Pattern Optimization of Fractional Diffusive Schnakenberg System by PD Control Strategy

Reaction-diffusion systems are widely used to describe pattern formation, and various control strategies have been applied to reaction-diffusion systems to achieve control objectives such as boundary control, output feedback stabilization, and synchronization. However, controlling pattern dynamics in reaction-diffusion systems with fractional-order diffusion remains an unresolved problem. This paper presents a proportional-derivative (PD) control strategy for the Schnakenberg system with fractional-order diffusion and cross-diffusion. Theoretical analysis explores the amplitude equation near the Turing bifurcation threshold, determining the selection and stability of pattern formations. Numerical simulations demonstrate that the PD controller accomplishes the modification of pattern structures and suppression of Turing instability by adjusting only two control parameters. Additionally, it is found that for smaller fractional diffusion order, the region can accommodate more hexagonal and stripe patterns in space. This work contributes to the control of complex pattern dynamics and offers a new approach to enhancing stability in fractional reaction-diffusion systems.