Zaman Skalasında Fonksiyonel Dinamik Denklemlerin Alternatif Parametreler Değişimi ile h-Kararlılığı

Abstract

Bu çalışmada, zaman skalalarında kurulmuş x^∆ (t)=a(t)x(t)+f(t,x(t)), t∈T, formundaki doğrusal olmayan fonksiyonel dinamik denklemlerin üzerinde durulmuş olup, bu tip denklemlerin h-kararlılığı çalışılmıştır. h-Kararlılık kavramı özel şartlar altında üstel kararlılığı, üniform kararlılığı ve Lipschitz kararlılığını kapsamaktadır. Makalenin analiz kısmında alternatif bir parametrelerin değişimi formülünün kullanılması ile odaklanılan denklemlerdeki regresiflik koşulu aranmamıştır, ve bu da daha geniş bir denklem sınıfının çalışılmasına imkan sağlamıştır. Elde edilen kararlılık sonuçlarına ek olarak, dinamik sistemlerin çözümlerinin üniform sınırlılığı ve h-kararlılığı arasındaki ilişki belirli şartlar altında elde edilmiştir.

Keywords

• h-Kararlılık
• Zaman Skalası
• Alternatif Parametrelerin Değişimi
• Üniform Sınırlılık

h- Stability of Functional Dynamic Equations on Time Scales by Alternative Variation of Parameters

Abstract

In this paper, we concentrate on nonlinear functional dynamic equations of the form x^∆ (t)=a(t)x(t)+f(t,x(t)), t∈T, on time scales and study h-stability, which implies uniform exponential stability, uniform Lipschitz stability, or uniform stability in particular cases. In our analysis, we use an alternative variation of parameters, which enables us to focus on a larger class of equations since the dynamic equations under the spotlight are not necessarily regressive. Also, we establish a linkage between uniform boundedness and h-stability notions for solutions of dynamic equations under sufficient conditions in addition to our stability results.

Keywords

• h-Stability
• Time Scale
• Alternative Variation of Parameters
• Uniform Boundedness

References

1. Hilger S. 1988. Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. Ph.D. Thesis, Universität Würzburg, Institut für Mathematik, Würzburg, Germany, 1-141.
2. Atıcı F.M., Biles D.C., Lebedinsky A. 2006. An application of time scales to economics. Mathematical and Computer Modelling, 43 (7-8): 718-726. https://doi.org/10.1016/j.mcm.2005.08.014.
3. Atıcı F.M., Turhan N. 2012. Sequential decision problems on isolated time domains. Journal of Mathematical Analysis and Applications, 388 (2): 753-759. https://doi.org/10.1016/j.jmaa.2011.09.068.
4. Chen J.Y., Zhang Y. 2021. Time-scale version of generalized Birkhoffian mechanics and its symmetries and conserved quantities of Noether type. Advances in Mathematical Physics, 2021: Article ID 9982975, 9 pages. https://doi.org/10.1155/2021/9982975.
5. Abraehim A.K., Jaber A.K., Al-Salih R. 2021. Flow optimization in dynamic networks on time scales. Journal of Physics: Conference Series, 1804 (1): 7 pages. https://doi.org/10.1088/1742-6596/1804/1/012025.
6. Bohner M., Streipert S., Torres D.F.M. 2019. Exact solution to a dynamic SIR model. Nonlinear Analysis: Hybrid Systems, 32: 228-238. https://doi.org/10.1016/j.nahs.2018.12.005.
7. Chen X., Shi C., Wang D. 2020. Dynamic behaviors for a delay Lasota-Wazewska model with feedback control on time scales. Advances in Difference Equations, 17: 13 pages. https://doi.org/10.1186/s13662-019-2483-8.
8. Pawłuszewicz E. 2008. Observability of Nonlinear Control Systems on Time Scales-Sufficient Conditions. In: Mathematical Control Theory and Finance, Edited by Sarychev A., Shiryaev A., Guerra M., Grossinho MR., Springer, Berlin, Heidelberg, 325-335. https://doi.org/10.1007/978-3-540-69532-5_18.

