A tempered fractional Halanay inequality with feedback stabilization applications

This paper develops a tempered fractional Halanay inequality for delay systems governed by the Caputo tempered fractional derivative. The proposed comparison principle establishes a sharp decay estimate combining exponential tempering with Mittag-Leffler relaxation, while explicitly characterizing the impact of delay through an amplified delay gain. Both homogeneous and inhomogeneous formulations are derived, the latter yielding a computable practical residual bound under bounded perturbations. The result is then embedded into a Lyapunov framework for tempered fractional delay systems and applied to the synthesis of stabilizing static state feedback controllers. The controller design is formulated as a tractable linear matrix inequality in transformed variables, enabling direct numerical implementation. The conservatism introduced by the Young-inequality treatment of the disturbance is quantified explicitly, including a closed-form expression for the splitting parameter that minimizes the practical residual level. A reproducible numerical study on an open-loop unstable delayed system confirms the theoretical decay rate, validates the practical stability bound under constant and time-varying delays, and verifies the homogeneous tempered Mittag-Leffler stability across several fractional orders and tempering levels. The results provide a rigorous and computationally implementable framework for stability analysis and control of tempered fractional-order systems with delays.