Letter Intermittent Control for Fixed-Time Synchronization of Coupled Networks Yongbao Wu, Ziyuan Sun, Guangtao Ran, and Lei Xue    Dear Editor, This letter deals with fixed-time synchronization (Fd-TS) of com- plex networks (CNs) under aperiodically intermittent control (AIC) for the first time. The average control rate and a new Lyapunov func- tion are proposed to overcome the difficulty of dealing with fixed- time stability/synchronization of CNs for AIC. Based on the Lya- punov and graph-theoretical methods, a Fd-TS criterion of CNs is given. Moreover, the method of this letter is also applicable to the study of finite-time synchronization of CNs for AIC. Finally, the the- oretical results are applied to study the Fd-TS of oscillator systems, and simulation results are given to verify the effectiveness of the results. Recently, the dynamics of CNs have attracted extensive attention due to their wide applications in real-world networks. As one of the most important collective behaviors of CNs, synchronization has received considerable interest in many fields [1]. It should be noted that the most existing results about the synchronization of CNs stud- ied asymptotic synchronization and exponential synchronization [2], and they are often classified into infinite-time synchronization. In many practical problems, achieving synchronization within a finite time is more desirable and useful. Therefore, finite-time synchroniza- tion (Fe-TS) has been investigated by many researchers. In contrast with infinite-time synchronization, Fe-TS has been reported to pos- sess faster convergence and better performance against uncertainties and disturbances. Nevertheless, a significant limitation of Fe-TS is that the settling time depends on the initial values. In many practical systems, the ini- tial values may be difficult to obtain in advance. Fortunately, this problem was overcome by Polyakov [3] through introducing the con- cept of fixed-time stability and presenting fundamental results on fixed-time stability. Inspired by Polyakov’s novel fixed-time stabil- ity theory, there are some follow-up works about fixed-time stability for various CNs [4]–[6]. Compared to Fe-TS, the settling time of Fd- TS is determined by the designed controller parameters, which do not rely on the initial values and can be estimated in advance. Further- more, many practical systems such as microgrid systems and space- craft dynamics usually desire to achieve fixed-time convergence. Consequently, it is meaningful and necessary to further explore the Fd-TS of CNs both in theory and methods. In general, it is difficult to realize self-synchronization for CNs due to the complexity of node dynamics and topologies. Therefore, many kinds of control techniques have been employed to achieve the syn- chronization of CNs in [7]. Different from the continuous control schemes, the discontinuous control methods such as intermittent con- trol (IC) [2], [8], and impulsive control have been extensively stud- ied because they can reduce control cost as well as the number of information exchanges. It is known that IC can be divided into peri- odically IC (PIC), and AIC [2], and AIC takes PIC as a special case; thus, it is more general to consider AIC. For AIC, scholars mainly considered asymptotic synchronization [2], exponential synchroniza- tion [9], and Fe-TS. However, there are few results focusing on the Fd-TS for CNs by AIC. The existing theory to study asymptotic syn- chronization, exponential synchronization, and Fe-TS cannot be directly extended to Fd-TS. Moreover, the Fd-TS theoretical frame- work based on AIC is not established. Therefore, it is urgently neces- sary to develop a new theory and methods to investigate the Fd-FS of CNs via AIC, which motivates this work. The main contributions of this letter are as follows. sup i {ζ i+1 - µ i }≤ con1 sup i {ζ i+1 - ζ i }≤ con2 con1 con2 ζ i µ i ϑ η i = (µ i - ζ i )/(ζ i+1 - ζ i ) ϑ ≥ lim inf i→∞ {η i } 1) Unlike the existing literature dealing with finite-time stability/ synchronization for IC, in this letter, we establish a theoretical frame- work of fixed-time stability/synchronization for AIC for the first time. 2) Compared with the existing literature [4], an auxiliary func- tion is introduced to consider the Fd-TS of CNs under IC, which allows the control function in the rest intervals of IC to be zero. In the existing literature [4], the control function in the rest intervals of IC is not zero, which may be regarded as a switching control rather than an IC in a general sense. Thus, the control strategy proposed in this letter is more general. 3) In [2], [9], some scholars mainly con- sidered the asymptotic synchronization or exponential synchroniza- tion for AIC. Moreover, the existing literature have certain restric- tions on the control intervals, such as , , where and are positive constants. For parame- ters and , see Fig. 1. This letter uses the average control rate in (5), not the infimum of the control rate , which is easy to satisfy the condition of the theorem because of . In addition, the idea of using the average control rate in this letter also can apply to the study of finite-time stability/syn- chronization for AIC, which is more general. Mathematical model: The CNs model is considered as follows: ˙ x k (t) = f k ( x k (t), t) + n ∑ h=1 b kh α kh ( x k (t), x h (t)), k ∈N (1) x k (t) = ( x k1 (t), x k2 (t),..., x km (t)) T ∈ R m N = {1, 2,..., n} α kh : R m × R m → R m f k : R m × R + → R m b kh ≥ 0 b kk = 0 k ∈N where and ; coupled function ; and is coupling weight and for all . System (1) is considered as a master system, and we consider the following system as slave system: ˙ z k (t) = f k (z k (t), t) + n ∑ h=1 b kh α kh (z k (t), z h (t)) + u k (t), k ∈N (2) Corresponding author: Yongbao Wu. Citation: Y. B. Wu, Z. Y. Sun, G. T. Ran, and L. Xue, “Intermittent control for fixed-time synchronization of coupled networks,” IEEE/CAA J. Autom. Sinica, vol. 10, no. 6, pp. 1488–1490, Jun. 2023. Y. B. Wu and L. Xue are with the School of Automation, Southeast University, Nanjing 210096, China (e-mail: yongbaowu199211@163.com; leixue@seu.edu.cn). Z. Y. Sun is with the Department of Applied Mathematics, University of Waterloo, Ontario N2J 4N3, Canada (e-mail: ziyuan.sun@uwaterloo.ca). G. T. Ran is with the Department of Control Science and Engineering, Harbin Institute of Technology, Harbin 150001, China (e-mail: ranguangtao@ hit.edu.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JAS.2023.123363 ζ i ζ i+1 μ i+1 μ i+2 ζ i+2 ζ i+3 μ i … … Fig. 1. Schematic diagram of AIC. Blue and yellow areas represent control and rest intervals, respectively. 1488 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 10, NO. 6, JUNE 2023