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

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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.

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Acknowledgements

The authors sincerely thank the editors and reviewers for their valuable time reviewing this manuscript and for their insightful comments.

Funding

This work was supported by the National Natural Science Foundation of China (12361035), Guizhou Provincial Science and Technology Projects (No. QKHJC-ZK[2024]YB061), and the Natural Science Special Research Fund Project of Guizhou University, China (202002).

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Correspondence to Danfeng Luo.

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An, W., Luo, D. & Huang, J. Relative Controllability and Hyers–Ulam Stability of Riemann–Liouville Fractional Delay Differential System. Qual. Theory Dyn. Syst. 23, 180 (2024). https://doi.org/10.1007/s12346-024-01046-4

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