In the past decades, complex systems with impulsive effects and logical dynamics have received much attention in both the natural and social sciences. This historical survey briefly introduces relevant studies on impulsive differential systems (IDSs) and logical networks (LNs), respectively. To begin with, we investigate five aspects of IDSs containing fundamental theory, Lyapunov stability, input-to-state stability, hybrid impulses and delay-dependent impulses. Next, we compactly summarize the research status of some problems of LNs including controllability, stability and stabilization, observability and current research. Moreover, some significant applications of proposed results are illustrated. Finally, based on this overview, we further discuss some future work on complex systems with impulsive effects and logical dynamics.
Citation: |
[1] |
T. Akutsu, M. Hayashida, W. Ching and M. Ng, Control of Boolean networks: Hardness results and algorithms for tree structured networks, Journal of Theoretical Biology, 244 (2007), 670-679.
doi: 10.1016/j.jtbi.2006.09.023.![]() ![]() ![]() |
[2] |
J. Benford and J. Swegle, Applications of high power microwaves, 1992 9th International Conference on High-Power Particle Beams, (1992).
doi: 10.1109/PLASMA.1992.697928.![]() ![]() |
[3] |
H. Chen and J. Liang, Local synchronization of interconnected Boolean networks with stochastic disturbances, IEEE Transactions on Neural Networks and Learning Systems, 31 (2019), 452-463.
![]() |
[4] |
W.-H. Chen, S. Luo and W. X. Zheng, Impulsive synchronization of reaction–diffusion neural networks with mixed delays and its application to image encryption, IEEE Transactions on Neural Networks and Learning Systems, 27 (2016), 2696-2710.
doi: 10.1109/TNNLS.2015.2512849.![]() ![]() |
[5] |
W.-H. Chen, D. Wei and W. Zheng, Delayed impulsive control of Takagi-Sugeno fuzzy delay systems, IEEE Transactions on Fuzzy Systems, 21 (2013), 516-526.
doi: 10.1109/TFUZZ.2012.2217147.![]() ![]() |
[6] |
W.-H. Chen and W.-X. Zheng, Exponential stability of nonlinear time-delay systems with delayed impulse effects, Automatica, 47 (2011), 1075-1083.
doi: 10.1016/j.automatica.2011.02.031.![]() ![]() ![]() |
[7] |
D. Cheng, Semi-tensor product of matrices and its application to Morgens problem, Science in China Series: Information Sciences, 44 (2001), 195-212.
![]() ![]() |
[8] |
D. Cheng, Disturbance decoupling of Boolean control networks, IEEE Transactions on Automatic Control, 56 (2011), 2-10.
doi: 10.1109/TAC.2010.2050161.![]() ![]() ![]() |
[9] |
D. Cheng, F. He, H. Qi and T. Xu, Modeling, analysis and control of networked evolutionary games, IEEE Transactions on Automatic Control, 60 (2015), 2402-2415.
doi: 10.1109/TAC.2015.2404471.![]() ![]() ![]() |
[10] |
D. Cheng, C. Li and F. He, Observability of Boolean networks via set controllability approach, Systems & Control Letters, 115 (2018), 22-25.
doi: 10.1016/j.sysconle.2018.03.004.![]() ![]() ![]() |
[11] |
D. Cheng and H. Qi, Controllability and observability of Boolean control networks, Automatica, 45 (2009), 1659-1667.
doi: 10.1016/j.automatica.2009.03.006.![]() ![]() ![]() |
[12] |
D. Cheng, H. Qi, Z. Li and J. Liu, Stability and stabilization of Boolean networks, International Journal of Robust and Nonlinear Control, 21 (2011), 134-156.
doi: 10.1002/rnc.1581.![]() ![]() ![]() |
[13] |
D. Cheng, H. Qi and Z. Liu, From STP to game-based control, Science China Information Sciences, 61 (2018), 010201.
doi: 10.1007/s11432-017-9265-2.![]() ![]() ![]() |
[14] |
W. Ching, S. Zhang, Y. Jiao, T. Akutsu, N. Tsing and A. Wong, Optimal control policy for probabilistic Boolean networks with hard constraints, IET Systems Biology, 3 (2009), 90-99.
doi: 10.1049/iet-syb.2008.0120.![]() ![]() |
[15] |
S. Dashkovskiy and P. Feketa, Input-to-state stability of impulsive systems and their networks, Nonlinear Analysis: Hybrid Systems, 26 (2017), 190-200.
doi: 10.1016/j.nahs.2017.06.004.![]() ![]() ![]() |
[16] |
S. Dashkovskiy and A. Mironchenko, Input-to-state stability of nonlinear impulsive systems, SIAM Journal on Control and Optimization, 51 (2013), 1962–1987.
doi: 10.1137/120881993.![]() ![]() ![]() |
[17] |
E. Dubrova, Finding matching initial states for equivalent NLFSRs in the Fibonacci and the Galois configurations, IEEE Transactions on Information Theory, 56 (2010), 2961-2966.
doi: 10.1109/TIT.2010.2046250.![]() ![]() ![]() |
[18] |
B. Faryabi, A. Datta and E. Dougherty, On approximate stochastic control in genetic regulatory networks, IET Systems Biology, 1 (2007), 361-368.
doi: 10.1049/iet-syb:20070015.![]() ![]() |
[19] |
E. Fornasini and M. Valcher, Observability, reconstructibility and state observers of Boolean control networks, IEEE Transactions on Automatic Control, 58 (2013), 1390-1401.
doi: 10.1109/TAC.2012.2231592.![]() ![]() ![]() |
[20] |
L. Gao, D. Wang and G. Wang, Further results on exponential stability for impulsive switched nonlinear time-delay systems with delayed impulse effects, Applied Mathematics and Computation, 268 (2015), 186-200.
doi: 10.1016/j.amc.2015.06.023.![]() ![]() ![]() |
[21] |
K. Gopalsamy and B. Zhang, On delay differential equations with impulses, Journal of Mathematical Analysis and Applications, 139 (1989), 110-122.
doi: 10.1016/0022-247X(89)90232-1.![]() ![]() ![]() |
[22] |
Z.-H. Guan, G. Chen and T. Ueta, On impulsive control of a periodically forced chaotic pendulum system, IEEE Transactions on Automatic Control, 45 (2000), 1724-1727.
doi: 10.1109/9.880633.![]() ![]() ![]() |
[23] |
Z.-H. Guan and N. Liu, Generating chaos for discrete time-delayed systems via impulsive control, Chaos, 20 (2010), 013135.
doi: 10.1063/1.3266929.![]() ![]() ![]() |
[24] |
Z.-H. Guan, Z.-W. Liu, G. Feng and W. Yan-Wu, Synchronization of complex dynamical networks with time-varying delays via impulsive distributed control, IEEE Transactions on Circuits and Systems I: Regular Papers, 57 (2010), 2182-2195.
