Secure control for simplicial complexes under hybrid cyber attacks–an event-triggered impulsive approach

This paper investigates delayed event-triggered impulsive control for simplicial complexes under stochastic hybrid cyber attacks, such as denial-of-service (DoS) and deception attacks. Firstly, leveraging the properties of d-simplices and link-dependent topology, an equivalent adjacency matrix is introduced to model D-simplicial complexes as a linear system. Secondly, a delayed event-triggered mechanism is developed to ensure efficient resource utilisation while preventing the Zeno phenomenon through a delayed waiting time. The proposed control strategy framework combines an event-triggered mechanism with topology-dependent impulsive control, where impulsive instants are dynamically determined. Additionally, the impact of hybrid attacks is modelled by incorporating DoS and deception attacks that follow a Bernoulli distribution. Thirdly, sufficient conditions for mean-square exponential synchronisation of simplicial complexes under attacks are derived using Lyapunov functions and inequality recursion techniques. Finally, numerical simulations and examples are presented to validate the effectiveness and robustness of the proposed approach.