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July  2022, 15(7): 1797-1821. doi: 10.3934/dcdss.2022080

A brief survey on stability and stabilization of impulsive systems with delayed impulses

1. 

School of Mathematics and Statistics, Shandong Normal University, Ji'nan 250014, China

2. 

School of Automation and Electrical Engineering & Key Laboratory of Complex systems, and Intelligent Computing in Universities of Shandong, Linyi University, Linyi 276005, China

3. 

School of Mathematics, Southeast University, Nanjing 211189, China

4. 

Yonsei Frontier Lab, Yonsei University, Seoul 03722, South Korea

* Corresponding author: Xiaodi Li

Received  December 2021 Revised  February 2022 Published  July 2022 Early access  March 2022

This survey addresses stability analysis for impulsive systems with delayed impulses, which constitute an important generalization of delayed impulsive systems. Fundamental issues such as the concept of a solution to an impulsive system with delayed impulses and methods to determine impulse instants are revisited and discussed. In view of the types of delays in impulses, impulsive systems with delayed impulses are classified into two categories including systems with time-dependent delayed impulses and systems with state-dependent delayed impulses. Then more efforts are devoted to the stability analysis of these two classes of impulsive systems, where corresponding Lyapunov-function-based sufficient conditions for Lyapunov stability, asymptotic stability, exponential stability, input-to-state stability and finite-time stability are presented, respectively. Moreover, the double effects of time-dependent delayed impulses on system performance are reemphasized, and recent applications of delayed impulses in synchronization control are discussed in detail. Several challenges are suggested for future works.

Citation: Xinyi He, Jianlong Qiu, Xiaodi Li, Jinde Cao. A brief survey on stability and stabilization of impulsive systems with delayed impulses. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1797-1821. doi: 10.3934/dcdss.2022080
References:
[1]

F. AmatoM. Ariola and P. Dorato, Finite-time control of linear systems subject to parametric uncertainties and disturbances, Automatica, 37 (2001), 1459-1463.  doi: 10.1016/S0005-1098(01)00087-5.

[2]

F. AmatoG. De Tommasi and A. Pironti, Necessary and sufficient conditions for finite-time stability of impulsive dynamical linear systems, Automatica J. IFAC, 49 (2013), 2546-2550.  doi: 10.1016/j.automatica.2013.04.004.

[3]

G. Ballinger and X. Liu, Existence and uniqueness results for impulsive delay differential equations, Dynamics of Continuous Discrete and Impulsive Systems, 5 (1999), 579-591. 

[4]

C. Briat, Stability analysis and stabilization of stochastic linear impulsive, switched and sampled-data systems under dwell-time constraints, Automatica J. IFAC, 74 (2016), 279-287.  doi: 10.1016/j.automatica.2016.08.001.

[5]

C. Briat and A. Seuret, Convex dwell-time characterizations for uncertain linear impulsive systems, IEEE Trans. Automat. Control, 57 (2012), 3241-3246.  doi: 10.1109/TAC.2012.2200379.

[6]

W. Cao and Q. Zhu, Razumikhin-type theorem for pth exponential stability of impulsive stochastic functional differential equations based on vector lyapunov function, Nonlinear Anal. Hybrid Syst., 39 (2021), 100983, 10 pp. doi: 10.1016/j.nahs.2020.100983.

[7]

J. ChenB. ChenZ. Zeng and P. Jiang, Effects of subsystem and coupling on synchronization of multiple neural networks with delays via impulsive coupling, IEEE Trans. Neural Netw. Learn. Syst., 30 (2019), 3748-3758.  doi: 10.1109/TNNLS.2019.2898919.

[8]

J. Chen, X. Li and D. Wang, Asymptotic stability and exponential stability of impulsive delayed Hopfield neural networks, Abstr. Appl. Anal., (2013), Art. ID 638496, 10 pp. doi: 10.1155/2013/638496.

[9]

W.-H. ChenZ. Ruan and W. X. Zheng, Stability and ${L}_2$-gain analysis for linear time-delay systems with delayed impulses: An augmentation-based switching impulse approach, IEEE Trans. Automat. Control, 64 (2019), 4209-4216.  doi: 10.1109/TAC.2019.2893149.

[10]

W.-H. ChenD. Wei and X. Lu, Global exponential synchronization of nonlinear time-delay lur'e systems via delayed impulsive control, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 3298-3312.  doi: 10.1016/j.cnsns.2014.01.018.

[11]

W.-H. ChenD. Wei and W. X. 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.

[12]

W.-H. Chen and W. X. Zheng, Exponential stability of nonlinear time-delay systems with delayed impulse effects, Automatica J. IFAC, 47 (2011), 1075-1083.  doi: 10.1016/j.automatica.2011.02.031.

[13]

K. E. M. Church and X. Liu, Invariant manifold-guided impulsive stabilization of delay equations, IEEE Trans. Automat. Control, 66 (2021), 5997-6002.  doi: 10.1109/TAC.2021.3057988.

[14]

S. Dashkovskiy and P. Feketa, Input-to-state stability of impulsive systems and their networks, Nonlinear Anal. Hybrid Syst., 26 (2017), 190-200.  doi: 10.1016/j.nahs.2017.06.004.

[15]

S. DashkovskiyM. KosmykovA. Mironchenko and L. Naujok, Stability of interconnected impulsive systems with and without time delays, using Lyapunov methods, Nonlinear Anal. Hybrid Syst., 6 (2012), 899-915.  doi: 10.1016/j.nahs.2012.02.001.

[16]

S. Dashkovskiy and A. Mironchenko, Input-to-state stability of nonlinear impulsive systems, SIAM J. Control Optim., 51 (2013), 1962-1987.  doi: 10.1137/120881993.

[17]

K. H. DegueD. Efimov and J.-P. Richard, Stabilization of linear impulsive systems under dwell-time constraints: Interval observer-based framework, Eur. J. Control, 42 (2018), 1-14.  doi: 10.1016/j.ejcon.2018.01.001.

[18]

W. DuS. Y. S. LeungY. Tang and A. V. Vasilakos, Differential evolution with event-triggered impulsive control, IEEE Transactions on Cybernetics, 47 (2017), 244-257.  doi: 10.1109/TCYB.2015.2512942.

[19]

P. Feketa and N. Bajcinca, On robustness of impulsive stabilization, Automatica J. IFAC, 104 (2019), 48-56.  doi: 10.1016/j.automatica.2019.02.056.