![]() |
[25] |
G. Zhao, Y. Wang and H. Li, A matrix approach to the modeling and analysis of networked evolutionary games with finite memories, IEEE/CAA Journal of Automatica Sinica, 5 (2018), 818-826.
doi: 10.1109/JAS.2016.7510259.![]() ![]() |
[26] |
Y. Guo, Observability of Boolean control networks using parallel extension and set reachability, IEEE Transactions on Neural Networks and Learning Systems, 29 (2018), 6402-6408.
doi: 10.1109/TNNLS.2018.2826075.![]() ![]() ![]() |
[27] |
Y. Guo, P. Wang, W. Gui and C. Yang, Set stability and set stabilization of Boolean control networks based on invariant subsets, Automatica, 61 (2015), 106-112.
doi: 10.1016/j.automatica.2015.08.006.![]() ![]() ![]() |
[28] |
D. Haltara and G. Ankhbayar, Using the maximum principle of impulse control for ecology-economical models, Ecological Modelling, 216 (2008), 150-156.
doi: 10.1016/j.ecolmodel.2008.03.025.![]() ![]() |
[29] |
W. He, G. Chen, Q. L. Han and F. Qian, Network-based leader-following consensus of nonlinear multi-agent systems via distributed impulsive control, Information Sciences, 380 (2017), 145-158.
![]() |
[30] |
J. P. Hespanha, D. Liberzon and A. R. Teel, Lyapunov conditions for input-to-state stability of impulsive systems, Automatica, 44 (2008), 2735-2744.
doi: 10.1016/j.automatica.2008.03.021.![]() ![]() ![]() |
[31] |
J. Hu, G. Sui, X. Lu and X. Li, Fixed-time control of delayed neural networks with impulsive perturbations, Nonlinear Analysis: Modelling and Control, 23 (2018), 904-920.
doi: 10.15388/NA.2018.6.6.![]() ![]() ![]() |
[32] |
M.-J. Hu, J.-W. Xiao, R.-B. Xiao and W.-H. Chen, Impulsive effects on the stability and stabilization of positive systems with delays, Journal of the Franklin Institute, 354 (2017), 4034-4054.
doi: 10.1016/j.jfranklin.2017.03.019.![]() ![]() ![]() |
[33] |
C. Huang, J. Lu, D. W. Ho, G. Zhai and J. Cao, Stabilization of probabilistic Boolean networks via pinning control strategy, Information Sciences, 510 (2020), 205-217.
doi: 10.1016/j.ins.2019.09.029.![]() ![]() ![]() |
[34] |
A. Ignatyev and A. Soliman, Asymptotic stability and instability of the solutions of systems with impulse action, Mathematical Notes, 80 (2006), 491-499.
doi: 10.1007/s11006-006-0167-7.![]() ![]() ![]() |
[35] |
G. Jia, M. Meng and J. Feng, Function perturbation of mix-valued logical networks with impacts on limit sets, Neurocomputing, 207 (2016), 428-436.
![]() |
[36] |
B. Jiang, J. Lu, X. Li and J. Qiu, Input/output-to-state stability of nonlinear impulsive delay systems based on a new impulsive inequality, International Journal of Robust and Nonlinear Control, 29 (2019), 6164-6178.
doi: 10.1002/rnc.4712.![]() ![]() ![]() |
[37] |
B. Jiang, J. Lu, J. Lou and J. Qiu, Synchronization in an array of coupled neural networks with delayed impulses: Average impulsive delay method, Neural Networks, 121 (2020), 452-460.
doi: 10.1016/j.neunet.2019.09.019.![]() ![]() |
[38] |
S. Kauffman, Metabolic stability and epigenesis in randomly constructed genetic nets, Journal of Theoretical Biology, 22 (1969), 437-467.
doi: 10.1016/0022-5193(69)90015-0.![]() ![]() ![]() |
[39] |
A. Khadra, X. Liu and X. Shen, Impulsively synchronizing chaotic systems with delay and applications to secure communication, Automatica, 41 (2005), 1491-1502.
doi: 10.1016/j.automatica.2005.04.012.![]() ![]() ![]() |
[40] |
J. Ladyman, J. Lambert and K. Wiesner, What is a complex system?, European Journal for Philosophy of Science, 3 (2013), 33-67.
doi: 10.1007/s13194-012-0056-8.![]() ![]() |
[41] |
V. Lakshmikantham and S. Leela, On perturbing Lyapunov functions, Mathematical Systems Theory, 10 (1976), 85-90.
doi: 10.1007/BF01683265.![]() ![]() ![]() |
[42] |
V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6, World Scientific, 1989.
doi: 10.1142/0906.![]() ![]() ![]() |
[43] |
D. Laschov and M. Margaliot, Controllability of Boolean control networks via perron-frobenius theory, Automatica J. IFAC, 48 (2012), 1218-1223.
doi: 10.1016/j.automatica.2012.03.022.![]() ![]() ![]() |
[44] |
D. Laschov, M. Margaliot and G. Even, Observability of Boolean networks: A graph-theoretic approach, Automatica, 49 (2013), 2351-2362.
doi: 10.1016/j.automatica.2013.04.038.![]() ![]() ![]() |
[45] |
Y.-J. Lee, J. H. Olof and N. Karin, Spatio-temporal dynamics of impulse responses to figure motion in optic flow neurons, Plos One, 10 (2015), e0126265.
