Bohnenblust–Karlin Fixed Point Approach to Controllability of Coupled Impulsive Fractional Integro-Differential Systems Involving Hilfer–Hadamard Derivatives and Nonlocal Initial Conditions

This study investigates the controllability of coupled systems governed by impulsive Hilfer–Hadamard fractional integro-differential equations subject to nonlocal coupled integro-type initial conditions. By employing the Bohnenblust–Karlin fixed-point theorem for multivalued maps, we establish sufficient conditions for the existence of solutions, thereby proving the system’s controllability from an arbitrary initial state to a prescribed final state via suitable control functions. The theoretical framework accommodates Hilfer–Hadamard derivatives with varying orders and modes, incorporates impulsive effects at discrete time points, and accounts for fractional integral terms. Rigorous assumptions are formulated for the involved nonlinear operators to ensure the applicability of the fixed-point results. Finally, a numerical example is provided to validate the theoretical findings, demonstrating that specific parameter choices satisfy the derived conditions and confirming the practical relevance of the proposed controllability criteria.