Abstract: We review the research on complex dynamical networks (CDNs) with impulsive effects. We provide a comprehensive and intuitive overview of the fundamental results and recent progress of CDNs with impulsive effects, where impulsive effects are considered from two aspects, i.e., impulsive control and impulsive perturbation. Five aspects of CDNs with impulsive effects are surveyed, including synchronizing impulses, desynchronizing impulses, adaptive-impulsive synchronization, pinning impulsive synchronization, and CDNs with stochastic and impulsive effects. Finally, conclusions and some future research directions are briefly addressed.

具有脉冲效应的复杂动态网络综述

[1]Amato F, de Tommasi G, Pironti A, 2013. Necessary and sufficient conditions for finite-time stability of impulsive dynamical linear systems. Automatica, 49(8):2546-2550.

[2]Buchanan M, 2002. Nexus: Small Worlds and the Groundbreaking Science of Networks. W. W. Norton, New York, USA.

[3]Cai SM, Zhou J, Xiang L, et al., 2008. Robust impulsive synchronization of complex delayed dynamical networks. Phys Lett A, 372(30):4990-4995.

[4]Chang YK, Nieto JJ, Zhao ZH, 2010. Existence results for a nondensely-defined impulsive neutral differential equation with state-dependent delay. Nonl Anal Hybr Syst, 4(3):593-599.

[5]Chen J, Li XD, Wang DQ, 2013. Asymptotic stability and exponential stability of impulsive delayed Hopfield neural networks. Abst Appl Anal, 2013:638496.

[6]Chen TP, Liu XW, Lu WL, 2007. Pinning complex networks by a single controller. IEEE Trans Circ Syst I, 54(6):1317-1326.

[7]Chen YS, Hwang RR, Chang CC, 2010. Adaptive impulsive synchronization of uncertain chaotic systems. Phys Lett A, 374(22):2254-2258.

[8]Farrow C, Heidel J, Maloney J, et al., 2004. Scalar equations for synchronous Boolean networks with biological applications. IEEE Trans Neur Netw, 15(2):348-354.

[9]Guan XP, Feng G, Chen CL, et al., 2007. A full delayed feedback controller design method for time-delay chaotic systems. Phys D, 227(1):36-42.

[10]Guan ZH, Chen GR, 1999. On delayed impulsive Hopfield neural networks. Neur Netw, 12(2):273-280.

[11]Guan ZH, Zhang H, 2008. Stabilization of complex network with hybrid impulsive and switching control. Chaos Sol Fract, 37(5):1372-1382.

[12]Guan ZH, Hill DJ, Shen XM, 2005. On hybrid impulsive and switching systems and application to nonlinear control. IEEE Trans Autom Contr, 50(7):1058-1062.

[13]Hong H, Choi MY, Kim BJ, 2002. Synchronization on small world networks. Phys Rev E, 65:026139.

[14]Hu JT, Sui GX, Lv XX, et al., 2018. Fixed-time control of delayed neural networks with impulsive perturbations. Nonl Anal Model Contr, 23(6):904-920.

[15]Huang C, Lu JQ, Ho DWC, et al., 2020. Stabilization of probabilistic Boolean networks via pinning control strategy. Inform Sci, 510:205-217.

[16]Huberman BA, Adamic LA, 1999. Growth dynamics of the World-Wide Web. Nature, 401(6749):131.

[17]Kauffman SA, 1969. Metabolic stability and epigenesist in randomly constructed genetic nets. J Theor Biol, 22(3):437-467.

[18]Khadra A, Liu XZ, Shen XM, 2009. Analyzing the robustness of impulsive synchronization coupled by linear delayed impulses. IEEE Trans Autom Contr, 54(4):923-928.

[19]Li BW, Lou JG, Liu Y, et al., 2019. Robust invariant set analysis of Boolean networks. Complexity, 2019:2731395.

[20]Li CX, Shi JP, Sun JT, 2011. Stability of impulsive stochastic differential delay systems and its application to impulsive stochastic neural networks. Nonl Anal Theory Methods Appl, 74(10):3099-3111.

