A robust uniform practical stability approach for Caputo fractional hybrid systems

Michael Precious Ineh; Umar Ishtiaq; Jackson Efiong Ante; Mubariz Garayev; Ioan-Lucian Popa; Michael Precious Ineh; Umar Ishtiaq; Jackson Efiong Ante; Mubariz Garayev; Ioan-Lucian Popa

doi:10.3934/math.2025320

AIMS Mathematics

2025, Volume 10, Issue 3: 7001-7021. doi: 10.3934/math.2025320

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A robust uniform practical stability approach for Caputo fractional hybrid systems

1.
Department of Mathematics and Computer Science, Ritman University, Ikot Ekpene, Akwa Ibom State, Nigeria; ineh.michael@ritmanuniversity.edu.ng
2.
Office of Research, Innovation and Commercialization, University of Management and Technology, Lahore, Pakistan
3.
Department of Mathematics, Topfaith University, Mkpatak, Akwa Ibom State, Nigeria; jackson.ante@topfaith.edu.ng
4.
Department of Mathematics, College of Science, King Saud University, P. O. Box 2455, Riyadh, Saudi Arabia; mgrayev@ksu.edu.sa
5.
Department of Computing, Mathematics and Electronics, "1 Decembrie 1918" University of Alba Iulia, Alba Iulia 510009, Romania
6.
Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, Brasov 500091, Romania

Received: 22 January 2025 Revised: 28 February 2025 Accepted: 05 March 2025 Published: 26 March 2025
MSC : 34A08, 34A34, 34D20, 34N05

This work introduces a novel approach for analyzing the uniform practical stability (UPS) and strong uniform practical stability (SUPS) of Caputo fractional dynamic equations on time scales, using two measures ($ m, m_0 $). The method employs an extended derivative, the Caputo fractional delta Dini derivative (CFr$ \Delta $DiD) of order $\zeta \in (0, 1)$, addressing the gap in unified stability frameworks for fractional hybrid systems that span both continuous and discrete time domains. This generalized framework not only unifies various stability concepts but also makes it applicable to hybrid systems with both gradual and abrupt changes. The UPS and SUPS results are demonstrated through illustrative examples.
- fractional calculus,
- uniform practical stability,
- Caputo fractional derivative,
- time scales,
- vector Lyapunov functions,
- dynamic systems
Citation: Michael Precious Ineh, Umar Ishtiaq, Jackson Efiong Ante, Mubariz Garayev, Ioan-Lucian Popa. A robust uniform practical stability approach for Caputo fractional hybrid systems[J]. AIMS Mathematics, 2025, 10(3): 7001-7021. doi: 10.3934/math.2025320

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Abstract

This work introduces a novel approach for analyzing the uniform practical stability (UPS) and strong uniform practical stability (SUPS) of Caputo fractional dynamic equations on time scales, using two measures ($ m, m_0 $). The method employs an extended derivative, the Caputo fractional delta Dini derivative (CFr$ \Delta $DiD) of order $\zeta \in (0, 1)$, addressing the gap in unified stability frameworks for fractional hybrid systems that span both continuous and discrete time domains. This generalized framework not only unifies various stability concepts but also makes it applicable to hybrid systems with both gradual and abrupt changes. The UPS and SUPS results are demonstrated through illustrative examples.