doi: 10.1371/journal.pone.0126265.![]() ![]() |
[46] |
B. Li, Y. Liu, K. Kou and L. Yu, Event-triggered control for the disturbance decoupling problem of Boolean control networks, IEEE Transactions on Cybernetics, 48 (2018), 2764-2769.
doi: 10.1109/TCYB.2017.2746102.![]() ![]() |
[47] |
B. Li, Y. Liu, J. Lou, J. Lu and J. Cao, The robustness of outputs with respect to disturbances for Boolean control networks, IEEE Transactions on Neural Networks and Learning Systems, 31 (2020), 1046-1051.
doi: 10.1109/TNNLS.2019.2910193.![]() ![]() |
[48] |
B. Li, J. Lu, Y. Liu and Z. Wu, The outputs robustness of Boolean control networks via pinning control, IEEE Transactions on Control of Network Systems, 7 (2020), 201-209.
doi: 10.1109/TCNS.2019.2913543.![]() ![]() |
[49] |
B. Li, J. Lu, Y. Liu and W. Zheng, The local convergence of Boolean networks with disturbances, IEEE Transactions on Circuits and Systems II: Express Briefs, 66 (2019), 667-671.
doi: 10.1109/TCSII.2018.2857841.![]() ![]() |
[50] |
B. Li, J. Lu, J. Zhong and Y. Liu, Fast-time stability of temporal Boolean networks, IEEE Transactions on Neural Networks and Learning Systems, 30 (2019), 2285-2294.
doi: 10.1109/TNNLS.2018.2881459.![]() ![]() ![]() |
[51] |
F. Li, Pinning control design for the stabilization of Boolean networks, IEEE Transactions on Neural Networks and Learning Systems, 27 (2016), 1585-1590.
doi: 10.1109/TNNLS.2015.2449274.![]() ![]() ![]() |
[52] |
F. Li, Pinning control design for the synchronization of two coupled Boolean networks, IEEE Transactions on Circuits and Systems II: Express Briefs, 63 (2016), 309-313.
doi: 10.1109/TCSII.2015.2482658.![]() ![]() |
[53] |
F. Li, Stability of Boolean networks with delays using pinning control, IEEE Transactions on Control of Network Systems, 5 (2018), 179-185.
doi: 10.1109/TCNS.2016.2585861.![]() ![]() ![]() |
[54] |
F. Li, H. Li, L. Xie and Q. Zhou, On stabilization and set stabilization of multivalued logical systems, Automatica, 80 (2017), 41-47.
doi: 10.1016/j.automatica.2017.01.032.![]() ![]() ![]() |
[55] |
F. Li and J. Sun, Observability analysis of Boolean control networks with impulsive effects, IET Control Theory & Applications, 5 (2011), 1609-1616.
doi: 10.1049/iet-cta.2010.0558.![]() ![]() ![]() |
[56] |
F. Li and J. Sun, Stability and stabilization of Boolean networks with impulsive effects, Systems & Control Letters, 61 (2012), 1-5.
doi: 10.1016/j.sysconle.2011.09.019.![]() ![]() ![]() |
[57] |
F. Li, J. Sun and Q. Wu, Observability of Boolean control networks with state time delays, IEEE Transactions on Neural Networks, 22 (2011), 948-954.
![]() |
[58] |
F. Li and Y. Tang, Set stabilization for switched Boolean control networks, Automatica, 78 (2017), 223-230.
doi: 10.1016/j.automatica.2016.12.007.![]() ![]() ![]() |
[59] |
F. Li, H. Yan and H. Karimi, Single-input pinning controller design for reachability of Boolean networks, IEEE Transactions on Neural Networks and Learning Systems, 29 (2018), 3264-3269.
![]() ![]() |
[60] |
H. Li and Y. Wang, Boolean derivative calculation with application to fault detection of combinational circuits via the semi-tensor product method, Automatica, 48 (2012), 688-693.
doi: 10.1016/j.automatica.2012.01.021.![]() ![]() ![]() |
[61] |
H. Li and Y. Wang, Controllability analysis and control design for switched Boolean networks with state and input constraints, SIAM Journal on Control and Optimization, 53 (2015), 2955-2979.
doi: 10.1137/120902331.![]() ![]() ![]() |
[62] |
H. Li and Y. Wang, Further results on feedback stabilization control design of Boolean control networks, Automatica, 83 (2017), 303-308.
doi: 10.1016/j.automatica.2017.06.043.![]() ![]() ![]() |
[63] |
H. Li and Y. Wang, Lyapunov-based stability and construction of Lyapunov functions for Boolean networks, SIAM Journal on Control and Optimization, 55 (2017), 3437-3457.
doi: 10.1137/16M1092581.![]() ![]() ![]() |
[64] |
H. Li, Y. Wang and Z. Liu, Stability analysis for switched Boolean networks under arbitrary switching signals, IEEE Transactions on Automatic Control, 59 (2014), 1978-1982.
doi: 10.1109/TAC.2014.2298731.![]() ![]() ![]() |
[65] |
H. Li, Y. Wang and L. Xie, Output tracking control of Boolean control networks via state feedback: Constant reference signal case, Automatica, 59 (2015), 54-59.
doi: 10.1016/j.automatica.2015.06.004.![]() ![]() ![]() |
[66] |
H. Li, Y. Wang, L. Xie and D. Cheng, Disturbance decoupling control design for switched Boolean control networks, Systems & Control Letters, 72 (2014), 1-6.
doi: 10.1016/j.sysconle.2014.07.008.![]() ![]() ![]() |
[67] |
P. Guo, Y. Wang and H. Li, Stable degree analysis for strategy profiles of evolutionary networked games, Science China Information Sciences, 59 (2016), 052204.
doi: 10.1007/s11432-015-5376-9.![]() ![]() |
[68] |
H. Li, Y. Wang and P. Guo, Output reachability analysis and output regulation control design of Boolean control networks, Science China Information Sciences, 60 (2017), 022202.