[21]Li F, Sun J, 2011. Observability analysis of Boolean control networks with impulsive effects. IET Contr Theory Appl, 5(14):1609-1616.

[22]Li HL, Jiang YL, Wang ZL, et al., 2015. Parameter identification and adaptive-impulsive synchronization of uncertain complex networks with nonidentical topological structures. Optik, 126(24):5771-5776.

[23]Li HT, Wang YZ, 2017. Lyapunov-based stability and construction of Lyapunov functions for Boolean networks. SIAM J Contr Optim, 55(6):3437-3457.

[24]Li K, Lai CH, 2008. Adaptive-impulsive synchronization of uncertain complex dynamical networks. Phys Lett A, 372(10):1601-1606.

[25]Li X, Wang XF, Chen GR, 2004. Pinning a complex dynamical network to its equilibrium. IEEE Trans Circ Syst I, 51(10):2074-2087.

[26]Li XD, 2012. Further analysis on uniform stability of impulsive infinite delay differential equations. Appl Math Lett, 25(2):133-137.

[27]Li XD, Bohner M, 2012. An impulsive delay differential inequality and applications. Comput Math Appl, 64(6):1875-1881.

[28]Li XD, Fu XL, 2012. Lag synchronization of chaotic delayed neural networks via impulsive control. IMA J Math Contr Inform, 29(1):133-145.

[29]Li XD, Rakkiyappan R, 2013. Impulsive controller design for exponential synchronization of chaotic neural networks with mixed delays. Commun Nonl Sci Numer Simul, 18(6):1515-1523.

[30]Li XD, Shen JH, 2010. LMI approach for stationary oscillation of interval neural networks with discrete and distributed time-varying delays under impulsive perturbations. IEEE Trans Neur Netw, 21(10):1555-1563.

[31]Li XD, Song SJ, 2013. Impulsive control for existence, uniqueness, and global stability of periodic solutions of recurrent neural networks with discrete and continuously distributed delays. IEEE Trans Neur Netw Learn Syst, 24(6):868-877.

[32]Li XD, Song SJ, 2014. Research on synchronization of chaotic delayed neural networks with stochastic perturbation using impulsive control method. Commun Nonl Sci Numer Simul, 19(10):3892-3900.

[33]Li XD, Akca H, Fu XL, 2013. Uniform stability of impulsive infinite delay differential equations with applications to systems with integral impulsive conditions. Appl Math Comput, 219(14):7329-7337.

[34]Li XD, O’Regan D, Akca H, 2015a. Global exponential stabilization of impulsive neural networks with unbounded continuously distributed delays. IMA J Appl Math, 80(1):85-99.

[35]Li XD, Bohner M, Wang CK, 2015b. Impulsive differential equations: periodic solutions and applications. Automatica, 52:173-178.

[36]Li XD, Shen JH, Akca H, et al., 2015c. LMI-based stability for singularly perturbed nonlinear impulsive differential systems with delays of small parameter. Appl Math Comput, 250:798-804.

[37]Li XD, Caraballo T, Rakkiyappan R, et al., 2015d. On the stability of impulsive functional differential equations with infinite delays. Math Methods Appl Sci, 38(14):3130-3140.

[38]Li XD, Shen JH, Rakkiyappan R, 2018. Persistent impulsive effects on stability of functional differential equations with finite or infinite delay. Appl Math Comput, 329:14-22.

[39]Li XD, Ho DWC, Cao JD, 2019a. Finite-time stability and settling-time estimation of nonlinear impulsive systems. Automatica, 99:361-368.

[40]Li XD, Yang XY, Huang TW, 2019b. Persistence of delayed cooperative models: impulsive control method. Appl Math Comput, 342:130-146.

[41]Li YY, Lou JG, Wang Z, et al., 2018. Synchronization of dynamical networks with nonlinearly coupling function under hybrid pinning impulsive controllers. J Franklin Inst, 355(14):6520-6530.

[42]Li YY, Liu RJ, Lou JG, et al., 2019. Output tracking of Boolean control networks driven by constant reference signal. IEEE Access, 7:112572-112577.

