Pattern Dynamics Optimization of the Schnakenberg System Based on State Feedback Control Strategy

This paper introduces a state feedback control strategy for the Schnakenberg system under the influence of cross-diffusion for the first time. Taking the cross-diffusion coefficient as bifurcation parameter, the necessary conditions for the Hiring instability of the controlled system are analyzed. Numerical simulations show that the introduction of state feedback control can not only alter the pattern structure, but also effectively suppress the occurrence of the Hiring instability. Changes in control parameter can lead to alterations in the Hiring bifurcation threshold.

I. Introduction

Pattern formation can be induced by diffusions which has been proposed by Turing in 1952s [1]. Therefore, these kinds of patterns can also be called Turing patterns. He believed that patterns could emerge in reaction-diffusion systems involving different substances with significant differences in diffusion rates, driven by the spontaneous breaking of symmetry. Many scholars studied pattern formation by reaction-diffusion systems. Many chemical reactions exhibit diffusion phenomena, and the diffusion processes of various chemical substances can be described by reaction-diffusion systems. A series of chemical reaction-diffusion systems have been developed, such as the Schnakenberg model [2], the Gray-Scott model [3], and the Gierer-Meinhardt model [4].