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.

This study introduces a unified analytical framework for investigating the transient thermoelastic vibration of a micro-scale beam resting on dual-parameter foundations. We apply the fractional three-phase-lag (FTPL) generalized thermoelastic model to accurately characterize scale-dependent coupled thermal and elastic responses by incorporating complex thermal relaxation effects through the fractional derivative order. By employing the Laplace transform technique and its numerical inversion, we derive the coupled distributions of temperature, displacement, bending moment, and deflection within the beams. A comprehensive parametric analysis is conducted to quantify the distinct influence of the fractional factor and the foundation’s shear and stiffness parameters on the beam’s dynamic stability and propagation characteristics. The calculated results are systematically compared with established classical theories to validate the model’s robustness while simultaneously demonstrating the enhanced predictive capacity of the (FTPL) approach, particularly for characterizing thermal wave dispersion at the micro-scale. This research provides critical design criteria for advanced micro-electromechanical systems (MEMSs) where foundation stiffness and thermal inertial effects are intrinsically linked, offering novel insights into the tailored design of microstructural components.

This paper investigates the stability analysis and stabilization problems for a class of nonlinear time-delay systems described by conformable derivatives. We consider systems with one-sided Lipschitz nonlinearities and quadratic inner-boundedness conditions, which encompass a broad range of practical applications. Through the construction of appropriate Lyapunov-Krasovskii functionals, we develop novel linear matrix inequality (LMI) conditions for 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.