Controllability of a system of fractional boundary value problem on time scales

Abstract This paper examines the controllability of a system of fractional dynamic equations that involve the Caputo fractional derivative with periodic boundary conditions on time scales. The study first demonstrates that the system possesses a unique fixed point, corresponding to its solution. The influence of the control input is then analyzed to establish the system’s controllability. The main theoretical results are derived using Schauder’s and Banach’s fixed point theorems. Green’s function plays a key role in the analysis, serving as the kernel of the integral equation associated with the proposed fractional dynamic system. To validate the theoretical findings, a detailed numerical example is provided, supported by MATLAB simulations that plot the trajectories of the system both with and without the control input.