”, Siberian Electronic Mathematical Reports, 17 (2020), 2204–2215