[20]

P. Getto and M. Waurick, A differential equation with state-dependent delay from cell population biology, J. Differential Equations, 260 (2016), 6176-6200.  doi: 10.1016/j.jde.2015.12.038.

[21]

K. Gu, V. L. Kharitonov and J. Chen, Stability of Time-Delay Systems, Birkhäuser Boston, Inc., Boston, MA, 2003. doi: 10.1007/978-1-4612-0039-0.

[22]

Z.-H. Guan and G. Chen, On delayed impulsive hopfield neural networks, Neural Networks, 12 (1999), 273-280.  doi: 10.1016/S0893-6080(98)00133-6.

[23]

Z.-H. GuanD. J. Hill and X. Shen, On hybrid impulsive and switching systems and application to nonlinear control, IEEE Trans. Automat. Control, 50 (2005), 1058-1062.  doi: 10.1109/TAC.2005.851462.

[24]

Z.-H. GuanZ.-W. LiuG. Feng and M. Jian, Impulsive consensus algorithms for second-order multi-agent networks with sampled information, Automatica J. IFAC, 48 (2012), 1397-1404.  doi: 10.1016/j.automatica.2012.05.005.

[25]

Z.-H. GuanZ.-W. LiuG. Feng and Y.-W. Wang, 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. 

[26] W. M. HaddadV. Chellaboina and S. G. Nersesov, Impulsive and Hybrid Dynamical Systems, Princeton University Press, 2006.  doi: 10.1515/9781400865246.
[27]

H. Haimovich and J. L. Mancilla-Aguilar, Nonrobustness of asymptotic stability of impulsive systems with inputs, Automatica J. IFAC, 122 (2020), 109238, 9 pp. doi: 10.1016/j.automatica.2020.109238.

[28]

H. Haimovich and J. L. Mancilla-Aguilar, Strong ISS implies strong iISS for time-varying impulsive systems, Automatica J. IFAC, 122 (2020), 109224, 12 pp. doi: 10.1016/j.automatica.2020.109224.

[29]

F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, in Handbook of Differential Equations: Ordinary Differential Equations, vol. 3, Elsevier, 2006,435–545. doi: 10.1016/S1874-5725(06)80009-X.

[30]

W. HeX. GaoW. Zhong and F. Qian, Secure impulsive synchronization control of multi-agent systems under deception attacks, Inform. Sci., 459 (2018), 354-368.  doi: 10.1016/j.ins.2018.04.020.

[31]

W. HeF. QianQ.-L. Han and G. Chen, Almost sure stability of nonlinear systems under random and impulsive sequential attacks, IEEE Trans. Automat. Control, 65 (2020), 3879-3886.  doi: 10.1109/TAC.2020.2972220.

[32]

W. HeF. QianJ. LamG. ChenQ.-L. Han and J. Kurths, Quasi-synchronization of heterogeneous dynamic networks via distributed impulsive control: Error estimation, optimization and design, Automatica J. IFAC, 62 (2015), 249-262.  doi: 10.1016/j.automatica.2015.09.028.

[33]

X. HeD. Peng and X. Li, Synchronization of complex networks with impulsive control involving stabilizing delay, J. Franklin Inst., 357 (2020), 4869-4886.  doi: 10.1016/j.jfranklin.2020.03.044.

[34]

X. He, Y. Wang and X. Li, Uncertain impulsive control for leader-following synchronization of complex networks, Chaos Solitons Fractals, 147 (2021), 110980, 7 pp. doi: 10.1016/j.chaos.2021.110980.

[35]

Z. He, C. Li, H. Li and Q. Zhang, Global exponential stability of high-order hopfield neural networks with state-dependent impulses, Phys. A, 542 (2020), 123434, 21 pp. doi: 10.1016/j.physa.2019.123434.

[36]

J. P. HespanhaD. Liberzon and A. R. Teel, Lyapunov conditions for input-to-state stability of impulsive systems, Automatica J. IFAC, 44 (2008), 2735-2744.  doi: 10.1016/j.automatica.2008.03.021.

[37]

J. HuG. SuiX. Lv and X. Li, Fixed-time control of delayed neural networks with impulsive perturbations, Nonlinear Anal. Model. Control, 23 (2018), 904-920.  doi: 10.15388/NA.2018.6.6.

[38]

B. JiangJ. Lu and Y. Liu, Exponential stability of delayed systems with average-delay impulses, SIAM J. Control Optim., 58 (2020), 3763-3784.  doi: 10.1137/20M1317037.

[39]

B. JiangJ. LuJ. 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.

[40]

A. KhadraX. Z. Liu and X. Shen, Analyzing the robustness of impulsive synchronization coupled by linear delayed impulses, IEEE Trans. Automat. Control, 54 (2009), 923-928.  doi: 10.1109/TAC.2009.2013029.

[41]

A. KhadraX. Z. Liu and X. Shen, Impulsively synchronizing chaotic systems with delay and applications to secure communication, Automatica J. IFAC, 41 (2005), 1491-1502.  doi: 10.1016/j.automatica.2005.04.012.

[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]

H. LiC. LiD. Ouyang and S. K. Nguang, Impulsive synchronization of unbounded delayed inertial neural networks with actuator saturation and sampled-data control and its application to image encryption, IEEE Trans. Neural Netw. Learn. Syst., 32 (2021), 1460-1473.  doi: 10.1109/TNNLS.2020.2984770.

[44]

P. LiX. Li and J. Lu, Input-to-state stability of impulsive delay systems with multiple impulses, IEEE Trans. Automat. Control, 66 (2021), 362-368.  doi: 10.1109/TAC.2020.2982156.

[45]

X. Li, Further analysis on uniform stability of impulsive infinite delay differential equations, Appl. Math. Lett., 25 (2012), 133-137.  doi: 10.1016/j.aml.2011.08.001.

[46]

X. LiH. Akca and X. Fu, Uniform stability of impulsive infinite delay differential equations with applications to systems with integral impulsive conditions, Appl. Math. Comput., 219 (2013), 7329-7337.  doi: 10.1016/j.amc.2012.12.033.

[47]

X. Li and M. Bohner, An impulsive delay differential inequality and applications, Comput. Math. Appl., 64 (2012), 1875-1881.  doi: 10.1016/j.camwa.2012.03.013.

