Approximate Controllability Results for a Mixed Fractional Hadamard‐Type System With a Weakly Singular Logarithmic Kernel on Banach Spaces

ABSTRACT This manuscript investigates the approximate controllability of a fractional system of mixed Hadamard‐type dynamic equations with a weakly singular logarithmic kernel in a Banach space. First, the existence and uniqueness of the solution are established by proving that the associated operator has a unique fixed point. Next, the effect of the control function is studied under some suitable assumptions. After introducing the control input, we establish that the system possesses a mild solution that steers it from the given initial state to a desired final state. The main results are derived using Banach's and Krasnoselskii's fixed point theorems. Finally, a numerical example is presented to illustrate and validate the theoretical results. The example is further explained graphically with the help of MATLAB simulations, which depict the evolution of the system state both in the presence and absence of the control input. These simulations confirm the theoretical findings and demonstrate the effectiveness of the proposed controllability results.