doi: 10.1007/s11432-015-0611-4.![]() ![]() |
[69] |
R. Li, M. Yang and T. Chu, State feedback stabilization for Boolean control networks, IEEE Transactions on Automatic Control, 58 (2013), 1853-1857.
doi: 10.1109/TAC.2013.2238092.![]() ![]() ![]() |
[70] |
R. Li, M. Yang and T. Chu, State feedback stabilization for probabilistic Boolean networks, Automatica, 50 (2014), 1272-1278.
doi: 10.1016/j.automatica.2014.02.034.![]() ![]() ![]() |
[71] |
S. Li, X. Song and A. Li, Strict practical stability of nonlinear impulsive systems by employing two Lyapunov-like functions, Nonlinear Analysis Series B: Real World Applications, 9 (2008), 2262-2269.
doi: 10.1016/j.nonrwa.2007.08.003.![]() ![]() ![]() |
[72] |
X. Li, J. Lu, J. Qiu, X. Chen, X. Li and F. Alsaadi, Set stability for switched Boolean networks with open-loop and closed-loop switching signals, Science China Information Sciences, 61 (2018), 092207.
![]() |
[73] |
X. Li, Further analysis on uniform stability of impulsive infinite delay differential equations, Applied Mathematics Letters, 25 (2012), 133-137.
doi: 10.1016/j.aml.2011.08.001.![]() ![]() ![]() |
[74] |
X. Li and M. Bohner, An impulsive delay differential inequality and applications, Computers & Mathematics with Applications, 64 (2012), 1875-1881.
doi: 10.1016/j.camwa.2012.03.013.![]() ![]() ![]() |
[75] |
X. Li, M. Bohner and C. K. Wang, Impulsive differential equations: Periodic solutions and applications, Automatica, 52 (2015), 173-178.
doi: 10.1016/j.automatica.2014.11.009.![]() ![]() ![]() |
[76] |
X. Li, J. Cao and D. W. C. Ho, Impulsive control of nonlinear systems with time-varying delay and applications, IEEE Transactions on Cybernetics, 50 (2020), 2661-2673.
doi: 10.1109/TCYB.2019.2896340.![]() ![]() |
[77] |
X. Li and F. Deng, Razumikhin method for impulsive functional differential equations of neutral type, Chaos Solitons and Fractals, 101 (2017), 41-49.
doi: 10.1016/j.chaos.2017.05.018.![]() ![]() ![]() |
[78] |
X. Li, D. W. C. Ho and J. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica, 99 (2019), 361-368.
doi: 10.1016/j.automatica.2018.10.024.![]() ![]() ![]() |
[79] |
X. Li, P. Li and Q. G. Wang, Input/output-to-state stability of impulsive switched systems, Systems & Control Letters, 116 (2018), 1-7.
doi: 10.1016/j.sysconle.2018.04.001.![]() ![]() ![]() |
[80] |
X. Li, D. O'Regan and H. Akca, Global exponential stabilization of impulsive neural networks with unbounded continuously distributed delays, IMA Journal of Applied Mathematics, 80 (2015), 85-99.
doi: 10.1093/imamat/hxt027.![]() ![]() ![]() |
[81] |
X. Li, R. Rakkiyappan and C. Pradeep, Robust $\mu-$stability analysis of Markovian switching uncertain stochastic genetic regulatory networks with unbounded time-varying delays, Communications in Nonlinear Science and Numerical Simulation, 17 (2012), 3894-3905.
doi: 10.1016/j.cnsns.2012.02.008.![]() ![]() ![]() |
[82] |
X. Li, J. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Applied Mathematics and Computation, 329 (2018), 14-22.
doi: 10.1016/j.amc.2018.01.036.![]() ![]() ![]() |
[83] |
X. Li and S. Song, Stabilization of delay systems: Delay-dependent impulsive control, IEEE Transactions on Automatic Control, 62 (2017), 406-411.
doi: 10.1109/TAC.2016.2530041.![]() ![]() ![]() |
[84] |
X. Li, S. Song and J. Wu, Exponential stability of nonlinear systems with delayed impulses and applications, IEEE Transactions on Automatic Control, 64 (2019), 4024-4034.
doi: 10.1109/TAC.2019.2905271.![]() ![]() ![]() |
[85] |
X. Li, X. Yang and T. Huang, Persistence of delayed cooperative models: Impulsive control method, Applied Mathematics and Computation, 342 (2019), 130-146.
doi: 10.1016/j.amc.2018.09.003.![]() ![]() ![]() |
[86] |
X. Li, X. Zhang and S. Song, Effect of delayed impulses on input-to-state stability of nonlinear systems, Automatica, 76 (2017), 378-382.
doi: 10.1016/j.automatica.2016.08.009.![]() ![]() ![]() |
[87] |
Y. Li, B. Li, Y. Liu and J. Lu, Set stability and stabilization of switched Boolean networks with state-based switching, IEEE Access, 6 (2018), 35624-35630.
doi: 10.1109/ACCESS.2018.2851391.![]() ![]() |
[88] |
Y. Li, R. Liu, J. Lou, J. Lu, Z. Wang and Y. Liu, Output tracking of Boolean control networks driven by constant reference signal, IEEE Access, 7 (2019), 112572-112577.
doi: 10.1109/ACCESS.2019.2934740.![]() ![]() |
[89] |
Y. Li, H. Li, X. Xu and Y. Li, Semi-tensor product approach to minimal-agent consensus control of networked evolutionary games, IET Control Theory & Applications, 12 (2018), 2269-2275.
doi: 10.1049/iet-cta.2018.5230.![]() ![]() ![]() |
[90] |
Y. Li, Impulsive synchronization of stochastic neural networks via controlling partial states, Neural Processing Letters, 46 (2017), 59-69.
doi: 10.1007/s11063-016-9568-0.![]() ![]() |
[91] |
Y. Li, J. Lou, Z. Wang and F. E. Alsaadi, Synchronization of dynamical networks with nonlinearly coupling function under hybrid pinning impulsive controllers, Journal of the Franklin Institute, 355 (2018), 6520-6530.