[48]

X. LiM. Bohner and C.-K. Wang, Impulsive differential equations: Periodic solutions and applications, Automatica J. IFAC, 52 (2015), 173-178.  doi: 10.1016/j.automatica.2014.11.009.

[49]

X. LiT. CaraballoR. Rakkiyappan and X. Han, On the stability of impulsive functional differential equations with infinite delays, Math. Methods Appl. Sci., 38 (2015), 3130-3140.  doi: 10.1002/mma.3303.

[50]

X. LiD. W. C. Ho and J. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica J. IFAC, 99 (2019), 361-368.  doi: 10.1016/j.automatica.2018.10.024.

[51]

X. Li and P. Li, Stability of time-delay systems with impulsive control involving stabilizing delays, Automatica J. IFAC, 124 (2021), 109336, 6 pp. doi: 10.1016/j.automatica.2020.109336.

[52]

X. LiD. O'Regan and H. Akca, Global exponential stabilization of impulsive neural networks with unbounded continuously distributed delays, IMA J. Appl. Math., 80 (2015), 85-99.  doi: 10.1093/imamat/hxt027.

[53]

X. LiD. Peng and J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Trans. Automat. Control, 65 (2020), 4908-4913.  doi: 10.1109/TAC.2020.2964558.

[54]

X. LiJ. ShenH. Akca and R. Rakkiyappan, LMI-based stability for singularly perturbed nonlinear impulsive differential systems with delays of small parameter, Appl. Math. Comput., 250 (2015), 798-804.  doi: 10.1016/j.amc.2014.10.113.

[55]

X. LiJ. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Appl. Math. Comput., 329 (2018), 14-22.  doi: 10.1016/j.amc.2018.01.036.

[56]

X. Li and S. Song, Stabilization of delay systems: Delay-dependent impulsive control, IEEE Trans. Automat. Control, 62 (2017), 406-411.  doi: 10.1109/TAC.2016.2530041.

[57]

X. Li and S. Song, Impulsive Systems with Delays: Stability and Control, Springer, Singapore, 2022. doi: 10.1007/978-981-16-4687-4.

[58]

X. LiS. Song and J. Wu, Exponential stability of nonlinear systems with delayed impulses and applications, IEEE Trans. Automat. Control, 64 (2019), 4024-4034.  doi: 10.1109/TAC.2019.2905271.

[59]

X. Li and J. Wu, Stability of nonlinear differential systems with state-dependent delayed impulses, Automatica J. IFAC, 64 (2016), 63-69.  doi: 10.1016/j.automatica.2015.10.002.

[60]

X. Li and J. Wu, Sufficient stability conditions of nonlinear differential systems under impulsive control with state-dependent delay, IEEE Trans. Automat. Control, 63 (2018), 306-311.  doi: 10.1109/TAC.2016.2639819.

[61]

X. Li, X. Yang and J. Cao, Event-triggered impulsive control for nonlinear delay systems, Automatica J. IFAC, 117 (2020), 108981, 7 pp. doi: 10.1016/j.automatica.2020.108981.

[62]

X. LiX. Yang and T. Huang, Persistence of delayed cooperative models: Impulsive control method, Appl. Math. Comput., 342 (2019), 130-146.  doi: 10.1016/j.amc.2018.09.003.

[63]

X. LiX. Yang and S. Song, Lyapunov conditions for finite-time stability of time-varying time-delay systems, Automatica J. IFAC, 103 (2019), 135-140.  doi: 10.1016/j.automatica.2019.01.031.

[64]

X. LiX. Zhang and S. Song, Effect of delayed impulses on input-to-state stability of nonlinear systems, Automatica J. IFAC, 76 (2017), 378-382.  doi: 10.1016/j.automatica.2016.08.009.

[65]

X. Li and Y. Zhao, Sliding mode control for linear impulsive systems with matched disturbances, IEEE Transactions on Automatic Control. doi: 10.1109/TAC.2021.3129735.

[66]

D. LinX. Li and D. O'Regan, $\mu$-stability of infinite delay functional differential systems with impulsive effects, Appl. Anal., 92 (2013), 15-26.  doi: 10.1080/00036811.2011.584185.

[67]

B. LiuD. J. Hill and Z. Sun, Stabilisation to input-to-state stability for continuous-time dynamical systems via event-triggered impulsive control with three levels of events, IET Control Theory Appl., 12 (2018), 1167-1179.  doi: 10.1049/iet-cta.2017.0820.

[68]

B. LiuX. LiuG. Chen and H. Wang, Robust impulsive synchronization of uncertain dynamical networks, IEEE Trans. Circuits Syst. I. Regul. Pap., 52 (2005), 1431-1441.  doi: 10.1109/TCSI.2005.851708.

[69]

B. Liu, Z. Sun, Y. Luo and Y. Zhong, Uniform synchronization for chaotic dynamical systems via event-triggered impulsive control, Phys. A, 531 (2019), 121725, 14 pp. doi: 10.1016/j.physa.2019.121725.

[70]

B. LiuB. Xu and T. Liu, Almost sure contraction for stochastic switched impulsive systems, IEEE Trans. Automat. Control, 66 (2021), 5393-5400.  doi: 10.1109/TAC.2020.3047554.

[71]

J. Liu and X. Li, Impulsive stabilization of high-order nonlinear retarded differential equations, Appl. Math., 58 (2013), 347-367.  doi: 10.1007/s10492-013-0017-3.

[72]

K. LiuA. Selivanov and E. Fridman, Survey on time-delay approach to networked control, Annu. Rev. Control, 48 (2019), 57-79.  doi: 10.1016/j.arcontrol.2019.06.005.

[73]

W. Liu, J. Sun, G. Wang and J. Chen, Quantized impulsive control of linear systems under bounded disturbances and DoS attacks, IEEE Transactions on Control of Network Systems. doi: 10.1109/TCNS.2021.3085759.

[74]

X. Liu, Practical stabilization of control systems with impulse effects, J. Math. Anal. Appl., 166 (1992), 563-576.  doi: 10.1016/0022-247X(92)90315-5.

[75]

X. Liu, Stability of impulsive control systems with time delay, Math. Comput. Modelling, 39 (2004), 511-519.  doi: 10.1016/S0895-7177(04)90522-5.

[76]

X. Liu and G. Ballinger, Uniform asymptotic stability of impulsive delay differential equations, Comput. Math. Appl., 41 (2001), 903-915.  doi: 10.1016/S0898-1221(00)00328-X.