References

[1]	J. E. Ante, J. U. Atsu, E. E. Abraham, O. O. Itam, E. J. Oduobuk, A. B. Inyang, On a class of piecewise continuous Lyapunov functions and asymptotic practical stability of nonlinear impulsive Caputo fractional differential equations via new modelled generalized Dini derivative, IEEE SEM, 12 (2024), 1–21.
[2]	W. W. Mohammed, C. Cesarano, F. M. Al-Askar, Solutions to the (4+1)-dimensional time-fractional Fokas equation with M-truncated derivative, Mathematics, 11 (2022), 194. https://doi.org/10.3390/math11010194 doi: 10.3390/math11010194
[3]	V. Lakshmikantham, S. Leela, A. Martynyuk, Practical stability of nonlinear systems, 1990. https://doi.org/10.1142/1192
[4]	V. Lakshmikantham, S. Sivasundaram, B. Kaymakçalan, Dynamic systems on measure chains, Springer Science & Business Media, 1996. https://doi.org/10.1007/978-1-4757-2449-3
[5]	R. P. Agarwal, S. Hristova, D. O'Regan, Practical stability of Caputo fractional differential equations by Lyapunov functions, Differ. Equations Appl., 8 (2012), 53–68. https://doi.org/10.7153/dea-08-04 doi: 10.7153/dea-08-04
[6]	J. O. Achuobi, E. P. Akpan, R. George, A. E. Ofem, Stability analysis of Caputo fractional time-dependent systems with delay using vector lyapunov functions, AIMS Math., 9 (2024), 28079–28099. https://doi.org/10.3934/math.20241362 doi: 10.3934/math.20241362
[7]	J. E. Ante, J. U. Atsu, A. Maharaj, O. K. Narain, E. E. Abraham, On a class of piecewise continuous Lyapunov functions and uniform eventual stability of nonlinear impulsive Caputo fractional differential equations via new modelled generalized Dini derivative, Asia Pac. J. Math., 99 (2024), 1–20. https://doi.org/10.28924/APJM/11-99 doi: 10.28924/APJM/11-99
[8]	M. P. Ineh, J. O. Achuobi, E. P. Akpan, J. E. Ante, CDq on the uniform stability of Caputo fractional differential equations using vector Lyapunov functions, J. Niger. Assoc. Math. Phys., 68 (2024), 51–64. https://doi.org/10.60787/jnamp.v68no1.416 doi: 10.60787/jnamp.v68no1.416
[9]	Y. Guo, T. Li, Fractional-order modeling and optimal control of a new online game addiction model based on real data, Commun. Nonlinear Sci. Numer. Simul., 121 (2023), 107221. https://doi.org/10.1016/j.cnsns.2023.107221 doi: 10.1016/j.cnsns.2023.107221
[10]	J. E. Ante, O. O. Itam, J. U. Atsu, S. O. Essang, E. E. Abraham, M. P. Ineh, On the novel auxiliary Lyapunov function and uniform asymptotic practical stability of nonlinear impulsive Caputo fractional differential equations via new modelled generalized Dini derivative, Afr. J. Math. Stat. Stud., 7 (2024), 11–33. https://doi.org/10.52589/AJMSS-VUNAIOBC doi: 10.52589/AJMSS-VUNAIOBC
[11]	M. P. Ineh, E. P. Akpan, Lyapunov uniform asymptotic stability of Caputo fractional dynamic equations on time scale using a generalized derivative, Trans. Niger. Assoc. Math. Phys., 20 (2024), 117–132. https://doi.org/10.60787/tnamp.v20.431 doi: 10.60787/tnamp.v20.431
[12]	J. E. Ante, M. P. Ineh, J. O. Achuobi, U. P. Akai, J. U. Atsu, N. A. O. Offiong, A novel Lyapunov asymptotic eventual stability approach for nonlinear impulsive Caputo fractional differential equations, Appl. Math., 4 (2024), 1600–1617. https://doi.org/10.3390/appliedmath4040085 doi: 10.3390/appliedmath4040085
[13]	M. P. Ineh, V. N. Nfor, M. I. Sampson, J. E. Ante, J. U. Atsu, O. O. Itam, A novel approach for Vector Lyapunov functions and practical stability of Caputo fractional dynamic equations on time scale in terms of two measures, Khayyam J. Math., 11 (2025), 61–89. https://doi.org/10.22034/kjm.2025.487084.3362 doi: 10.22034/kjm.2025.487084.3362