[43]Li Z, Fang JA, Huang TW, et al., 2017. Synchronization of stochastic discrete-time complex networks with partial mixed impulsive effects. J Franklin Inst, 354(10):4196-4214.

[44]Lin DW, Li XD, O’Regan D, 2013. μ-stability of infinite delay functional differential systems with impulsive effects. Appl Anal, 92(1):15-26.

[45]Liu B, Liu XZ, Chen GR, et al., 2005. Robust impulsive synchronization of uncertain dynamical networks. IEEE Trans Circ Syst I, 52(7):1431-1441.

[46]Liu B, Teo KL, Liu XZ, 2008. Robust exponential stabilization for large-scale uncertain impulsive systems with coupling time-delays. Nonl Anal Theory Methods Appl, 68(5):1169-1183.

[47]Liu DF, Wu ZY, Ye QL, 2014. Structure identification of an uncertain network coupled with complex-variable chaotic systems via adaptive impulsive control. Chin Phys B, 23(4):040504.

[48]Liu J, Li XD, 2013. Impulsive stabilization of high-order nonlinear retarded differential equations. Appl Math, 58:347-367.

[49]Liu Y, Li BW, Lu JQ, et al., 2017. Pinning control for the disturbance decoupling problem of Boolean networks. IEEE Trans Autom Contr, 62(12):6595-6601.

[50]Liu ZW, Guan ZH, Shen XM, et al., 2012. Consensus of multi-agent networks with aperiodic sampled communication via impulsive algorithms using position-only measurements. IEEE Trans Autom Contr, 57(10):2639-2643.

[51]Lu JG, Chen GR, 2009. Global asymptotical synchronization of chaotic neural networks by output feedback impulsive control: an LMI approach. Chaos Sol Fract, 41(5):2293-2300.

[52]Lu JQ, Ho DWC, Cao JD, 2010. A unified synchronization criterion for impulsive dynamical networks. Automatica, 46(7):1215-1221.

[53]Lu JQ, Ho DWC, Cao JD, et al., 2011. Exponential synchronization of linearly coupled neural networks with impulsive disturbances. IEEE Trans Neur Netw, 22(2):329-336.

[54]Lu JQ, Wang ZD, Cao JD, et al., 2012a. Pinning impulsive stabilization of nonlinear dynamical networks with timevarying delay. Int J Bifurc Chaos, 22(7):1250176.

[55]Lu JQ, Kurths J, Cao JD, et al., 2012b. Synchronization control for nonlinear stochastic dynamical networks: pinning impulsive strategy. IEEE Trans Neur Netw Learn Syst, 23(2):285-292.

[56]Lu JQ, Ho DWC, Cao JD, et al., 2013. Single impulsive controller for globally exponential synchronization of dynamical networks. Nonl Anal Real World Appl, 14(1):581-593.

[57]Lu JQ, Ding CD, Lou JG, et al., 2015. Outer synchronization of partially coupled dynamical networks via pinning impulsive controllers. J Franklin Inst, 352(11):5024-5041.

[58]Lu WL, Li X, Rong ZH, 2010. Global stabilization of complex networks with digraph topologies via a local pinning algorithm. Automatica, 46(1):116-121.

[59]Lv XX, Li XD, Cao JD, et al., 2018a. Exponential synchronization of neural networks via feedback control in complex environment. Complexity, 2018:4352714.

[60]Lv XX, Rakkiyappan R, Li X, 2018b. μ-stability criteria for nonlinear differential systems with additive leakage and transmission time-varying delays. Nonl Anal Model Contr, 23(3):380-400.

[61]Mei GF, Wu XQ, Wang YF, et al., 2018. Compressive sensing-based structure identification for multilayer networks. IEEE Trans Cybern, 48(2):754-764.

[62]Qin JH, Zheng WX, Gao HJ, 2011. On pinning synchronisability of complex networks with arbitrary topological structure. Int J Syst Sci, 42(9):1559-1571.

[63]Rakshit S, Majhi S, Bera BK, et al., 2017. Time-varying multiplex network: intralayer and interlayer synchronization. Phys Rev E, 96:062308.