doi: 10.1016/j.jfranklin.2018.06.021.![]() ![]() ![]() |
[92] |
Z. Li, W. Zhang, G. He and J.-A. Fang, Synchronisation of discrete-time complex networks with delayed heterogeneous impulses, Applications, 9 (2015), 2648-2656.
doi: 10.1049/iet-cta.2014.1281.![]() ![]() ![]() |
[93] |
J. Liang, H. Chen and J. Lam, An improved criterion for controllability of boolean control networks, IEEE Transactions on Automatic Control, 62 (2017), 6012-6018.
doi: 10.1109/TAC.2017.2702008.![]() ![]() ![]() |
[94] |
L. Lin, J. Cao and L. Rutkowski, Robust event-triggered control invariance of probabilistic Boolean control networks, IEEE Transactions on Neural Networks and Learning Systems, 31 (2020), 1060-1065.
doi: 10.1109/TNNLS.2019.2917753.![]() ![]() |
[95] |
H. Liu, Y. Liu, Y. Li, Z. Wang and F. Alsaadi, Observability of Boolean networks via STP and graph methods, IET Control Theory & Applications, 13 (2019), 1031-1037.
doi: 10.1049/iet-cta.2018.5279.![]() ![]() ![]() |
[96] |
J. Liu and Z. Zhao, Multiple solutions for impulsive problems with non-autonomous perturbations $\star$, Applied Mathematics Letters, 64 (2017), 143-149.
doi: 10.1016/j.aml.2016.08.020.![]() ![]() ![]() |
[97] |
J. Liu and X. Li, Impulsive stabilization of high-order nonlinear retarded differential equations, Applications of Mathematics, 58 (2013), 347-367.
doi: 10.1007/s10492-013-0017-3.![]() ![]() ![]() |
[98] |
R. Liu, J. Lu, Y. Liu, J. Cao and Z. Wu, Delayed feedback control for stabilization of Boolean control networks with state delay, IEEE Transactions on Neural Networks and Learning Systems, 29 (2018), 3283-3288.
![]() ![]() |
[99] |
R. Liu, J. Lu, W. Zheng and J. Kurths, Output feedback control for set stabilization of Boolean control networks, IEEE Transactions on Neural Networks and Learning Systems
![]() |
[100] |
X. Liu and J. Zhu, On potential equations of finite games, Automatica, 68 (2016), 245-253.
doi: 10.1016/j.automatica.2016.01.074.![]() ![]() ![]() |
[101] |
X. Liu, Stability results for impulsive differential systems with applications to population growth models, Dynamics and Stability of Systems, 9 (1994), 163-174.
doi: 10.1080/02681119408806175.![]() ![]() ![]() |
[102] |
X. Liu and G. Ballinger, Existence and continuability of solutions for differential equations with delays and state-dependent impulses, Nonlinear Analysis, 51 (2002), 633-647.
doi: 10.1016/S0362-546X(01)00847-1.![]() ![]() ![]() |
[103] |
X. Liu and K. Zhang, Input-to-state stability of time-delay systems with delay-dependent impulses, IEEE Transactions on Automatic Control, 65 (2012), 1676-1682.
doi: 10.1109/TAC.2019.2930239.![]() ![]() |
[104] |
Y. Liu, J. Cao, B. Li and J. Lu, Normalization and solvability of dynamic-algebraic Boolean networks, IEEE Transactions on Neural Networks and Learning Systems, 29 (2018), 3301-3306.
![]() ![]() |
[105] |
Y. Liu, J. Cao, L. Sun and J. Lu, Sampled-data state feedback stabilization of Boolean control networks, Neural Computation, 28 (2016), 778-799.
doi: 10.1162/NECO_a_00819.![]() ![]() ![]() |
[106] |
Y. Liu, H. Chen, J. Lu and B. Wu, Controllability of probabilistic Boolean control networks based on transition probability matrices, Automatica, 52 (2015), 340-345.
doi: 10.1016/j.automatica.2014.12.018.![]() ![]() ![]() |
[107] |
Y. Liu, B. Li, H. Chen and J. Cao, Function perturbations on singular Boolean networks, Automatica, 84 (2017), 36-42.
doi: 10.1016/j.automatica.2017.06.035.![]() ![]() ![]() |
[108] |
Y. Liu, B. Li and J. Lou, Disturbance decoupling of singular Boolean control networks, IEEE/ACM Transactions on Computational Biology and Bioinformatics, 13 (2016), 1194-1200.
doi: 10.1109/TCBB.2015.2509969.![]() ![]() |
[109] |
Y. Liu, B. Li, J. Lu and J. Cao, Pinning control for the disturbance decoupling problem of Boolean networks, IEEE Transactions on Automatic Control, 62 (2017), 6595-6601.
doi: 10.1109/TAC.2017.2715181.![]() ![]() ![]() |
[110] |
Y. Liu, L. Sun, J. Lu and J. Liang, Feedback controller design for the synchronization of Boolean control networks, IEEE Transactions on Neural Networks and Learning Systems, 27 (2016), 1991-1996.
doi: 10.1109/TNNLS.2015.2461012.![]() ![]() ![]() |
[111] |
Z. Liu, Y. Wang and H. Li, New approach to derivative calculation of multi-valued logical functions with application to fault detection of digital circuits, IET ControlTheory & Applications, 8 (2014), 554-560.
doi: 10.1049/iet-cta.2013.0104.![]() ![]() ![]() |
[112] |
J. Lu, H. Li, Y. Liu and F. Li, Survey on semi-tensor product method with its applications in logical networks and other finite-valued systems, IET Control Theory & Applications, 11 (2017), 2040-2047.
doi: 10.1049/iet-cta.2016.1659.![]() ![]() ![]() |
[113] |
J. Lu, M. Li, T. Huang, Y. Liu and J. Cao, The transformation between the Galois NLFSRs and the Fibonacci NLFSRs via semi-tensor product of matrices, Automatica, 96 (2018), 393-397.
doi: 10.1016/j.automatica.2018.07.011.![]() ![]() ![]() |
[114] |
J. Lu, M. Li, Y. Liu, D. Ho and J. Kurths, Nonsingularity of grain-like cascade FSRs via semi-tensor product, Science China Information Sciences, 61 (2018), 010204, 12 pp.
doi: 10.1007/s11432-017-9269-6.![]() ![]() ![]() |
[115] |
B. Li and J. Lu, Boolean-network-based approach for the construction of filter generators, Science China Information Sciences, in press.
