Synchronization of fractional-order delayed coupled networks with reaction-diffusion terms and Neumann boundary value conditions

This paper investigates the adaptive pinning strategies for achieving synchronization of fractional-order delayed coupled networks with reaction-diffusion terms and a digraph topology by incorporating Neumann boundary value conditions. By employing the inf-sup method, a novel fractional-order inequality is proved. The classical Poincare inequality is also extended by utilizing the Holder inequality. Two types of control laws are developed to achieve synchroniza-tion: one with control gains dependent solely on time, and another with control gains dependent on both space and time. For each case, adaptive control laws and synchronization criteria based on matrix inequalities are proposed. Finally, the effectiveness of the synchronization results is demonstrated through two numerical examples.