9. Öztürk Ö., Güzey H.M. 2018. Optimal control of quadrotor unmanned aerial vehicles on time scales. International Journal of Differential Equations, 13 (1): 41-54.
10. Poulsen D.R., Davis J.M., Gravagne I.A. 2017. Optimal control on stochastic time scales. IFAC-PapersOnLine, 50 (1): 14861-14866. https://doi.org/10.1016/j.ifacol.2017.08.2518.
11. Hoffacker C., Tisdell C.C. 2005. Stability and instability for dynamic equations on time scales. Computers & Mathematics with Applications, 49 (9-10): 1327-1334. https://doi.org/10.1016/j.camwa.2005.01.016.
12. Wu H., Zhou Z. 2007. Stability for first order delay dynamic equations on time scales. Computers & Mathematics with Applications, 53 (12): 1820-1831. https://doi.org/10.1016/j.camwa.2006.09.011.
13. Zhang J., Fan M. 2012. Boundedness and stability of semi-linear dynamic equations on time scales. Progress in Qualitative Analysis of Functional Equations, 1786: 45-56.
14. Akın-Bohner E., Raffoul Y. N., Tisdell C. 2010. Exponential stability in functional dynamic equations on time scales. Communications in Mathematical Analysis, 9 (1): 93-108.
15. Peterson A.C., Raffoul Y.N. 2005. Exponential stability of dynamic equations on time scales. Advances in Difference Equations, 2005 (2): Article ID 858671, 1-12. https://doi.org/10.1155/ADE.2005.133.
16. Pinto M. 1984. Perturbations of asymptotically stable differential systems, Analysis, 4 (1-2): 161-175.
17. Choi S.K., Koo N.J., Ryu R.S. 1997. h-stability of differential systems via t_∞-similarity. Bulletin of the Korean Mathematical Society, 34 (3): 371-383.
18. Damak H., Hammmami M.A., Kicha A. 2021. h-stability and boundedness results for solutions to certain nonlinear perturbed systems. Mathematics for Applications, 10: 9-23. https://doi.org/10.13164/ma.2021.02.
19. Choi S.K., Koo N.J., Song S.M. 2004. h-stability for nonlinear perturbed difference systems. Bulletin of the Korean Mathematical Society, 41 (3): 435-450. https://doi.org/10.4134/BKMS.2004.41.3.435.
20. Choi S.K., Koo N.J., Im D.M. 2006. h-stability for linear dynamic equations on time scales. Journal of Mathematical Analysis and Applications, 324 (1): 707-720. https://doi.org/10.1016/j.jmaa.2005.12.046.
21. Choi S.K., Goo Y.H., Koo N. 2008. h-stability of dynamic equations on time scales with nonregressivity. Abstract and Applied Analysis, 2008: Article ID 632473, 13 pages. https://doi.org/10.1155/2008/632473.
22. Choi S.K., Cui Y., Koo N. 2012. Variationally stable dynamic systems on time scales. Advances in Difference Equations, 2012 (1): Article ID 129, 1-17. https://doi.org/10.1186/1687-1847-2012-129.
23. Nasser B.B., Djemai M., Defoort M., Laleg-Kirati T. M. 2021. Time scale state feedback h-stabilisation of linear systems under Lipschitz-type disturbances. International Journal of Systems Science, 52(8): 1719-1729. https://doi.org/10.1080/00207721.2020.1869345.
24. Neggal B., Boukerrioua K., Kilani B., Meziri I. 2020. h-stability for nonlinear abstract dynamic equations on time scales and applications. Rendiconti del Circolo Matematico di Palermo Series 2, 69 (3): 1017-1031. https://doi.org/10.1007/s12215-019-00452-x.
25. Nasser B.B., Boukerrioua K., Defoort M., Djemai M., Hammmami M.A., Laleg-Kirati T.M. 2019. Sufficient conditions for uniform exponential stability and h-stability of some classes of dynamic equations on arbitrary time scales. Nonlinear Analysis: Hybrid Systems, 32: 54-64. https://doi.org/10.1016/j.nahs.2018.10.009.
26. Raffoul Y.N. 2019. Stability and boundedness in nonlinear and neutral difference equations using new variation of parameters formula and fixed point theory. Cubo A Mathematical Journal, 21(3): 39-61. https://doi.org/10.4067/s0719-06462019000300039.
27. Raffoul Y.N. 2019. Nonlinear functional delay differential equations arising from population models. Advances in Dynamical Systems and Applications, 14(1): 67-81. https://doi.org/10.37622/ADSA/14.1.2019.67-81.
28. Larraín-Hubach A., Raffoul Y.N. 2020. Boundedness, periodicity and stability in nonlinear delay differential equations. Advances in Dynamical Systems and Applications, 15(1): 29-37.
29. Koyuncuoğlu H.C. 2021. Some qualitative results for functional delay dynamic equations on time scales. Turkish Journal of Mathematics, 45(1): 1985-2007. https://doi.org/10.3906/mat-2102-106.
30. Bohner M., Peterson A. 2001. Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 1-358.
31. Khalil H.K. 2002. Nonlinear Systems, Prentice Hall, New York, 1-734.
32. Adıvar M., Raffoul Y.N. 2009. Stability and periodicity in dynamic delay equations. Computers & Mathematics with Applications, 58(2): 264-272. https://doi.org/10.1016/j.camwa.2009.03.065.