New results on controllability analysis of nonlinear fractional order integrodifferential Langevin system with multiple delays

Nowadays, the study of fractional differential equations (FDEs) is one of the most interesting research topics. In this article, we study a novel nonlinear fractional order Langevin system involving delays. The primary focus of this article is to analyze the relative controllability of a nonlinear integrodifferential fractional Langevin dynamical system with multiple delays in control for finite-dimensional spaces. Fundamentals of the fixed-point theory are used to achieve the desired results. Using Schauder's fixed theorem, sufficient conditions for the controllability are established and the Mittag-Leffler matrix function defines the controllability Grammian matrix. Additionally, the derived results are validated through an application.