![]() |
[116] |
J. Lu, L. Sun, Y. Liu, D. Ho and J. Cao, Stabilization of Boolean control networks under aperiodic sampled-data control, SIAM Journal on Control and Optimization, 56 (2018), 4385-4404.
doi: 10.1137/18M1169308.![]() ![]() ![]() |
[117] |
J. Lu, J. Zhong, D. W. C. Ho, Y. Tang and J. Cao, On controllability of delayed Boolean control networks, SIAM Journal on Control and Optimization, 54 (2016), 475-494.
doi: 10.1137/140991820.![]() ![]() ![]() |
[118] |
J. Lu, J. Zhong, C. Huang and J. Cao, On pinning controllability of Boolean control networks, IEEE Transactions on Automatic Control, 61 (2016), 1658-1663.
doi: 10.1109/TAC.2015.2478123.![]() ![]() ![]() |
[119] |
J. Lu, D. W. C. Ho and J. Cao, A unified synchronization criterion for impulsive dynamical networks, Automatica, 46 (2010), 1215-1221.
doi: 10.1016/j.automatica.2010.04.005.![]() ![]() ![]() |
[120] |
J. Lu, J. Kurths, J. Cao, N. Mahdavi and C. Huang, Synchronization control for nonlinear stochastic dynamical networks: Pinning impulsive strategy, IEEE Transactions on Neural Networks and Learning Systems, 23 (2012), 285-292.
![]() |
[121] |
J. Lu, R. Liu, J. Lou and Y. Liu, Pinning stabilization of Boolean control networks via a minimum number of controllers, IEEE Transactions on Cybernetics, (2019), 1–9.
doi: 10.1109/TCYB.2019.2944659.![]() ![]() |
[122] |
J. Lu, J. Yang, J. Lou and J. Qiu, Event-triggered sampled feedback synchronization in an array of output-coupled Boolean control networks, IEEE Transactions on Cybernetics, (2019), 1–6.
doi: 10.1109/TCYB.2019.2939761.![]() ![]() |
[123] |
Y. Mao, L. Wang, Y. Liu, J. Lu and Z. Wang, Stabilization of evolutionary networked games with length-r information, Applied Mathematics and Computation, 337 (2018), 442-451.
doi: 10.1016/j.amc.2018.05.027.![]() ![]() ![]() |
[124] |
M. Meng, J. Lam, J. Feng and K. Cheung, Stability and guaranteed cost analysis of time-triggered Boolean networks, IEEE Transactions on Neural Networks and Learning Systems, 29 (2018), 3893-3899.
doi: 10.1109/TNNLS.2017.2737649.![]() ![]() ![]() |
[125] |
M. Meng, J. Lam, J. Feng and K. Cheung, Stability and stabilization of Boolean networks with stochastic delays, IEEE Transactions on Automatic Control, 64 (2019), 790-796.
![]() ![]() |
[126] |
M. Meng, L. Liu and G. Feng, Stability and ${L}_1$ gain analysis of Boolean networks with Markovian jump parameters, IEEE Transactions on Automatic Control, 62 (2017), 4222-4228.
doi: 10.1109/TAC.2017.2679903.![]() ![]() ![]() |
[127] |
V. Milman and A. Myshkis, On the stability of motion in the presence of impulses, Sib. Math. J, 1 (1960), 233-237.
![]() ![]() |
[128] |
P. Naghshtabrizi, J. P. Hespanha and A. R. Teel, Exponential stability of impulsive systems with application to uncertain sampled-data systems, Systems & Control Letters, 57 (2008), 378-385.
doi: 10.1016/j.sysconle.2007.10.009.![]() ![]() ![]() |
[129] |
P. Naghshtabrizi, J. P. Hespanha and A. R. Teel, Stability of delay impulsive systems with application to networked control systems, Transactions of the Institute of Measurement and Control, 32 (2010), 511-528.
doi: 10.1109/ACC.2007.4282847.![]() ![]() |
[130] |
F. G. Pfeiffer and M. O. Foerg, On the structure of multiple impact systems, Nonlinear Dynamics, 42 (2005), 101-112.
doi: 10.1007/s11071-005-1910-4.![]() ![]() ![]() |
[131] |
I. Shmulevich, E. R. Dougherty, S. Kim and W. Zhang, Probabilistic Boolean networks: A rule-based uncertainty model for gene regulatory networks, Bioinformatics, 18 (2002), 261-274.
doi: 10.1093/bioinformatics/18.2.261.![]() ![]() |
[132] |
P. S. Simeonov and D. D. Bainov, Exponential stability of the solutions of singularly perturbed systems with impulse effect, J. Math. Anal. Appl., 151 (1990), 462-487.
doi: 10.1016/0022-247X(90)90161-8.![]() ![]() ![]() |
[133] |
X. Song and A. Li, Stability and boundedness criteria of nonlinear impulsive systems employing perturbing Lyapunov functions $\star$, Applied Mathematics and Computation, 217 (2011), 10166-10174.
doi: 10.1016/j.amc.2011.05.011.![]() ![]() ![]() |
[134] |
E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Transactions on Automatic Control, 34 (1989), 435-443.
doi: 10.1109/9.28018.![]() ![]() ![]() |
[135] |
E. D. Sontag, Comments on integral variants of ISS, Systems & Control Letters, 34 (1998), 93-100.
doi: 10.1016/S0167-6911(98)00003-6.![]() ![]() ![]() |
[136] |
I. Stamova and G. Stamov, Stability analysis of impulsive functional systems of fractional order, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 702-709.
doi: 10.1016/j.cnsns.2013.07.005.![]() ![]() ![]() |
[137] |
I. M. Stamova and G. T. Stamov, Lyapunov-Razumikhin method for impulsive functional differential equations and applications to the population dynamics, Journal of Computational and Applied Mathematics, 130 (2001), 163-171.