[77]

X. Liu and G. Ballinger, On boundedness of solutoins for impulsive systems in terms of two measures, Nonlinear World, 4 (1997), 417-434. 

[78]

X. Liu and G. Ballinger, Existence and continuability of solutions for differential equations with delays and state-dependent impulses, Nonlinear Anal., 51 (2002), 633-647.  doi: 10.1016/S0362-546X(01)00847-1.

[79]

X. Liu and K. Rohlf, Impulsive control of a Lotka-Volterra system, IMA J. Math. Control Inform., 15 (1998), 269-284.  doi: 10.1093/imamci/15.3.269.

[80]

X. Liu and K. Zhang, Synchronization of linear dynamical networks on time scales: Pinning control via delayed impulses, Automatica J. IFAC, 72 (2016), 147-152.  doi: 10.1016/j.automatica.2016.06.001.

[81]

X. Liu and K. Zhang, Input-to-state stability of time-delay systems with delay-dependent impulses, IEEE Trans. Automat. Control, 65 (2020), 1676-1682.  doi: 10.1109/TAC.2019.2930239.

[82]

X. LiuK. Zhang and W.-C. Xie, Consensus seeking in multi-agent systems via hybrid protocols with impulse delays, Nonlinear Anal. Hybrid Syst., 25 (2017), 90-98.  doi: 10.1016/j.nahs.2017.03.002.

[83]

Y. LiuS. Zhao and J. Lu, A new fuzzy impulsive control of chaotic systems based on T–S fuzzy model, IEEE Transactions on Fuzzy Systems, 19 (2011), 393-398.  doi: 10.1109/TFUZZ.2010.2090162.

[84]

Z.-W. LiuG. WenX. YuZ.-H. Guan and T. Huang, Delayed impulsive control for consensus of multiagent systems with switching communication graphs, IEEE Transactions on Cybernetics, 50 (2020), 3045-3055.  doi: 10.1109/TCYB.2019.2926115.

[85]

J. LuD. W. C. Ho and J. Cao, A unified synchronization criterion for impulsive dynamical networks, Automatica J. IFAC, 46 (2010), 1215-1221.  doi: 10.1016/j.automatica.2010.04.005.

[86]

J. LuD. W. C. HoJ. Cao and J. Kurths, Exponential synchronization of linearly coupled neural networks with impulsive disturbances, IEEE Transactions on Neural Networks, 22 (2011), 329-336.  doi: 10.1109/TNN.2010.2101081.

[87]

S. Luo, F. Deng and W.-H. Chen, Stability and stabilization of linear impulsive systems with large impulse-delays: A stabilizing delay perspective, Automatica J. IFAC, 127 (2021), 109533, 7 pp. doi: 10.1016/j.automatica.2021.109533.

[88]

X. LvJ. CaoX. LiM. Abdel-Aty and U. A. Al-Juboori, Synchronization analysis for complex dynamical networks with coupling delay via event-triggered delayed impulsive control, IEEE Transactions on Cybernetics, 51 (2021), 5269-5278.  doi: 10.1109/TCYB.2020.2974315.

[89]

X. Lv and X. Li, Finite time stability and controller design for nonlinear impulsive sampled-data systems with applications, ISA Transactions, 70 (2017), 30-36.  doi: 10.1016/j.isatra.2017.07.025.

[90]

J. L. Mancilla-Aguilar, H. Haimovich and P. Feketa, Uniform stability of nonlinear time-varying impulsive systems with eventually uniformly bounded impulse frequency, Nonlinear Anal. Hybrid Syst., 38 (2020), 100933, 16 pp. doi: 10.1016/j.nahs.2020.100933.

[91]

P. NaghshtabriziJ. P. Hespanha and A. R. Teel, Exponential stability of impulsive systems with application to uncertain sampled-data systems, Systems Control Lett., 57 (2008), 378-385.  doi: 10.1016/j.sysconle.2007.10.009.

[92]

S. G. Nersesov and W. M. Haddad, Finite-time stabilization of nonlinear impulsive dynamical systems, Nonlinear Anal. Hybrid Syst., 2 (2008), 832-845.  doi: 10.1016/j.nahs.2007.12.001.

[93]

S.-I. Niculescu, Delay Effects on Stability: A Robust Control Approach, vol. 269, Springer-Verlag London, Ltd., London, 2001.

[94]

S. Peng and F. Deng, New criteria on $p$th moment input-to-state stability of impulsive stochastic delayed differential systems, IEEE Trans. Automat. Control, 62 (2017), 3573-3579.  doi: 10.1109/TAC.2017.2660066.

[95]

R. RakkiyappanP. Balasubramaniam and J. Cao, Global exponential stability results for neutral-type impulsive neural networks, Nonlinear Anal. Real World Appl., 11 (2010), 122-130.  doi: 10.1016/j.nonrwa.2008.10.050.

[96]

W. Ren and J. Xiong, Vector-Lyapunov-function-based input-to-state stability of stochastic impulsive switched time-delay systems, IEEE Trans. Automat. Control, 64 (2019), 654-669. 

[97]

W. Ren and J. Xiong, Stability analysis of impulsive switched time-delay systems with state-dependent impulses, IEEE Trans. Automat. Control, 64 (2019), 3928-3935.  doi: 10.1109/TAC.2018.2890768.

[98]

W. Ren and J. Xiong, Stability analysis of stochastic impulsive switched systems with deterministic state-dependent impulses and switches, SIAM J. Control Optim., 59 (2021), 2068-2092.  doi: 10.1137/20M1353460.

[99]

H. RíosL. Hetel and D. Efimov, Nonlinear impulsive systems: 2D stability analysis approach, Automatica J. IFAC, 80 (2017), 32-40.  doi: 10.1016/j.automatica.2017.01.010.

[100]

H. RíosL. Hetel and D. Efimov, Robust output-feedback control for uncertain linear sampled-data systems: A 2D impulsive system approach, Nonlinear Anal. Hybrid Syst., 32 (2019), 177-201.  doi: 10.1016/j.nahs.2018.11.005.

[101]

J. Shen and J. Li, Existence and global attractivity of positive periodic solutions for impulsive predator–prey model with dispersion and time delays, Nonlinear Anal. Real World Appl., 10 (2009), 227-243.  doi: 10.1016/j.nonrwa.2007.08.026.

