Controllability and Modeling Perspectives of Tempered Ψ-Caputo Fractional Systems

In this article, we investigated the controllability of fractional dynamical systems (FDS) involving the tempered Ψ-Caputo fractional derivative (FD). First, we derived the solution representation for this generalized FD with the help of Laplace transform and Mittag–Leffler (M-L) function. Furthermore, the controllability of the linear system is established using the controllability Gramian matrix. For nonlinear systems, we employ a Schauder’s fixed-point theorem to determine sufficient conditions for controllability. The major innovation of this paper is the introduction and study of controllability based on the tempered Ψ-Caputo FD that generalizes and combines some widely used FDs such as the tempered Caputo derivatives. These derivatives are particularly advantageous as they effectively model memory effects and enhance the accuracy of real-world dynamical systems. To support the theoretical results, a numerical illustration is presented.