Research Article
Year 2023,
, 1121 - 1132, 12.12.2023
Abstract
The impulsive fractional delayed differential system with the Caputo derivative with respect to another function is considered. An explicit solution to the system in the light of the available studies on this subject is determined and its existence and uniqueness are debated. Lastly, the stability and controllability of the given system are investigated.
Keywords
Existence uniqueness impulsive fractional delayed system relative controllability Ulam-Hyers stability
References
- Aydin, M., Mahmudov, N. I., Aktuğlu, H., Baytunç, E., & Atamert, M. S. (2022). On a study of the representation of solutions of a Ψ-Caputo fractional differential equations with a single delay. Electronic Research Archive, 30, 1016–1034.
- Aydin, M., & Mahmudov, N. I. (2022). Iterative learning control for impulsive fractional order time-delay systems with nonpermutable constant coefficient matrices. International Journal of Adaptive Control and Signal Processing, 36(1), 1419–1438.
- Bainov, D. D., & Simeonov, P. S. (1989). Systems with Impulse Effect. Ellis Horwood Series:Mathematics and Its Applications, Ellis Horwood, Chichester.
- Bainov, D. D., & Simeonov, P. S. (1993). Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics. (66 vol.). Longman Scientific & Technical, Harlow; JohnWiley & Sons, New York.
- Elshenhab, A. M., & Wang, X. T. (2021a). Representation of solutions for linear fractional systems with pure delay and multiple delays. Mathematical Methods in the Applied Sciences, 44, 12835–12850.
- Elshenhab, A. M., & Wang, X. T. (2021b). Representation of solutions of linear differential systems with pure delay and multiple delays with linear parts given by non-permutable matrices. Applied Mathematics and Computation, 410, 126443. https://doi.org/10.1016/j.amc.2021.126443
- Khusainov, D. Y., & Shuklin, G. V. (2005). Relative controllability in systems with pure delay. International Journal of Applied Mathematics, 2, 210–221. https://doi.org/10.1007/s10778-005-0079-3
- Khusainov, D. Y., & Shuklin, G. V. (2003). Linear autonomous time-delay system with permutation matrices solving. Stud. Univ. Zilina, 17, 101–108.
- Lakshmikantham, V., Bainov, D. D., & Simeonov, P. S. (1989). Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. (6 vol.). World Scientific, New Jersy. https://doi.org/10.1142/0906
- Liang, C., Wang, J., & O’Regan, D. (2017). Controllability of nonlinear delay oscillating systems. Electronic Journal of Qualitative Theory of Differential Equations, 2017, 1–18. https://doi.org/10.14232/ejqtde.2017.1.47
- Li, M., & Wang, J. R. (2018). Exploring delayed Mittag-Leffler type matrix functions to study finite time stability of fractional delay differential equations. Applied Mathematics and Computation, 324, 254–265. https://doi.org/10.1016/j.amc.2017.11.063
- Liu, L., Dong, Q., & Li, G. (2021). Exact solutions and Hyers–Ulam stability for fractional oscillation equations with pure delay. Applied Mathematics Letters, 112, 106666. https://doi.org/10.1016/j.aml.2020.106666
- Mahmudov, N. I., & Aydın, M. (2021). Representation of solutions of nonhomogeneous conformable fractional delay differential equations. Chaos Solitons Fractals, 150, 111190. https://doi.org/10.1016/j.chaos.2021.111190
- Mahmudov, N. I. (2022). Multi-delayed perturbation of Mittag-Leffler type matrix functions. Journal of Mathematical Analysis and Applications, 505, 125589. https://doi.org/10.1016/j.jmaa.2021.125589
- Mahmudov, N. I. (2019). Delayed perturbation of Mittag-Leffler functions and their applications to fractional linear delay differential equations. Mathematical Methods in the Applied Sciences, 42, 5489–5497. https://doi.org/10.1002/mma.5446
- Mahmudov, N. I. (2018). Representation of solutions of discrete linear delay systems with non permutable matrices. Applied Mathematics Letters, 85, 8–14. https://doi.org/10.1016/j.aml.2018.05.015
- Samoilenko, A. M., & Perestyuk, N. A. (1995). Impulsive Differential Equations, World Scientific Serieson Nonlinear Science. Series A:Monographs and Treatises, vol. 14, World Scientific, New Jersey, ISBN: 978-981-02-2416-5. https://doi.org/10.1142/2892
- Wang, J., Luo, Z., & Feckan, M. (2017). Relative controllability of semilinear delay differential systems with linear parts defined by permutable matrices. European Journal of Control, 38, 39–46. https://doi.org/10.1016/j.ejcon.2017.08.002
- You, Z., Feckan, M., & Wang, J. (2020). Relative Controllability of Fractional Delay Differential Equations via Delayed Perturbation of Mittag-Leffler Functions. Journal of Computational and Applied Mathematics, 378, 112939. https://doi.org/10.1016/j.cam.2020.112939.
