phone +7 (3412) 91 60 92
mail imi@udsu.ru
ru-RU Russian
Home » Issues » Archive of Issues » 2024 - 2 issue » pp. 222-247

Archive of Issues

Russia; Tunisia Izhevsk; Sfax
Year
2024
Volume
34
Issue
2
Pages
222-247
<<
>>
Section Mathematics
Title On the stability in variation of non-autonomous differential equations with perturbations
Author(-s) Hammami M.A.a, Hamlili R.a, Zaitsev V.A.b
Affiliations University of Sfaxa, Udmurt State Universityb
Abstract In this paper, we investigate the problem of stability in variation of solutions for nonautonomous differential equations. Some new sufficient conditions for the asymptotic or exponential stability for some classes of nonlinear time-varying differential equations are presented by using Lyapunov functions that are not necessarily smooth. The proposed approach for stability analysis is based on the determination of the bounds that characterize the asymptotic convergence of the solutions to a certain closed set containing the origin. Furthermore, some illustrative examples are given to prove the validity of the main results.
Keywords nonautonomous differential equations, perturbation, Lyapunov functions, asymptotic stability
UDC 517.9
MSC 34D20, 34D10, 34D05
DOI 10.35634/vm240204
Received 19 October 2023
Language English
Citation Hammami M.A., Hamlili R., Zaitsev V.A. On the stability in variation of non-autonomous differential equations with perturbations, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2024, vol. 34, issue 2, pp. 222-247.
References
  1. Alekseev V.M. An estimate for the perturbations of the solutions of ordinary differential equations. I, Vestnik Moskovskogo Universiteta. Ser. Matematicheskaya, 1961, no. 2, pp. 28–36 (in Russian).
  2. Athanassov Z.S. Perturbation theorems for nonlinear systems of ordinary differential equations, Journal of Mathematical Analysis and Applications, 1982, vol. 86, issue 1, pp. 194–207. https://doi.org/10.1016/0022-247x(82)90264-5
  3. BenAbdallah A., Hammami M.A. On the output feedback stability for non-linear uncertain control systems, International Journal of Control, 2001, vol. 74, issue 6, pp. 547–551. https://doi.org/10.1080/00207170010017383
  4. BenAbdallah A., Dlala M., Hammami M.A. Exponential stability of perturbed nonlinear systems, Nonlinear Dynamics and Systems Theory, 2005, vol. 5, issue 4, pp. 357–367.
  5. BenAbdallah A., Dlala M., Hammami M.A. A new Lyapunov function for stability of time-varying nonlinear perturbed systems, Systems and Control Letters, 2007, vol. 56, issue 3, pp. 179–187. https://doi.org/10.1016/j.sysconle.2006.08.009
  6. BenAbdallah A., Ellouze I., Hammami M.A. Practical exponential stability of perturbed triangular systems and a separation principle, Asian Journal of Control, 2011, vol. 13, issue 3, pp. 445–448. https://doi.org/10.1002/asjc.325
  7. Ben Hamed B., Ellouze I., Hammami M.A. Practical uniform stability of nonlinear differential delay equations, Mediterranean Journal of Mathematics, 2011, vol. 8, issue 4, pp. 603–616. https://doi.org/10.1007/s00009-010-0083-7
  8. Ben Makhlouf A., Hammami M.A. A nonlinear inequality and application to global asymptotic stability of perturbed systems, Mathematical Methods in the Applied Sciences, 2015, vol. 38, issue 12, pp. 2496–2505. https://doi.org/10.1002/mma.3236
  9. Brauer F., Strauss A. Perturbations of nonlinear systems of differential equations, III, Journal of Mathematical Analysis and Applications, 1970, vol. 31, issue 1, pp. 37–48. https://doi.org/10.1016/0022-247x(70)90118-6
  10. Brauer F. Perturbations of nonlinear systems of differential equations, IV, Journal of Mathematical Analysis and Applications, 1972, vol. 37, issue 1, pp. 214–222. https://doi.org/10.1016/0022-247x(72)90269-7
  11. Brauer F. Nonlinear differential equations with forcing terms, Proceedings of the American Mathematical Society, 1964, vol. 15, no. 5, pp. 758–765. https://doi.org/10.1090/s0002-9939-1964-0166452-8
  12. Caraballo T., Hammami M.A., Mchiri L. Practical stability of stochastic delay evolution equations, Acta Applicandae Mathematicae, 2016, vol. 142, issue 1, pp. 91–105. https://doi.org/10.1007/s10440-015-0016-3
  13. Caraballo T., Hammami M.A., Mchiri L. Practical exponential stability of impulsive stochastic functional differential equations, Systems and Control Letters, 2017, vol. 109, pp. 43–48. https://doi.org/10.1016/j.sysconle.2017.09.009
  14. Corless M., Glielmo L. On the exponential stability of singularly perturbed systems, SIAM Journal on Control and Optimization, 1992, vol. 30, issue 6, pp. 1338–1360. https://doi.org/10.1137/0330071
  15. Corless M., Garofalo F., Glielmo L. Robust stabilization of singularly perturbed nonlinear systems, International Journal of Robust and Nonlinear Control, 1993, vol. 3, issue 2, pp. 105–114. https://doi.org/10.1002/rnc.4590030204
  16. Damak H., Hammami M.A., Kalitine B. On the global uniform asymptotic stability of time-varying systems, Differential Equations and Dynamical Systems, 2014, vol. 22, issue 2, pp. 113–124. https://doi.org/10.1007/s12591-012-0157-z
  17. Dannan F.M., Elaydi S. Lipschitz stability of nonlinear systems of differential equations, Journal of Mathematical Analysis and Applications, 1986, vol. 113, issue 2, pp. 562–577. https://doi.org/10.1016/0022-247x(86)90325-2
  18. Dannan F.M., Elaydi S. Lipschitz stability of nonlinear systems of differential equations. II. Liapunov functions, Journal of Mathematical Analysis and Applications, 1989, vol. 143, issue 2, pp. 517–529. https://doi.org/10.1016/0022-247x(89)90057-7
  19. Demidovich B.P. Lektsii po matematicheskoi teorii ustoichivosti (Lectures on the mathematical stability theory), Moscow: Moscow State University, 1998.
