Abstract
Linear and nonlinear fractional-delay systems are studied. As an application, we derive the controllability and Hyers-Ulam stability results using the representation of solutions of these systems with the help of their delayed Mittag-Leffler matrix functions. We provide some sufficient and necessary conditions for the controllability of linear fractional-delay systems by introducing a fractional delay Gramian matrix. Furthermore, we establish some sufficient conditions of controllability and Hyers-Ulam stability of nonlinear fractional-delay systems by applying Krasnoselskii's fixed-point theorem. Our results improve, extend, and complement some existing ones. Finally, numerical examples of linear and nonlinear fractional-delay systems are presented to demonstrate the theoretical results.