[102]

Q. Song and J. Zhang, Global exponential stability of impulsive Cohen–Grossberg neural network with time-varying delays, Nonlinear Anal. Real World Appl., 9 (2008), 500-510.  doi: 10.1016/j.nonrwa.2006.11.015.

[103]

G. StamovE. Gospodinova and I. Stamova, Practical exponential stability with respect to $h-$manifolds of discontinuous delayed cohen–grossberg neural networks with variable impulsive perturbations, Mathematical Modelling and Control, 1 (2021), 26-34.  doi: 10.3934/mmc.2021003.

[104]

G. T. Stamov and I. M. Stamova, Almost periodic solutions for impulsive neural networks with delay, Applied Mathematical Modelling, 31 (2007), 1263-1270.  doi: 10.1016/j.apm.2006.04.008.

[105]

J. SunQ.-L. Han and X. Jiang, Impulsive control of time-delay systems using delayed impulse and its application to impulsive master–slave synchronization, Phys. Lett. A, 372 (2008), 6375-6380.  doi: 10.1016/j.physleta.2008.08.067.

[106]

X. TanJ. Cao and X. Li, Consensus of leader-following multiagent systems: A distributed event-triggered impulsive control strategy, IEEE Transactions on Cybernetics, 49 (2019), 792-801.  doi: 10.1109/TCYB.2017.2786474.

[107]

Y. TangH. GaoW. Zhang and J. Kurths, Leader-following consensus of a class of stochastic delayed multi-agent systems with partial mixed impulses, Automatica J. IFAC, 53 (2015), 346-354.  doi: 10.1016/j.automatica.2015.01.008.

[108]

Y. Tang, X. Wu, P. Shi and F. Qian, Input-to-state stability for nonlinear systems with stochastic impulses, Automatica J. IFAC, 113 (2020), 108766, 12 pp. doi: 10.1016/j.automatica.2019.108766.

[109]

Y. TangX. XingH. R. KarimiL. Kocarev and J. Kurths, Tracking control of networked multi-agent systems under new characterizations of impulses and its applications in robotic systems, IEEE Transactions on Industrial Electronics, 63 (2016), 1299-1307.  doi: 10.1109/TIE.2015.2453412.

[110]

L. Wang and X. Li, $\mu$-stability of impulsive differential systems with unbounded time-varying delays and nonlinear perturbations, Math. Methods Appl. Sci., 36 (2013), 1140-1446.  doi: 10.1002/mma.2696.

[111]

X. WangC. LiT. Huang and X. Pan, Impulsive control and synchronization of nonlinear system with impulse time window, Nonlinear Dynam., 78 (2014), 2837-2845.  doi: 10.1007/s11071-014-1629-1.

[112]

Y. Wang and J. Lu, Some recent results of analysis and control for impulsive systems, Commun. Nonlinear Sci. Numer. Simul., 80 (2020), 104862, 15 pp. doi: 10.1016/j.cnsns.2019.104862.

[113]

Y. WangJ. LuX. Li and J. Liang, Synchronization of coupled neural networks under mixed impulsive effects: A novel delay inequality approach, Neural Networks, 127 (2020), 38-46.  doi: 10.1016/j.neunet.2020.04.002.

[114]

Y. Wang, J. Lu and Y. Lou, Halanay-type inequality with delayed impulses and its applications, Sci. China Inf. Sci., 62 (2019), 192206, 10 pp. doi: 10.1007/s11432-018-9809-y.

[115]

T. WeiX. Xie and X. Li, Persistence and periodicity of survival red blood cells model with time-varying delays and impulses, Mathematical Modelling and Control, 1 (2021), 12-25.  doi: 10.3934/mmc.2021002.

[116]

S. WuX. SunX. Li and H. Wang, On controllability and observability of impulsive control systems with delayed impulses, Math. Comput. Simulation, 171 (2020), 65-78.  doi: 10.1016/j.matcom.2019.03.013.

[117]

X. WuP. ShiY. Tang and W. Zhang, Input-to-state stability of nonlinear stochastic time-varying systems with impulsive effects, Internat. J. Robust Nonlinear Control, 27 (2017), 1792-1809.  doi: 10.1002/rnc.3637.

[118]

X. WuY. Tang and W. Zhang, Input-to-state stability of impulsive stochastic delayed systems under linear assumptions, Automatica J. IFAC, 66 (2016), 195-204.  doi: 10.1016/j.automatica.2016.01.002.

[119]

X. Wu, L. Yan, W. Zhang and Y. Tang, Exponential stability of stochastic differential delay systems with delayed impulse effects, J. Math. Phys., 52 (2011), 092702, 14 pp. doi: 10.1063/1.3638037.

[120]

D. Xu and Z. Yang, Impulsive delay differential inequality and stability of neural networks, J. Math. Anal. Appl., 305 (2005), 107-120.  doi: 10.1016/j.jmaa.2004.10.040.

[121]

D. XuZ. Yang and Z. Yang, Exponential stability of nonlinear impulsive neutral differential equations with delays, Nonlinear Anal., 67 (2007), 1426-1439.  doi: 10.1016/j.na.2006.07.043.

[122]

F. XuL. DongD. WangX. Li and R. Rakkiyappan, Globally exponential stability of nonlinear impulsive switched systems, Math. Notes, 97 (2015), 803-810.  doi: 10.1134/S0001434615050156.

[123]

Z. XuX. Li and P. Duan, Synchronization of complex networks with time-varying delay of unknown bound via delayed impulsive control, Neural Networks, 125 (2020), 224-232.  doi: 10.1016/j.neunet.2020.02.003.

[124]

Z. Xu, X. Li and V. Stojanovic, Exponential stability of nonlinear state-dependent delayed impulsive systems with applications, Nonlinear Anal. Hybrid Syst., 42 (2021), 101088, 12 pp. doi: 10.1016/j.nahs.2021.101088.

[125]

T. Yang, Impulsive Control Theory, vol. 272, Springer-Verlag, Berlin, 2001.

[126]

X. YangJ. Cao and J. Lu, Stochastic synchronization of complex networks with nonidentical nodes via hybrid adaptive and impulsive control, IEEE Trans. Circuits Syst. I. Regul. Pap., 59 (2012), 371-384.  doi: 10.1109/TCSI.2011.2163969.

