Stability of solutions to a class of systems of integro–differential equations

A class of systems of nonlinear integro-differential equations is considered and the exponential stability of the zero solution is studied. With the use of a special Lyapunov–Krasovskii functional, we establish stability conditions, estimates for attraction sets, and estimates characterizing decay rates of solutions at infinity.

1. Elsgolts L. E. and Norkin S. B., Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Acad. Press, New York; London (1973).

2. Hale J. K. Theory of Functional Differential Equations, Springer, New York; Heidelberg; Berlin (1977).

3. Korenevskii D. G., Stability of Dynamical Systems under Random Perturbations of Parameters, Algebraic Criteria [in Russian], Nauk. Dumka, Kiev (1989).

4. Azbelev N. V., Selected Works [in Russian], Inst. Komp’ut. Issled., Moscow; Izhevsk (2012).

5. Dolgii Yu. F., Stability of Periodic Differential-Difference Equations [in Russian], Ural. Gos. Univ., Yekaterinburg (1996).

6. 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).

7. Michiels W. and Niculescu S. I., Stability, Control, and Computation for Time-delay Systems, An Eigenvalue-Based Approach, SIAM, Philadelphia, PA (2014) (Adv. Des. Control; vol. 27).

8. Agarwal R. P., Berezansky L., Braverman E., and Domoshnitsky A., Nonoscillation Theory of Functional Differential Equations with Applications, Springer, New York (2012).

9. Kharitonov V. L., Time-Delay Systems, Lyapunov Functionals and Matrices, Control Engineering, Birkha¨user; Springer, New York (2013).

10. Gil’ M. I., Stability of Neutral Functional Differential Equations, Atlantis Press, Paris (2014) (Atlantis Stud. Differ. Equ.; vol. 3).

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

12. Demidenko G. V. and Matveeva I. I., “Asymptotic properties of solutions to delay differential equations [in Russian],” Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform., 5, No. 3, 20–28 (2005).

13. Demidenko G. V. and Matveeva I. I., “Stability of solutions to delay differential equations with periodic coefficients of linear terms,” Sib. Math. J., 48, No. 5, 824–836 (2007).

14. Demidenko G. V., Matveeva I. I., and Skvortsova M. A., “Estimates for solutions to neutral differential equations with periodic coefficients of linear terms,” Sib. Math. J., 60, No. 5, 828–841 (2019).

15. Matveeva I. I., “Estimates of exponential decay of solutions to one class of nonlinear systems of neutral type with periodic coefficients,” Comput. Math. Math. Phys., 60, No. 4, 601–609 (2020).

16. Matveeva I. I., “Estimates for solutions to a class of nonautonomous systems of neutral type with unbounded delay,” Sib. Math. J., 62, No. 3, 468–481 (2021).

17. Matveeva I. I., “Estimates of solutions for a class of nonautonomous systems of neutral type with concentrated and distributed delays,” Comput. Math. Math. Phys., 64, No. 8, 1796–1808 (2024).

18. 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., 83, 1–22 (2015).

19. Matveeva I. I., “On the exponential stability of solutions of periodic systems of the neutral type with several delays,” Differ. Equ., 53, No. 6, 725–735 (2017).