  20. Dlala M., Hammami M.A. Uniform exponential practical stability of impulsive perturbed systems, Journal of Dynamical and Control Systems, 2007, vol. 13, issue 3, pp. 373–386. https://doi.org/10.1007/s10883-007-9020-x
  21. Dlala M., Ghanmi B., Hammami M.A. Exponential practical stability of nonlinear impulsive systems: Converse theorem and applications, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 2014, vol. 21, no. 1, pp. 37–64.
  22. Dragomir S.S. Some Grönwall type inequalities and applications, 2003. Science Direct Working Paper No. S1574-0358(04)70847-3. https://ssrn.com/abstract=3158353
  23. Ellouze I., Hammami M.A. A separation principle of time-varying dynamical systems: A practical stability approach, Mathematical Modelling and Analysis, 2007, vol. 12, no. 3, pp. 297–308. https://doi.org/10.3846/1392-6292.2007.12.297-308
  24. Ghanmi B., Hadj Taieb N., Hammami M.A. Growth conditions for exponential stability of time-varying perturbed systems, International Journal of Control, 2013, vol. 86, issue 6, pp. 1086–1097. https://doi.org/10.1080/00207179.2013.774464
  25. Hahn W. Stability of motion, New York: Springer, 1967.
  26. HajSalem Z., Hammami M.A., Mabrouk M. On the global uniform asymptotic stability of time-varying dynamical systems, Studia Universitatis Babeş-Bolyai Mathematica, 2014, vol. 59, no. 1, pp. 57–67.
  27. Hammi M., Hammami M.A. Non-linear integral inequalities and applications to asymptotic stability, IMA Journal of Mathematical Control and Information, 2015, vol. 32, issue 4, pp. 717–736. https://doi.org/10.1093/imamci/dnu016
  28. Hammami M.A. On the stability of nonlinear control systems with uncertainty, Journal of Dynamical and Control Systems, 2001, vol. 7, issue 2, pp. 171–179. https://doi.org/10.1023/A:1013099004015
  29. Khalil H.K. Nonlinear systems, Upper Saddle River: Prentice Hall, 2002.
  30. Lyapunov A.M. The general problem of the stability of motion, International Journal of Control, 1992, vol. 55, issue 3, pp. 531–534. https://doi.org/10.1080/00207179208934253
  31. Rouche N., Habets P., Laloy M. Stability theory by Liapunov's direct method, New York: Springer, 1977. https://doi.org/10.1007/978-1-4684-9362-7
  32. Teel A.R., Zaccarian L. On “uniformity” in definitions of global asymptotic stability for time-varying nonlinear systems, Automatica, 2006, vol. 42, issue 12, pp. 2219–2222. https://doi.org/10.1016/j.automatica.2006.07.012
  33. Tunç C. Stability and boundedness of solutions of non-autonomous differential equations of second order, Journal of Computational Analysis and Applications, 2011, vol. 13, no. 6, pp. 1067–1074.
  34. Tunç C., Tunç O. A note on certain qualitative properties of a second order linear differential system, Applied Mathematics and Information Sciences, 2015, vol. 9, no. 2, pp. 953–956.
  35. Tunç C., Tunç O. New qualitative criteria for solutions of Volterra integro-differential equations, Arab Journal of Basic and Applied Sciences, 2018, vol. 25, issue 3, pp. 158–165. https://doi.org/10.1080/25765299.2018.1509554
  36. Vidyasagar M. Nonlinear systems analysis, Englewood Cliffs: Prentice Hall, 1993.
  37. Yoshizawa T. Stability theory by Liapunov's second method, Tokyo: The Mathematical Society of Japan, 1966.
  38. Zaitsev V.A., Kim I.G. On the stability of linear time-varying differential equations, Proceedings of the Steklov Institute of Mathematics, 2022, vol. 319, suppl. 1, pp. S298–S317. https://doi.org/10.1134/s0081543822060268
  39. Zaitsev V., Kim I. Exponential stabilization of linear time-varying differential equations with uncertain coefficients by linear stationary feedback, Mathematics, 2020, vol. 8, issue 5, article 853. https://doi.org/10.3390/math8050853
Full text
/doc/issues-2024/24-02-04.pdf
<< Previous article
Next article >>