1 Guangdong Provincial Key Laboratory of Intelligent Decision and Cooperative Control, School of Automation, Guangdong University of Technology, Guangzhou 510006, China. Electronic address: yanglx@mail2.gdut.edu.cn.
2 Guangdong Provincial Key Laboratory of Intelligent Decision and Cooperative Control, School of Automation, Guangdong University of Technology, Guangzhou 510006, China. Electronic address: taojiedyx@163.com.
3 Guangdong Provincial Key Laboratory of Intelligent Decision and Cooperative Control, School of Automation, Guangdong University of Technology, Guangzhou 510006, China. Electronic address: yonghua.liu@outlook.com.
4 Guangdong Provincial Key Laboratory of Intelligent Decision and Cooperative Control, School of Automation, Guangdong University of Technology, Guangzhou 510006, China. Electronic address: xuyong809@163.com.
5 Department of Mechanical and Industrial Engineering, Concordia University, Montreal,

This paper studies the energy scheduling for Denial-of-Service (DoS) attack against remote state estimation over multi-hop networks. A smart sensor observes a dynamic system, and transmits its local state estimate to a remote estimator. Due to the limited communication range of the sensor, some relay nodes are employed to deliver data packets from the sensor to the remote estimator, which constitutes a multi-hop network. To maximize the estimation error covariance with energy constraint, a DoS attacker needs to determine the energy level implemented on each channel. This problem is formulated as an associated Markov decision process (MDP), and the existence of an optimal deterministic and stationary policy (DSP) is proved for the attacker. Besides, a simple threshold structure of the optimal policy is obtained, which significantly reduces the computational complexity. Furthermore, an up-to-date deep reinforcement learning (DRL) algorithm, dueling double Q-network (D3QN), is introduced to approximate the optimal policy. Finally, a simulation example illustrates the developed results and verifies the effectiveness of D3QN for optimal DoS attack energy scheduling.