Delay-dependent impulsive synchronization control for coupled dynamic networks under hybrid cyberattack

This paper studies the secure impulsive synchronization of dynamic networks (DNs) under hybrid deception attacks. First, considering that the secure impulsive control is limited to the single deception attack and the same attack probability in existing literature, a new hybrid cyberattack model is proposed, and deception attacks are described by Bernoulli distribution with different probabilities. Second, a delayed secure impulsive law is designed, and the average impulsive delay method is adopted to extract the time delay existing in impulses and then this information is integrated to mitigate the malicious information injected by hybrid deception attacks. It is shown that the delay can effectively weaken the effects of hybrid deception attacks and bring a synchronizing effect to DNs. The relaxation conditions of the secure bounded synchronization for coupled DNs are established by using the average impulsive interval method. The proposed results does not have any restriction on the lower/upper bounds of continuous impulsive control signals, which is more flexible than it used to be. Finally, our delayed secure impulsive control results are demonstrated by numerical examples.