[127]

X. YangJ. Cao and J. Lu, Synchronization of delayed complex dynamical networks with impulsive and stochastic effects, Nonlinear Anal. Real World Appl., 12 (2011), 2252-2266.  doi: 10.1016/j.nonrwa.2011.01.007.

[128]

X. YangJ. Cao and J. Qiu, Pth moment exponential stochastic synchronization of coupled memristor-based neural networks with mixed delays via delayed impulsive control, Neural Networks, 65 (2015), 80-91. 

[129]

X. Yang and X. Li, Finite-time stability of nonlinear impulsive systems with applications to neural networks, IEEE Transactions on Neural Networks and Learning Systems. doi: 10.1109/TNNLS.2021.3093418.

[130]

X. YangX. LiQ. Xi and P. Duan, Review of stability and stabilization for impulsive delayed systems, Math. Biosci. Eng., 15 (2018), 1495-1515.  doi: 10.3934/mbe.2018069.

[131]

X. YangC. LiQ. SongH. Li and J. Huang, Effects of state-dependent impulses on robust exponential stability of quaternion-valued neural networks under parametric uncertainty, IEEE Trans. Neural Netw. Learn. Syst., 30 (2019), 2197-2211.  doi: 10.1109/TNNLS.2018.2877152.

[132]

X. Yang and J. Lu, Finite-time synchronization of coupled networks with markovian topology and impulsive effects, IEEE Trans. Automat. Control, 61 (2016), 2256-2261.  doi: 10.1109/TAC.2015.2484328.

[133]

X. YangJ. LuD. W. C. Ho and Q. Song, Synchronization of uncertain hybrid switching and impulsive complex networks, Appl. Math. Model., 59 (2018), 379-392.  doi: 10.1016/j.apm.2018.01.046.

[134]

X. YangZ. Yang and X. Nie, Exponential synchronization of discontinuous chaotic systems via delayed impulsive control and its application to secure communication, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 1529-1543.  doi: 10.1016/j.cnsns.2013.09.012.

[135]

Z. Yang and D. Xu, Stability analysis of delay neural networks with impulsive effects, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 13 (2006), 563-573. 

[136]

Z. Yang and D. Xu, Stability analysis and design of impulsive control systems with time delay, IEEE Trans. Automat. Control, 52 (2007), 1448-1454.  doi: 10.1109/TAC.2007.902748.

[137]

H. ZhangZ.-H. Guan and G. Feng, Reliable dissipative control for stochastic impulsive systems, Automatica J. IFAC, 44 (2008), 1004-1010.  doi: 10.1016/j.automatica.2007.08.018.

[138]

H. ZhangT. MaG.-B. Huang and Z. Wang, Robust global exponential synchronization of uncertain chaotic delayed neural networks via dual-stage impulsive control, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 40 (2010), 831-844.  doi: 10.1109/TSMCB.2009.2030506.

[139]

K. Zhang and E. Braverman, Time-delay systems with delayed impulses: A unified criterion on asymptotic stability, Automatica J. IFAC, 125 (2021), 109470, 8 pp. doi: 10.1016/j.automatica.2020.109470.

[140]

L. ZhangX. YangC. Xu and J. Feng, Exponential synchronization of complex-valued complex networks with time-varying delays and stochastic perturbations via time-delayed impulsive control, Appl. Math. Comput., 306 (2017), 22-30.  doi: 10.1016/j.amc.2017.02.004.

[141]

W. ZhangY. TangJ.-A. Fang and X. Wu, Stability of delayed neural networks with time-varying impulses, Neural Networks, 36 (2012), 59-63.  doi: 10.1016/j.neunet.2012.08.014.

[142]

X. Zhang and C. Li, Finite-time stability of nonlinear systems with state-dependent delayed impulses, Nonlinear Dynamics, 102 (2020), 197-210.  doi: 10.1007/s11071-020-05953-4.

[143]

X. ZhangC. Li and H. Li, Finite-time stabilization of nonlinear systems via impulsive control with state-dependent delay, J. Franklin Inst., 359 (2022), 1196-1214.  doi: 10.1016/j.jfranklin.2021.11.013.

[144]

Y. Zhang and J. Sun, Stability of impulsive neural networks with time delays, Physics Letters A, 348 (2005), 44-50.  doi: 10.1016/j.physleta.2005.08.030.

[145]

Y. ZhangJ. Sun and G. Feng, Impulsive control of discrete systems with time delay, IEEE Trans. Automat. Control, 54 (2009), 871-875.  doi: 10.1109/TAC.2008.2010968.

[146]

Y. Zhao, X. Li and J. Cao, Global exponential stability for impulsive systems with infinite distributed delay based on flexible impulse frequency, Appl. Math. Comput., 386 (2020), 125467, 10 pp. doi: 10.1016/j.amc.2020.125467.

[147]

Y. ZhouH. Zhang and Z. Zeng, Quasi-synchronization of delayed memristive neural networks via a hybrid impulsive control, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 51 (2021), 1954-1965.  doi: 10.1109/TSMC.2019.2911366.

[148]

C. Zhu, X. Li and J. Cao, Finite-time ${H}_\infty$ dynamic output feedback control for nonlinear impulsive switched systems, Nonlinear Anal. Hybrid Syst., 39 (2021), 100975, 13 pp. doi: 10.1016/j.nahs.2020.100975.

[149]

W. ZhuD. WangL. Liu and G. Feng, Event-based impulsive control of continuous-time dynamic systems and its application to synchronization of memristive neural networks, IEEE Trans. Neural Netw. Learn. Syst., 29 (2018), 3599-3609.  doi: 10.1109/TNNLS.2017.2731865.

show all references

References:
[1]

F. AmatoM. Ariola and P. Dorato, Finite-time control of linear systems subject to parametric uncertainties and disturbances, Automatica, 37 (2001), 1459-1463.  doi: 10.1016/S0005-1098(01)00087-5.

[2]

F. AmatoG. De Tommasi and A. Pironti, Necessary and sufficient conditions for finite-time stability of impulsive dynamical linear systems, Automatica J. IFAC, 49 (2013), 2546-2550.  doi: 10.1016/j.automatica.2013.04.004.

[3]

G. Ballinger and X. Liu, Existence and uniqueness results for impulsive delay differential equations, Dynamics of Continuous Discrete and Impulsive Systems, 5 (1999), 579-591. 

