Distributed Control for Mittag–Leffler Synchronization of Variable-Order Fractional Gierer–Meinhardt Reaction-Diffusion Systems

This paper investigates Mittag–Leffler synchronization (MLSY) for variable-order fractional Gierer–Meinhardt reaction-diffusion systems (VFO-GM-RDs). We introduce, for the first time, a distributed control scheme specifically designed to achieve MLSY in these complex systems. A key contribution is the derivation of an explicit analytical expression for the finite settling time (ST), which is expressed in terms of system parameters and the bounds of the VFO exponent. Rigorous Lyapunov-based proofs, tailored to VFO-RD dynamics and leveraging new fractional inequalities, are provided to establish the MLSY conditions. Numerical simulations, employing a finite-difference approximation (FDA) coupled with an implicit time-stepping method, are performed to validate both the theoretical MLSY criteria and the accuracy of the predicted ST. The results demonstrate the effectiveness of the proposed control strategy in ensuring robust synchronization of VFO-GM-RDs within a predictable finite time frame.