How to regulate pattern formations for malware propagation in cyber-physical systems

Malware propagation can be fatal to cyber-physical systems. How to detect and prevent the spatiotemporal evolution of malware is the major challenge we are facing now. This paper is concerned with the control of Turing patterns arising in a malware propagation model depicted by partial differential equations for the first time. From the control theoretic perspective, the goal is not only to predict the formation and evolution of patterns but also to design the spatiotemporal state feedback scheme to modulate the switch of patterns between different modes. The Turing instability conditions are obtained for the controlled malware propagation model with cross-diffusion. Then, the multi-scale analysis is carried out to explore the amplitude equations near the threshold of Turing bifurcation. The selection and stability of pattern formations are determined based on the established amplitude equations. It is proved that the reaction-diffusion propagation model has three types of patterns: hexagonal pattern, striped pattern, and mixed pattern, and selecting the appropriate control parameters can make the pattern transform among the three patterns. The results of the analysis are numerically verified and provide valuable insights into dynamics and control of patterns embedded in reaction-diffusion systems.