[4]

C. Briat, Stability analysis and stabilization of stochastic linear impulsive, switched and sampled-data systems under dwell-time constraints, Automatica J. IFAC, 74 (2016), 279-287.  doi: 10.1016/j.automatica.2016.08.001.

[5]

C. Briat and A. Seuret, Convex dwell-time characterizations for uncertain linear impulsive systems, IEEE Trans. Automat. Control, 57 (2012), 3241-3246.  doi: 10.1109/TAC.2012.2200379.

[6]

W. Cao and Q. Zhu, Razumikhin-type theorem for pth exponential stability of impulsive stochastic functional differential equations based on vector lyapunov function, Nonlinear Anal. Hybrid Syst., 39 (2021), 100983, 10 pp. doi: 10.1016/j.nahs.2020.100983.

[7]

J. ChenB. ChenZ. Zeng and P. Jiang, Effects of subsystem and coupling on synchronization of multiple neural networks with delays via impulsive coupling, IEEE Trans. Neural Netw. Learn. Syst., 30 (2019), 3748-3758.  doi: 10.1109/TNNLS.2019.2898919.

[8]

J. Chen, X. Li and D. Wang, Asymptotic stability and exponential stability of impulsive delayed Hopfield neural networks, Abstr. Appl. Anal., (2013), Art. ID 638496, 10 pp. doi: 10.1155/2013/638496.

[9]

W.-H. ChenZ. Ruan and W. X. Zheng, Stability and ${L}_2$-gain analysis for linear time-delay systems with delayed impulses: An augmentation-based switching impulse approach, IEEE Trans. Automat. Control, 64 (2019), 4209-4216.  doi: 10.1109/TAC.2019.2893149.

[10]

W.-H. ChenD. Wei and X. Lu, Global exponential synchronization of nonlinear time-delay lur'e systems via delayed impulsive control, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 3298-3312.  doi: 10.1016/j.cnsns.2014.01.018.

[11]

W.-H. ChenD. Wei and W. X. 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.

[12]

W.-H. Chen and W. X. Zheng, Exponential stability of nonlinear time-delay systems with delayed impulse effects, Automatica J. IFAC, 47 (2011), 1075-1083.  doi: 10.1016/j.automatica.2011.02.031.

[13]

K. E. M. Church and X. Liu, Invariant manifold-guided impulsive stabilization of delay equations, IEEE Trans. Automat. Control, 66 (2021), 5997-6002.  doi: 10.1109/TAC.2021.3057988.

[14]

S. Dashkovskiy and P. Feketa, Input-to-state stability of impulsive systems and their networks, Nonlinear Anal. Hybrid Syst., 26 (2017), 190-200.  doi: 10.1016/j.nahs.2017.06.004.

[15]

S. DashkovskiyM. KosmykovA. Mironchenko and L. Naujok, Stability of interconnected impulsive systems with and without time delays, using Lyapunov methods, Nonlinear Anal. Hybrid Syst., 6 (2012), 899-915.  doi: 10.1016/j.nahs.2012.02.001.

[16]

S. Dashkovskiy and A. Mironchenko, Input-to-state stability of nonlinear impulsive systems, SIAM J. Control Optim., 51 (2013), 1962-1987.  doi: 10.1137/120881993.

[17]

K. H. DegueD. Efimov and J.-P. Richard, Stabilization of linear impulsive systems under dwell-time constraints: Interval observer-based framework, Eur. J. Control, 42 (2018), 1-14.  doi: 10.1016/j.ejcon.2018.01.001.

[18]

W. DuS. Y. S. LeungY. Tang and A. V. Vasilakos, Differential evolution with event-triggered impulsive control, IEEE Transactions on Cybernetics, 47 (2017), 244-257.  doi: 10.1109/TCYB.2015.2512942.

[19]

P. Feketa and N. Bajcinca, On robustness of impulsive stabilization, Automatica J. IFAC, 104 (2019), 48-56.  doi: 10.1016/j.automatica.2019.02.056.

[20]

P. Getto and M. Waurick, A differential equation with state-dependent delay from cell population biology, J. Differential Equations, 260 (2016), 6176-6200.  doi: 10.1016/j.jde.2015.12.038.

[21]

K. Gu, V. L. Kharitonov and J. Chen, Stability of Time-Delay Systems, Birkhäuser Boston, Inc., Boston, MA, 2003. doi: 10.1007/978-1-4612-0039-0.

[22]

Z.-H. Guan and G. Chen, On delayed impulsive hopfield neural networks, Neural Networks, 12 (1999), 273-280.  doi: 10.1016/S0893-6080(98)00133-6.

[23]

Z.-H. GuanD. J. Hill and X. Shen, On hybrid impulsive and switching systems and application to nonlinear control, IEEE Trans. Automat. Control, 50 (2005), 1058-1062.  doi: 10.1109/TAC.2005.851462.

[24]

Z.-H. GuanZ.-W. LiuG. Feng and M. Jian, Impulsive consensus algorithms for second-order multi-agent networks with sampled information, Automatica J. IFAC, 48 (2012), 1397-1404.  doi: 10.1016/j.automatica.2012.05.005.

[25]

Z.-H. GuanZ.-W. LiuG. Feng and Y.-W. Wang, 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. 

[26] W. M. HaddadV. Chellaboina and S. G. Nersesov, Impulsive and Hybrid Dynamical Systems, Princeton University Press, 2006.  doi: 10.1515/9781400865246.
[27]

H. Haimovich and J. L. Mancilla-Aguilar, Nonrobustness of asymptotic stability of impulsive systems with inputs, Automatica J. IFAC, 122 (2020), 109238, 9 pp. doi: 10.1016/j.automatica.2020.109238.

[28]

H. Haimovich and J. L. Mancilla-Aguilar, Strong ISS implies strong iISS for time-varying impulsive systems, Automatica J. IFAC, 122 (2020), 109224, 12 pp. doi: 10.1016/j.automatica.2020.109224.

[29]

F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, in Handbook of Differential Equations: Ordinary Differential Equations, vol. 3, Elsevier, 2006,435–545. doi: 10.1016/S1874-5725(06)80009-X.

[30]

W. HeX. GaoW. Zhong and F. Qian, Secure impulsive synchronization control of multi-agent systems under deception attacks, Inform. Sci., 459 (2018), 354-368.  doi: 10.1016/j.ins.2018.04.020.

