Relative Controllability and Hyers–Ulam Stability of Riemann–Liouville Fractional Delay Differential System

Abstract

In this work, we focus on the relative controllability and Hyers–Ulam stability of Riemann–Liouville fractional delay differential system of order \(\alpha \in (1,2)\). Firstly, for the linear system based on Mittag-Laffler matrix function, we define a controllability Grammian matrix to judge whether the system is relatively controllable. Additionally, with the aid of Krasnoselskii’s fixed point theorem, sufficient conditions for the relative controllability of the corresponding semilinear system is also studied. Furthermore, we used Grönwall’s inequality to investigate Hyers–Ulam stability for Riemann–Liouville fractional semilinear delay differential equations. Lastly, three instances are provided to verify that our theoretical results are accurate.