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Symmetry
Volume 17
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10.3390/sym17122141
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Article

Conformable Time-Delay Systems: Stability and Stabilization Under One-Sided Lipschitz Conditions

by
Raouf Fakhfakh
1,*,
Abdellatif Ben Makhlouf
2,3,
Ibrahim-Elkhalil Ahmed
1,
Husam E. Dargail
1 and
Omar Naifar
4,5,*
1
Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka 72388, Saudi Arabia
2
Mathematics Education, Section Faculty of Education and Arts, Sohar University, P.O. Box 44, Sohar 311, Oman
3
Department of Mathematics, Faculty of Oman Sciences of Sfax, Sfax University, Sfax 3029, Tunisia
4
Control and Energy Management Laboratory, National School of Engineering, Sfax University, Sfax 3038, Tunisia
5
Higher Institute of Applied Sciences and Technology of Kairouan, University of Kairouan, Kairouan 3100, Tunisia
*
Authors to whom correspondence should be addressed.
Symmetry 2025, 17(12), 2141; https://doi.org/10.3390/sym17122141
Submission received: 14 October 2025 / Revised: 10 November 2025 / Accepted: 2 December 2025 / Published: 12 December 2025
(This article belongs to the Section Mathematics)
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Abstract

This study looks at the stability and stabilization issues concerning the nonlinear time-delay systems specified by conformable derivatives. These requirements can be used for many useful applications. Through the construction of appropriate Lyapunov–Krasovskii functionals, we develop novel linear matrix inequality (LMI) conditions for the exponential stability of autonomous systems and practical exponential stability for systems subject to bounded perturbations. Furthermore, we propose state-feedback stabilization strategies that transform the controller design problem into a convex optimization framework solvable via efficient LMI techniques. The theoretical developments are comprehensively validated through numerical examples that demonstrate the effectiveness of the proposed stability and stabilization criteria. The results establish a rigorous framework for analyzing and controlling conformable fractional-order systems with time delays, bridging theoretical advances with practical implementation considerations.
Keywords: conformable derivatives; nonlinear time-delay systems; exponential stability; practical stability; Lyapunov–Krasovskii functionals; linear matrix inequalities; state-feedback stabilization; one-sided Lipschitz nonlinearities; quadratic inner-boundedness conformable derivatives; nonlinear time-delay systems; exponential stability; practical stability; Lyapunov–Krasovskii functionals; linear matrix inequalities; state-feedback stabilization; one-sided Lipschitz nonlinearities; quadratic inner-boundedness

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MDPI and ACS Style

Fakhfakh, R.; Ben Makhlouf, A.; Ahmed, I.-E.; Dargail, H.E.; Naifar, O. Conformable Time-Delay Systems: Stability and Stabilization Under One-Sided Lipschitz Conditions. Symmetry 2025, 17, 2141. https://doi.org/10.3390/sym17122141

AMA Style

Fakhfakh R, Ben Makhlouf A, Ahmed I-E, Dargail HE, Naifar O. Conformable Time-Delay Systems: Stability and Stabilization Under One-Sided Lipschitz Conditions. Symmetry. 2025; 17(12):2141. https://doi.org/10.3390/sym17122141

Chicago/Turabian Style

Fakhfakh, Raouf, Abdellatif Ben Makhlouf, Ibrahim-Elkhalil Ahmed, Husam E. Dargail, and Omar Naifar. 2025. "Conformable Time-Delay Systems: Stability and Stabilization Under One-Sided Lipschitz Conditions" Symmetry 17, no. 12: 2141. https://doi.org/10.3390/sym17122141

APA Style

Fakhfakh, R., Ben Makhlouf, A., Ahmed, I.-E., Dargail, H. E., & Naifar, O. (2025). Conformable Time-Delay Systems: Stability and Stabilization Under One-Sided Lipschitz Conditions. Symmetry, 17(12), 2141. https://doi.org/10.3390/sym17122141

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