[14]	Q. Zhang, A class of vector Lyapunov functions for stability analysis of nonlinear impulsive differential systems, Math. Probl. Eng., 1 (2014), 649012. https://doi.org/10.1155/2014/649012 doi: 10.1155/2014/649012
[15]	D. K. Igobi, E. Ndiyo, M. P. Ineh, Variational stability results of dynamic equations on time-scales using generalized ordinary differential equations, World J. Appl. Sci. Technol., 15 (2023), 2. https://doi.org/10.4314/wojast.v15i2.14 doi: 10.4314/wojast.v15i2.14
[16]	K. Liu, W. Jiang, Stability of nonlinear Caputo fractional differential equations, Appl. Math. Model., 40 (2016), 3919–3924. https://doi.org/10.1016/j.apm.2015.10.048 doi: 10.1016/j.apm.2015.10.048
[17]	K. Hattaf, Stability of fractional differential equations with new generalized hattaf fractional derivative, Math. Prob. Eng., 1 (2021), 8608447. https://doi.org/10.1155/2021/8608447 doi: 10.1155/2021/8608447
[18]	N. Laledj, A. Salim, J. E. Lazreg, S. Abbas, B. Ahmad, M. Benchohra, On implicit fractional $q$‐difference equations: analysis and stability, Math. Methods Appl. Sci., 45 (2022), 10775–10797. https://doi.org/10.1002/mma.8417 doi: 10.1002/mma.8417
[19]	S. Hilger, Analysis on measure chains – a unified approach to continuous and discrete calculus, Results Math., 18 (1990), 18–56. https://doi.org/10.1007/BF03323153 doi: 10.1007/BF03323153
[20]	R. E. Orim, M. P. Ineh, D. K. Igobi, A. Maharaj, O. K. Narain, A novel approach to Lyapunov uniform stability of Caputo fractional dynamic equations on time scale using a new generalized derivative, Asia Pac. J. Math., 12 (2025), 6. https://doi.org/10.28924/APJM/12-6 doi: 10.28924/APJM/12-6
[21]	J. A. Ugboh, C. F. Igiri, M. P. Ineh, A. Maharaj, O. K. Narain, A novel approach to Lyapunov eventual stability of Caputo fractional dynamic equations on time scale, Asia Pac. J. Math., 12 (2025), 3. https://doi.org/10.28924/APJM/12-3 doi: 10.28924/APJM/12-3
[22]	J. Oboyi, M. P. Ineh, A. Maharaj, J. O. Achuobi, O. K. Narain, Practical stability of Caputo fractional dynamic equations on time scale, Adv. Fixed Point Theory, 3 (2025), 3. https://doi.org/10.28919/afpt/8959 doi: 10.28919/afpt/8959
[23]	M. P. Ineh, E. P. Akpan, H. Nabwey, A novel approach to Lyapunov stability of Caputo fractional dynamic equations on time scale using a new generalized derivative, AIMS Math., 9 (2024), 34406–34434. https://doi.org/10.3934/math.20241639 doi: 10.3934/math.20241639
[24]	A. Ahmadkhanlu, A. Jahanshahi, On the existence and uniqueness of solution of initial value problem for fractional order differential equations on time scales, Bull. Iran. Math. Soc., 38 (2012), 241–252.
[25]	V. Kumar, M. Malik, Existence, stability and controllability results of fractional dynamic system on time scales with application to population dynamics, Int. J. Nonlinear Sci. Numer. Simul., 6 (2022), 741–766. https://doi.org/10.1515/ijnsns-2019-0199 doi: 10.1515/ijnsns-2019-0199
[26]	A. B. Makhlouf, Partial practical stability for fractional-order nonlinear systems, Math. Methods Appl. Sci., 45 (2022), 5135–5148. https://doi.org/10.1002/mma.8097 doi: 10.1002/mma.8097
[27]	M. Bohner, P. Allan, Dynamic equations on time scales: an introduction with applications, Springer Science & Business Media, 2001. https://doi.org/10.1007/978-1-4612-0201-1
[28]	J. Hoffacker, C. C. Tisdell, Stability and instability for dynamic equations on time scales, Comput. Math. Appl., 49 (2005), 1327–1334. https://doi.org/10.1016/j.camwa.2005.01.016 doi: 10.1016/j.camwa.2005.01.016

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