Spatiotemporal Evolution Control of Malicious Virus Propagation in Cyber Physical Systems via PD Feedback Control

This article puts forward a malicious virus propagation model with diffusion and time delay in cyber physical systems (CPSs). The appearance of diffusion and time delay may lead to Turing instability and Hopf bifurcation, respectively, which are two significant features for the spread of malicious viruses in CPS. Therefore, understanding the evolution mechanism of malicious virus propagation and controlling its Turing instability and Hopf bifurcation are the overriding problems of paramount importance in CPS. However, studies on control of Turing instability and Hopf bifurcation for time-delay reaction-diffusion systems have not been reported. We first propose a proportional derivative (PD) feedback control program, which provides strategic guidance for the prediction and control of malware spread. The specific conditions of Turing instability and Hopf bifurcation are constructed. The stability and direction of Hopf bifurcation are explored for the controlled malicious virus propagation model. It is proven that the objective of suppressing Turing instability and regulating Hopf bifurcation is successfully accomplished through the PD control scheme. Finally, some numerical simulation cases are supplied to authenticate the theoretical results.