BAŞKA BİR FONKSİYONA BAĞLI CAPUTO KESİRLİ ANİ DEĞİŞİMLİ GECİKMELİ SİSTEMİN GÖRECELİ KONTOL EDİLEBİLİRLİĞİ
Abstract
Herhangi bir fonskiyona göre tanımlanmış Caputo türevli ani değişmeli kesirli gecikmeli bir sistem dikkate alınmaktadır. Bu konuda mevcut çalışmaların ışığında sistemin sarih bir çözümü belirlenmekte ve çözümün varlığı ve tekliği tartışılmaktadır. Son olarak, verilen sistemin kararlılığı ve kontrol edilebilirliği araştırılmaktadır.
Keywords
Ani değişmeli kesirli gecikmeli sistem varlık teklik Ulam-Hyers kararlılığı göreceli kontrol edilebilirliği
References
- Aydin, M., Mahmudov, N. I., Aktuğlu, H., Baytunç, E., & Atamert, M. S. (2022). On a study of the representation of solutions of a Ψ-Caputo fractional differential equations with a single delay. Electronic Research Archive, 30, 1016–1034.
- Aydin, M., & Mahmudov, N. I. (2022). Iterative learning control for impulsive fractional order time-delay systems with nonpermutable constant coefficient matrices. International Journal of Adaptive Control and Signal Processing, 36(1), 1419–1438.
- Bainov, D. D., & Simeonov, P. S. (1989). Systems with Impulse Effect. Ellis Horwood Series:Mathematics and Its Applications, Ellis Horwood, Chichester.
- Bainov, D. D., & Simeonov, P. S. (1993). Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics. (66 vol.). Longman Scientific & Technical, Harlow; JohnWiley & Sons, New York.
- Elshenhab, A. M., & Wang, X. T. (2021a). Representation of solutions for linear fractional systems with pure delay and multiple delays. Mathematical Methods in the Applied Sciences, 44, 12835–12850.
- Elshenhab, A. M., & Wang, X. T. (2021b). Representation of solutions of linear differential systems with pure delay and multiple delays with linear parts given by non-permutable matrices. Applied Mathematics and Computation, 410, 126443. https://doi.org/10.1016/j.amc.2021.126443
- Khusainov, D. Y., & Shuklin, G. V. (2005). Relative controllability in systems with pure delay. International Journal of Applied Mathematics, 2, 210–221. https://doi.org/10.1007/s10778-005-0079-3
- Khusainov, D. Y., & Shuklin, G. V. (2003). Linear autonomous time-delay system with permutation matrices solving. Stud. Univ. Zilina, 17, 101–108.
- Lakshmikantham, V., Bainov, D. D., & Simeonov, P. S. (1989). Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. (6 vol.). World Scientific, New Jersy. https://doi.org/10.1142/0906
- Liang, C., Wang, J., & O’Regan, D. (2017). Controllability of nonlinear delay oscillating systems. Electronic Journal of Qualitative Theory of Differential Equations, 2017, 1–18. https://doi.org/10.14232/ejqtde.2017.1.47
- Li, M., & Wang, J. R. (2018). Exploring delayed Mittag-Leffler type matrix functions to study finite time stability of fractional delay differential equations. Applied Mathematics and Computation, 324, 254–265. https://doi.org/10.1016/j.amc.2017.11.063
- Liu, L., Dong, Q., & Li, G. (2021). Exact solutions and Hyers–Ulam stability for fractional oscillation equations with pure delay. Applied Mathematics Letters, 112, 106666. https://doi.org/10.1016/j.aml.2020.106666
- Mahmudov, N. I., & Aydın, M. (2021). Representation of solutions of nonhomogeneous conformable fractional delay differential equations. Chaos Solitons Fractals, 150, 111190. https://doi.org/10.1016/j.chaos.2021.111190
- Mahmudov, N. I. (2022). Multi-delayed perturbation of Mittag-Leffler type matrix functions. Journal of Mathematical Analysis and Applications, 505, 125589. https://doi.org/10.1016/j.jmaa.2021.125589
- Mahmudov, N. I. (2019). Delayed perturbation of Mittag-Leffler functions and their applications to fractional linear delay differential equations. Mathematical Methods in the Applied Sciences, 42, 5489–5497. https://doi.org/10.1002/mma.5446
- Mahmudov, N. I. (2018). Representation of solutions of discrete linear delay systems with non permutable matrices. Applied Mathematics Letters, 85, 8–14. https://doi.org/10.1016/j.aml.2018.05.015
- Samoilenko, A. M., & Perestyuk, N. A. (1995). Impulsive Differential Equations, World Scientific Serieson Nonlinear Science. Series A:Monographs and Treatises, vol. 14, World Scientific, New Jersey, ISBN: 978-981-02-2416-5. https://doi.org/10.1142/2892
- Wang, J., Luo, Z., & Feckan, M. (2017). Relative controllability of semilinear delay differential systems with linear parts defined by permutable matrices. European Journal of Control, 38, 39–46. https://doi.org/10.1016/j.ejcon.2017.08.002
- You, Z., Feckan, M., & Wang, J. (2020). Relative Controllability of Fractional Delay Differential Equations via Delayed Perturbation of Mittag-Leffler Functions. Journal of Computational and Applied Mathematics, 378, 112939. https://doi.org/10.1016/j.cam.2020.112939.
There are 19 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Mechanical Engineering (Other) |
| Journal Section | Mechanical Engineering |
| Authors | |
| Publication Date | December 12, 2023 |
| Submission Date | August 8, 2023 |
| Published in Issue | Year 2023 |