Event-Triggered and Self-Triggered L ∞ Control for Markov Jump Stochastic Nonlinear Systems Under DoS Attacks

This article investigates event-triggered and self-triggered $\mathcal {L}_{\infty }$ control problems for the Markov jump stochastic nonlinear systems subject to denial-of-service (DoS) attacks. When attacks prevent system devices from obtaining valid information over networks, a new switched model with unstable subsystems is constructed to characterize the effect of DoS attacks. On the basis of the switched model, a multiple Lyapunov function method is utilized and a set of sufficient conditions incorporating the event-triggering scheme (ETS) and restriction of DoS attacks are provided to preserve $\mathcal {L}_{\infty }$ performance. In particular, considering that ETS based on mathematical expectation is difficult to be implemented on a practical platform, a self-triggering scheme (STS) without mathematical expectation is presented. Meanwhile, to avoid the Zeno behavior resulted from general exogenous disturbance, a positive lower bound is fixed in STS in advance. In addition, the exponent parameters are designed in STS to reduce triggering frequency. Based on the STS, the mean-square asymptotical stability and almost sure exponential stability are both discussed when the system is in the absence of exogenous disturbance. Finally, two examples are given to substantiate the effectiveness of the proposed method.