Investigation of Controllability and Synchronization of Fractional Order Dynamical

Abstract

This investigation delves into the qualitative theory of fractional dynamical systems, emphasizing their controllability and synchronization properties, particularly in systems characterized by different fractional derivatives and subject to time delays. The study is motivated by the growing importance of fractional calculus in accurately describing memory and hereditary properties in complex dynamical systems. This work examines the controllability of systems with distributed delays, employing the and#968;-Hilfer fractional derivative and damped dynamical system, then it is extended to the analysis of systems modelled by the 2nd level Hilfer fractional derivative. The solution for the above dynamical systems is obtained through Laplace transform, Mittag-Leffler function (MLF). Controllability analysis is then conducted to examine the system s ability to steer from one state to another using external inputs, known as controls. In addition to controllability, this investigation also addresses the synchronization of fractional order neural networks in the Caputo sense. Two types of synchronization strategies are developed and analysed. The first approach achieves finite-time synchronization using sampled data control with time delays, ensuring rapid and precise alignment of neural network states. The second approach utilizes a matrix measure technique to synchronize time-delayed newlineneural networks, offering a robust method for dealing with the inherent complexities of fractional order systems. Finally, various theories are verified with simulation results to ensure they align with the proposed controllability and synchronization problems. newline