This study investigated the impact of the fractional Caputo-tempered two-phase-lag (FCT-TPL) heat conduction model on thermoelastic vibrations within a medium containing spherical cavities and voids. In the proposed model, the nonlocality of time and space is integrated to unify classical and generalized thermoelastic theories, enabling a thorough investigation of size-dependent phenomena and the scattering characteristics of thermo-mechanical waves. Also, by integrating fractional calculus with tempered derivatives, the proposed model adeptly captures the complex interaction between localized thermal effects and nonlocal mechanical responses, particularly in materials with pronounced microstructural features. The fractional order and tempering parameter are shown to play a crucial role in controlling thermal relaxation times and the amplitude of thermoelastic vibrations. The findings reveal that the integration of the fractional Caputo-tempered derivative, along with temporal and spatial nonlocal effects, into the two-phase-lag model significantly improves the accuracy of predicting transient thermoelastic responses in materials with cavities and voids.

This work introduces a novel control framework using the Caputo fractional derivative (CFD) with respect to another function—a derivative that has not been thoroughly treated in control theory. By extending the widely recognized Caputo-Hadamard (CH) fractional-order derivative, we address its utility in nonlinear systems. The core of our contribution is the practical stability for systems governed by this derivative, which ensures convergence toward a bounded region around the origin. Additionally, we extend the Lipschitz condition (LC) to the one-sided Lipschitz (OSL) condition for observer design and observer based-control design in fractional-order systems, ensuring its practical stability. Finally, three numerical examples validate the effectiveness of our proposed framework, providing practical insights for control theory advancements.