[64]Shmulevich I, Dougherty ER, Kim S, et al., 2002. Probabilistic Boolean networks: a rule-based uncertainty model for gene regulatory networks. Bioinformatics, 18(2):261-274.

[65]Sun JT, Zhang YP, Qiao F, et al., 2004. Some impulsive synchronization criterions for coupled chaotic systems via unidirectional linear error feedback approach. Chaos Sol Fract, 19(5):1049-1055.

[66]Sun L, Lu J, Ching W, 2020. Switching-based stabilization of aperiodic sampled-data Boolean control networks with all subsystems unstable. Front Inform Technol Electron Eng, 21(2):260-267.

[67]Tan X, Cao J, Li X, 2019. Consensus of leader-following multiagent systems: a distributed event-triggered impulsive control strategy. IEEE Trans Cybern, 49(3):792-801.

[68]Tang Y, Qian F, Gao HJ, et al., 2014. Synchronization in complex networks and its application—a survey of recent advances and challenges. Ann Rev Contr, 38(2):184-198.

[69]Tong LY, Liu Y, Li YY, et al., 2018. Robust control invariance of probabilistic Boolean control networks via event-triggered control. IEEE Access, 6:37767-37774.

[70]Um J, Minnhagen P, Kim BJ, 2011. Synchronization in interdependent networks. Chaos, 21(2):025106.

[71]Vinodkumar A, Senthilkumar T, Li XD, 2018. Robust exponential stability results for uncertain infinite delay differential systems with random impulsive moments. Adv Differ Equat, 2018:39.

[72]Wang HL, Chen GR, 2015. On the initial function space of time-delayed systems: a time-delayed feedback control perspective. J Franklin Inst, 352(8):3243-3249.

[73]Wang JL, Wu HN, 2012. Synchronization criteria for impulsive complex dynamical networks with time-varying delay. Nonl Dynam, 70(1):13-24.

[74]Wang L, Li XD, 2013. μ-stability of impulsive differential systems with unbounded time-varying delays and nonlinear perturbations. Math Methods Appl Sci, 36(11):1440-1446.

[75]Wang X, She K, Zhong SM, et al., 2017a. Pinning cluster synchronization of delayed complex dynamical networks with nonidentical nodes and impulsive effects. Nonl Dynam, 88(4):2771-2782.

[76]Wang X, Liu XZ, She K, et al., 2017b. Pinning impulsive synchronization of complex dynamical networks with various time-varying delay sizes. Nonl Anal Hybr Syst, 26:307-318.

[77]Wang XF, Chen GR, 2002a. Pinning control of scale-free dynamical networks. Phys A, 310(3-4):521-531.

[78]Wang XF, Chen GR, 2002b. Synchronization in scale-free dynamical networks: robustness and fragility. IEEE Trans Circ Syst I, 49(1):54-62.

[79]Wang YQ, Lu JQ, Liang JL, et al., 2019. Pinning synchronization of nonlinear coupled Lur’e networks under hybrid impulses. IEEE Trans Circ Syst II, 66(3):432-436.

[80]Wei X, Wu XQ, Chen SH, et al., 2018. Cooperative epidemic spreading on a two-layered interconnected network. SIAM J Appl Dynam Syst, 17(2):1503-1520.

[81]Wu B, Liu Y, Lu JQ, 2012. New results on global exponential stability for impulsive cellular neural networks with any bounded time-varying delays. Math Comput Model, 55(3-4):837-843.

[82]Wu ZY, 2015. Complex hybrid synchronization of complex variable dynamical network via impulsive control. Optik Int J Light Electron Opt, 126(19):2110-2114.

[83]Xing W, Shi P, Agarwal RK, et al., 2019. A survey on global pinning synchronization of complex networks. J Franklin Inst, 356(6):3590-3611.

[84]Xu F, Dong L, Wang D, et al., 2015. Globally exponential stability of nonlinear impulsive switched systems. Math Notes, 97(5-6):803-810.

[85]Xu XJ, Li HT, Li YL, et al., 2018a. Output tracking control of Boolean control networks with impulsive effects. Math Methods Appl Sci, 41(4):1554-1564.