doi: 10.1016/S0377-0427(99)00385-4.![]() ![]() ![]() |
[138] |
J. Sun, Q. L. Han and X. Jiang, Impulsive control of time-delay systems using delayed impulse and its application to impulsive master-slave synchronization, Physics Letters A, 372 (2008), 6375-6380.
doi: 10.1016/j.physleta.2008.08.067.![]() ![]() ![]() |
[139] |
L. Sun, J. Lu and W. Ching, Switching-based stabilization of aperiodic sampled-data Boolean control networks with all modes unstable, Frontiers of Information Technology & Electronic Engineering, 21 (2020), 260-267.
doi: 10.1631/FITEE.1900312.![]() ![]() |
[140] |
L. Tong, Y. Liu, Y. Li, J. Lu, Z. Wang and F. Alsaadi, Robust control invariance of probabilistic Boolean control networks via event-triggered control, IEEE Access, 6 (2018), 37767-37774.
doi: 10.1109/ACCESS.2018.2828128.![]() ![]() |
[141] |
G. Wang and Y. Shen, Second-order cluster consensus of multi-agent dynamical systems with impulsive effects, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 3220-3228.
doi: 10.1016/j.cnsns.2014.02.021.![]() ![]() ![]() |
[142] |
N. Wang, X. Li, J. Lu and F. E. Alsaadi, Unified synchronization criteria in an array of coupled neural networks with hybrid impulses, Neural Networks, 101 (2018), 25-32.
doi: 10.1016/j.neunet.2018.01.017.![]() ![]() |
[143] |
Y. Wang and H. Li, On definition and construction of lyapunov functions for Boolean networks, in Proceedings of the 10th World Congress on Intelligent Control and Automation, IEEE, (2012), 1247–1252.
doi: 10.1109/WCICA.2012.6358072.![]() ![]() |
[144] |
Y. Wang, J. Lu, J. Liang, J. Cao and M. Perc, Pinning synchronization of nonlinear coupled Lur'e networks under hybrid impulses, IEEE Transactions on Circuits and Systems II: Express Briefs, 66 (2019), 432-436.
![]() |
[145] |
Y. Wang, J. Lu, J. Lou, C. Ding, F. E. Alsaadi and T. Hayat, Synchronization of heterogeneous partially coupled networks with heterogeneous impulses, Neural Processing Letters, 48 (2018), 557-575.
doi: 10.1007/s11063-017-9735-y.![]() ![]() |
[146] |
Y. Wang, J. Lu and Y. Lou, Halanay-type inequality with delayed impulses and its applications, Science China Information Sciences, 62 (2019), 192206, 10 pp.
doi: 10.1007/s11432-018-9809-y.![]() ![]() ![]() |
[147] |
E. Weiss and M. Margaliot, A polynomial-time algorithm for solving the minimal observability problem in conjunctive Boolean networks, IEEE Transactions on Automatic Control, 64 (2019), 2727-2736.
doi: 10.1109/TAC.2018.2882154.![]() ![]() ![]() |
[148] |
Q. Wu, J. Zhou and L. Xiang, Impulses-induced exponential stability in recurrent delayed neural networks, Neurocomputing, 74 (2011), 3204-3211.
doi: 10.1016/j.neucom.2011.05.001.![]() ![]() |
[149] |
Y. Wu and T. Shen, Policy iteration approach to control residual gas fraction in ic engines under the framework of stochastic logical dynamics, IEEE Transactions on Control Systems Technology, 25 (2017), 1100-1107.
doi: 10.1109/TCST.2016.2587247.![]() ![]() |
[150] |
Y. Wu and T. Shen, A finite convergence criterion for the discounted optimal control of stochastic logical networks, IEEE Transactions on Automatic Control, 63 (2018), 262-268.
doi: 10.1109/TAC.2017.2720730.![]() ![]() ![]() |
[151] |
D. Xie, H. Peng, L. Li and Y. Yang, Semi-tensor compressed sensing, Digital Signal Processing, 58 (2016), 85-92.
doi: 10.1016/j.dsp.2016.07.003.![]() ![]() |
[152] |
F. Xu, L. Dong, D. Wang, X. Li and R. Rakkiyappan, Globally exponential stability of nonlinear impulsive switched systems, Mathematical Notes, 97 (2015), 803-810.
doi: 10.1134/S0001434615050156.![]() ![]() ![]() |
[153] |
M. Xu, Y. Liu, J. Lou, Z.-G. Wu and J. Zhong, Set stabilization of probabilistic boolean control networks: A sampled-data control approach, IEEE Transactions on Cybernetics, (2019), 1–8.
doi: 10.1109/TCYB.2019.2940654.![]() ![]() |
[154] |
J. Yang, J. Lu, L. Li, Y. Liu, Z. Wang and F. Alsaadi, Event-triggered control for the synchronization of Boolean control networks, Nonlinear Dynamics, 96 (2019), 1335-1344.
doi: 10.1007/s11071-019-04857-2.![]() ![]() |
[155] |
M. Yang, R. Li and T. Chu, Controller design for disturbance decoupling of Boolean control networks, Automatica, 49 (2013), 273-277.
doi: 10.1016/j.automatica.2012.10.010.![]() ![]() ![]() |
[156] |
T. Yang, Impulsive Control Theory Lecture Notes in Control and Information Sciences, Springer Science and Business Media, 2001.
