Controllability analysis of a class of (1,2)‐Caputo time‐fractional systems

Abstract The main purpose of this paper is to examine the controllability of linear time‐fractional systems involving the Caputo fractional derivative of order . We first define the concept of controllability and then develop a theoretical framework to demonstrate certain exact and approximate controllability properties of time‐fractional systems. This is achieved using a novel expression for the controllability operator based on the formulation of the mild solution of our problem, fractional calculus theory, and the cosine family. Subsequently, we extend the Hilbert uniqueness method (HUM) approach to determine the optimal control of our problem. Finally, we present numerical simulations for one‐dimensional systems with different types of actuators, either zonal or pointwise, to support the theoretical results discussed in the paper. These simulations achieve a satisfactory error margin, which determines the effectiveness of our method.