## 1 Introduction
However, the above achievements are research on the security of deterministic CPSs, ignoring the research on CPSs with random jumps under cyber attacks. Generally, it is difficult for the system to obtain measurement or control signals in a continuous manner. In order to overcome persistent communication, many sampled-data control policies based on a time-triggered communication mechanism have been widely investigated, see [29; 30; 31; 32]. However, due to the limited communication bandwidth, this implies that the time-triggered sampling strategies can generate unnecessary consumption and computing resources. Consequently, the oversampling problem needs to be eliminated. As an effective solution to alleviate the communication burden, the event-triggered mechanism (ETM) regarded as _\"on-demand communication\"_ was reported in [33; 34; 35; 36; 37], where the authors in [33] proposed a co-design method for the resilient event-triggered control (ETC) strategy to tolerate DoS attacks as much as possible. To obtain a higher communication efficiency, a novel switching-like ETC strategy for continuous CPSs was developed in [34] to balance the desired communication rate and security performance. In [35], a fully distributed secure cooperative control protocol for CPSs was developed to guarantee asymptotic consensus against distributed DoS attacks from multiple adversaries. The authors in [36] formulated a stochastic ETC scheme to overcome the stochastic DoS attacks by fully using the dynamic features of communication in the open network. Particularly, CPSs are vulnerable to environmental mutations or random failures of physical components, resulting in their potential to be composed of multiple subsystems with different structures and parameters. Therefore, it is necessary to develop appropriate security control strategies to ensure that jump CPSs still have an acceptable level of operation under cyber attacks. In addition, the authors in [37] developed a finite-time ETC strategy for nonlinear semi-Markov jump CPS to quickly defend against FDI attacks in finite time. Note that most ETC schemes adjust their event-triggering parameters in advance to counter the negative impact of DoS attacks on data transmission. This indicates that the traditional strategies have certain limitations when dealing with unpredictable DoS jamming attacks. Recently,the impact of DoS attacks was transformed into the uncertainty in triggering rule reported in [38], which plays a positive role in solving the mentioned problem. However, very few works are available to solve similar topics for stochastic CPSs under DoS attacks and Markovian switching, to our knowledge. These facts inspire us to proceed with the present work. This paper proposes a novel dual-mode resilient event-triggered control strategy for MJCPSs under DoS attacks. The salient contributions are as follows:  
1. A novel resilient event-triggering rule that relies on DoS signals is designed to select corresponding control strategies on demand based on the current network service quality. 2. By analyzing the inner relationship between the system mode and the triggering instant under DoS attacks, the minimum and maximum inter-execution intervals are calculated to avoid Zeno behavior and singular triggering. 3. Based on the LMI method and Lyapunov stability theory, sufficient conditions for the existence of security controller gains and resilient triggering parameters are given in the form of concise LMIs simultaneously. The outline of this paper is organized as follows. Section 2 presents preliminaries and problem formulation. In Section 3, stability analysis under resilient event-triggered rule and DoS attacks is investigated in detail. The dual-mode security controller is designed in Section 4. In Section 5, a simulation example is presented to illustrate the effectiveness of the proposed method. Finally, Section 6 summarizes this paper.