[86]Xu XJ, Liu YS, Li HT, et al., 2018b. Synchronization of switched Boolean networks with impulsive effects. Int J Biomath, 11(6):1850080.

[87]Xu ZL, Peng DX, Li XD, 2019. Synchronization of chaotic neural networks with time delay via distributed delayed impulsive control. Neur Netw, 118:332-337.

[88]Yang D, Li XD, Shen JH, et al., 2018. State-dependent switching control of delayed switched systems with stable and unstable modes. Math Methods Appl Sci, 41(16):6968-6983.

[89]Yang JJ, Lu JQ, Lou JG, et al., 2020. Synchronization of drive-response Boolean control networks with impulsive disturbances. Appl Math Comput, 364:124679.

[90]Yang T, 2001. Impulsive Control Theory. Springer, Berlin.

[91]Yang XS, Lu JQ, 2016. Finite-time synchronization of coupled networks with Markovian topology and impulsive effects. IEEE Trans Autom Contr, 61(8):2256-2261.

[92]Yang XS, Cao JD, Lu JQ, 2011a. Synchronization of delayed complex dynamical networks with impulsive and stochastic effects. Nonl Anal Real World Appl, 12(4):2252-2266.

[93]Yang XS, Huang CX, Zhu QX, 2011b. Synchronization of switched neural networks with mixed delays via impulsive control. Chaos Sol Fract, 44(10):817-826.

[94]Yang XY, Li XD, 2018. Finite-time stability of linear nonautonomous systems with time-varying delays. Adv Differ Equat, 2018:101.

[95]Yang XY, Li XD, Xi Q, et al., 2018. Review of stability and stabilization for impulsive delayed systems. Math Biosci Eng, 15(6):1495-1515.

[96]Yang ZC, Xu DY, 2005. Stability analysis of delay neural networks with impulsive effects. IEEE Trans Circ Syst II, 52(8):517-521.

[97]Zhang G, Liu ZR, Ma ZJ, 2007. Synchronization of complex dynamical networks via impulsive control. Chaos, 17:043126.

[98]Zhang QJ, Lu JA, 2009. Impulsively control complex networks with different dynamical nodes to its trivial equilibrium. Comput Math Appl, 57(7):1073-1079.

[99]Zhang QJ, Luo J, Wan L, 2013. Parameter identification and synchronization of uncertain general complex networks via adaptive-impulsive control. Nonl Dynam, 71:353-359.

[100]Zhang XY, Li XD, Han XP, 2017a. Design of hybrid controller for synchronization control of Chen chaotic system. J Nonl Sci Appl, 10(6):3320-3327.

[101]Zhang XY, Lv XX, Li XD, 2017b. Sampled-data-based lag synchronization of chaotic delayed neural networks with impulsive control. Nonl Dynam, 90:2199-2207.

[102]Zhang XY, Li XD, Cao JD, et al., 2018. Design of memory controllers for finite-time stabilization of delayed neural networks with uncertainty. J Franklin Inst, 355(13):5394-5413.

[103]Zhang Y, Sun JT, Feng G, 2009. Impulsive control of discrete systems with time delay. IEEE Trans Autom Contr, 54(4):830-834.

[104]Zheng S, 2017. Pinning and impulsive synchronization control of complex dynamical networks with non-derivative and derivative coupling. J Franklin Inst, 354(14):6341-6363.

[105]Zhou J, Xiang L, Liu ZR, 2007. Synchronization in complex delayed dynamical networks with impulsive effects. Phys A, 384(2):684-692.

[106]Zhu QX, Liu Y, Lu JQ, et al., 2018. On the optimal control of Boolean control networks. SIAM J Contr Optim, 56(2):1321-1341.

[107]Zhu QX, Liu Y, Lu JQ, et al., 2019. Further results on the controllability of Boolean control networks. IEEE Trans Autom Contr, 64(1):440-442.

[108]Zhu SY, Lou JG, Liu Y, et al., 2018. Event-triggered control for the stabilization of probabilistic Boolean control networks. Complexity, 2018:9259348.