@article{Hammad2024, 
author = {Hasanen A. Hammad and Mohammed E. Dafaalla and Kottakkaran Sooppy Nisar},
title = {A Grammian matrix and controllability study of fractional delay integro-differential Langevin systems},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {6},
pages = {15469-15485},
keywords = {Langevin system, integro-differential equation, fixed point technique, controllability, delay term, fractional derivatives},
url = {https://www.sciopen.com/article/10.3934/math.2024748},
doi = {10.3934/math.2024748},
abstract = {This study focused on introducing a fresh model of fractional operators incorporating multiple delays, termed fractional integro-differential Langevin equations with multiple delays. Additionally, the research evaluated the relative controllability of this model within finite-dimensional spaces. Employing fixed-point theory yields the desired outcomes, with the controllability assessment facilitated by Schauder's fixed point and the Grammian matrix defined through the Mittag-Leffler matrix function. Validation of the results was conducted through an application.}
}