![]() ![]() |
[157] |
X. Yang, B. Chen, Y. Li, Y. Liu and F. E. Alsaadi, Stabilization of dynamic-algebraic Boolean control networks via state feedback control, Journal of the Franklin Institute, 355 (2018), 5520-5533.
doi: 10.1016/j.jfranklin.2018.05.049.![]() ![]() ![]() |
[158] |
X. Yang and J. Lu, Finite-time synchronization of coupled networks with markovian topology and impulsive effects, IEEE Transactions on Automatic Control, 61 (2016), 2256-2261.
doi: 10.1109/TAC.2015.2484328.![]() ![]() ![]() |
[159] |
Z. Yang and D. Xu, Stability analysis and design of impulsive control systems with time delay, IEEE Transactions on Automatic Control, 52 (2007), 1448-1454.
doi: 10.1109/TAC.2007.902748.![]() ![]() ![]() |
[160] |
Y. Yu, J. Feng, J. Pan and D. Cheng, Block decoupling of Boolean control networks, IEEE Transactions on Automatic Control, 64 (2019), 3129-3140.
doi: 10.1109/TAC.2018.2880411.![]() ![]() ![]() |
[161] |
K. Zhang and L. Zhang, Observability of Boolean control networks: A unified approach based on finite automata, IEEE Transactions on Automatic Control, 61 (2016), 2733-2738.
doi: 10.1109/TAC.2015.2501365.![]() ![]() ![]() |
[162] |
K. Zhang, L. Zhang and L. Xie, Finite automata approach to observability of switched Boolean control networks, Nonlinear Analysis: Hybrid Systems, 19 (2016), 186-197.
doi: 10.1016/j.nahs.2015.10.002.![]() ![]() ![]() |
[163] |
L. Zhang and K. Zhang, Controllability and observability of Boolean control networks with time-variant delays in states, IEEE Transactions on Neural Networks and Learning Systems, 24 (2013), 1478-1484.
doi: 10.1109/TNNLS.2013.2246187.![]() ![]() |
[164] |
W. Zhang, C. Li, T. Huang and X. He, Synchronization of memristor-based coupling recurrent neural networks with time-varying delays and impulses, IEEE Transactions on Neural Networks and Learning Systems, 26 (2015), 3308-3313.
doi: 10.1109/TNNLS.2015.2435794.![]() ![]() ![]() |
[165] |
X. Zhang and X. Li, Input-to-state stability of nonlinear systems with distributed-delayed impulses, IET Control Theory and Applications, 11 (2017), 81–89.
doi: 10.1049/iet-cta.2016.0469.![]() ![]() ![]() |
[166] |
Y. Zhang and Y. Liu, Nonlinear second-order multi-agent systems subject to antagonistic interactions without velocity constraints, Applied Mathematics and Computation, 364 (2020), 124667, 14 pp.
doi: 10.1016/j.amc.2019.124667.![]() ![]() ![]() |
[167] |
J. Zhong, D. Ho, J. Lu and W. Xu, Global robust stability and stabilization of Boolean network with disturbances, Automatica, 84 (2017), 142-148.
doi: 10.1016/j.automatica.2017.07.013.![]() ![]() ![]() |
[168] |
J. Zhong, B. Li, Y. Liu and W. Gui, Output feedback stabilizer design of Boolean networks based on network structure, Frontiers of Information Technology & Electronic Engineering, 21 (2020), 247-259.
doi: 10.1631/FITEE.1900229.![]() ![]() |
[169] |
J. Zhong and D. Lin, Stability of nonlinear feedback shift registers, Science China Information Sciences, 59 (2016), 1-12.
doi: 10.1109/ICInfA.2014.6932738.![]() ![]() |
[170] |
J. Zhong and D. Lin, On minimum period of nonlinear feedback shift registers in grain-like structure, IEEE Transactions on Information Theory, 64 (2018), 6429-6442.
doi: 10.1109/TIT.2018.2849392.![]() ![]() ![]() |
[171] |
J. Zhong, Y. Liu, K. Kou, L. Sun and J. Cao, On the ensemble controllability of Boolean control networks using STP method, Applied Mathematics and Computation, 358 (2019), 51-62.
doi: 10.1016/j.amc.2019.03.059.![]() ![]() ![]() |
[172] |
J. Zhong, J. Lu, T. Huang and D. Ho, Controllability and synchronization analysis of identical-hierarchy mixed-valued logical control networks, IEEE Transactions on Cybernetics, 47 (2017), 3482-3493.
doi: 10.1109/TCYB.2016.2560240.![]() ![]() |
[173] |
J. Zhong, J. Lu, Y. Liu and J. Cao, Synchronization in an array of output-coupled Boolean networks with time delay, IEEE Transactions on Neural Networks and Learning Systems, 25 (2014), 2288-2294.
doi: 10.1109/TNNLS.2014.2305722.![]() ![]() |
[174] |
B. Zhu, X. Xia and Z. Wu, Evolutionary game theoretic demand-side management and control for a class of networked smart grid, Automatica, 70 (2016), 94-100.
doi: 10.1016/j.automatica.2016.03.027.![]() ![]() ![]() |
[175] |
Q. Zhu, Y. Liu, J. Lu and J. Cao, Observability of Boolean control networks, Science China Information Sciences, 61 (2018), 092201, 12 pp.
doi: 10.1007/s11432-017-9135-4.![]() ![]() ![]() |
[176] |
Q. Zhu, Y. Liu, J. Lu and J. Cao, On the optimal control of Boolean control networks, SIAM Journal on Control and Optimization, 56 (2018), 1321-1341.
doi: 10.1137/16M1070281.![]() ![]() ![]() |
[177] |
Q. Zhu, Y. Liu, J. Lu and J. Cao, Further results on the controllability of Boolean control networks, IEEE Transactions on Automatic Control, 64 (2019), 440-442.
doi: 10.1109/TAC.2018.2830642.![]() ![]() ![]() |
[178] |
S. Zhu, Y. Liu, J. Lou, J. Lu and F. Alsaadi, Sampled-data state feedback control for the set stabilization of Boolean control networks, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 50 (2020), 1580-1589.
doi: 10.1109/TSMC.2018.2852703.![]() ![]() |
[179] |
S. Zhu, J. Lou, Y. Liu, Y. Li and Z. Wang, Event-triggered control for the stabilization of probabilistic Boolean control networks, Complexity, 2018 (2018), 1-7.
doi: 10.1155/2018/9259348.![]() ![]() |
[180] |
S. Zhu, J. Lu and Y. Liu, Asymptotical stability of probabilistic Boolean networks with state delays, IEEE Transactions on Automatic Control, 65 (2020), 1779-1784.
doi: 10.1109/TAC.2019.2934532.![]() ![]() |