[31]

W. HeF. QianQ.-L. Han and G. Chen, Almost sure stability of nonlinear systems under random and impulsive sequential attacks, IEEE Trans. Automat. Control, 65 (2020), 3879-3886.  doi: 10.1109/TAC.2020.2972220.

[32]

W. HeF. QianJ. LamG. ChenQ.-L. Han and J. Kurths, Quasi-synchronization of heterogeneous dynamic networks via distributed impulsive control: Error estimation, optimization and design, Automatica J. IFAC, 62 (2015), 249-262.  doi: 10.1016/j.automatica.2015.09.028.

[33]

X. HeD. Peng and X. Li, Synchronization of complex networks with impulsive control involving stabilizing delay, J. Franklin Inst., 357 (2020), 4869-4886.  doi: 10.1016/j.jfranklin.2020.03.044.

[34]

X. He, Y. Wang and X. Li, Uncertain impulsive control for leader-following synchronization of complex networks, Chaos Solitons Fractals, 147 (2021), 110980, 7 pp. doi: 10.1016/j.chaos.2021.110980.

[35]

Z. He, C. Li, H. Li and Q. Zhang, Global exponential stability of high-order hopfield neural networks with state-dependent impulses, Phys. A, 542 (2020), 123434, 21 pp. doi: 10.1016/j.physa.2019.123434.

[36]

J. P. HespanhaD. Liberzon and A. R. Teel, Lyapunov conditions for input-to-state stability of impulsive systems, Automatica J. IFAC, 44 (2008), 2735-2744.  doi: 10.1016/j.automatica.2008.03.021.

[37]

J. HuG. SuiX. Lv and X. Li, Fixed-time control of delayed neural networks with impulsive perturbations, Nonlinear Anal. Model. Control, 23 (2018), 904-920.  doi: 10.15388/NA.2018.6.6.

[38]

B. JiangJ. Lu and Y. Liu, Exponential stability of delayed systems with average-delay impulses, SIAM J. Control Optim., 58 (2020), 3763-3784.  doi: 10.1137/20M1317037.

[39]

B. JiangJ. LuJ. 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.

[40]

A. KhadraX. Z. Liu and X. Shen, Analyzing the robustness of impulsive synchronization coupled by linear delayed impulses, IEEE Trans. Automat. Control, 54 (2009), 923-928.  doi: 10.1109/TAC.2009.2013029.

[41]

A. KhadraX. Z. Liu and X. Shen, Impulsively synchronizing chaotic systems with delay and applications to secure communication, Automatica J. IFAC, 41 (2005), 1491-1502.  doi: 10.1016/j.automatica.2005.04.012.

[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]

H. LiC. LiD. Ouyang and S. K. Nguang, Impulsive synchronization of unbounded delayed inertial neural networks with actuator saturation and sampled-data control and its application to image encryption, IEEE Trans. Neural Netw. Learn. Syst., 32 (2021), 1460-1473.  doi: 10.1109/TNNLS.2020.2984770.

[44]

P. LiX. Li and J. Lu, Input-to-state stability of impulsive delay systems with multiple impulses, IEEE Trans. Automat. Control, 66 (2021), 362-368.  doi: 10.1109/TAC.2020.2982156.

[45]

X. Li, Further analysis on uniform stability of impulsive infinite delay differential equations, Appl. Math. Lett., 25 (2012), 133-137.  doi: 10.1016/j.aml.2011.08.001.

[46]

X. LiH. Akca and X. Fu, Uniform stability of impulsive infinite delay differential equations with applications to systems with integral impulsive conditions, Appl. Math. Comput., 219 (2013), 7329-7337.  doi: 10.1016/j.amc.2012.12.033.

[47]

X. Li and M. Bohner, An impulsive delay differential inequality and applications, Comput. Math. Appl., 64 (2012), 1875-1881.  doi: 10.1016/j.camwa.2012.03.013.

[48]

X. LiM. Bohner and C.-K. Wang, Impulsive differential equations: Periodic solutions and applications, Automatica J. IFAC, 52 (2015), 173-178.  doi: 10.1016/j.automatica.2014.11.009.

[49]

X. LiT. CaraballoR. Rakkiyappan and X. Han, On the stability of impulsive functional differential equations with infinite delays, Math. Methods Appl. Sci., 38 (2015), 3130-3140.  doi: 10.1002/mma.3303.

[50]

X. LiD. W. C. Ho and J. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica J. IFAC, 99 (2019), 361-368.  doi: 10.1016/j.automatica.2018.10.024.

[51]

X. Li and P. Li, Stability of time-delay systems with impulsive control involving stabilizing delays, Automatica J. IFAC, 124 (2021), 109336, 6 pp. doi: 10.1016/j.automatica.2020.109336.

[52]

X. LiD. O'Regan and H. Akca, Global exponential stabilization of impulsive neural networks with unbounded continuously distributed delays, IMA J. Appl. Math., 80 (2015), 85-99.  doi: 10.1093/imamat/hxt027.

[53]

X. LiD. Peng and J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Trans. Automat. Control, 65 (2020), 4908-4913.  doi: 10.1109/TAC.2020.2964558.

[54]

X. LiJ. ShenH. Akca and R. Rakkiyappan, LMI-based stability for singularly perturbed nonlinear impulsive differential systems with delays of small parameter, Appl. Math. Comput., 250 (2015), 798-804.  doi: 10.1016/j.amc.2014.10.113.

[55]

X. LiJ. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Appl. Math. Comput., 329 (2018), 14-22.  doi: 10.1016/j.amc.2018.01.036.

[56]

X. Li and S. Song, Stabilization of delay systems: Delay-dependent impulsive control, IEEE Trans. Automat. Control, 62 (2017), 406-411.  doi: 10.1109/TAC.2016.2530041.

[57]

X. Li and S. Song, Impulsive Systems with Delays: Stability and Control, Springer, Singapore, 2022. doi: 10.1007/978-981-16-4687-4.

[58]

X. LiS. Song and J. Wu, Exponential stability of nonlinear systems with delayed impulses and applications, IEEE Trans. Automat. Control, 64 (2019), 4024-4034.  doi: 10.1109/TAC.2019.2905271.

[59]

X. Li and J. Wu, Stability of nonlinear differential systems with state-dependent delayed impulses, Automatica J. IFAC, 64 (2016), 63-69.  doi: 10.1016/j.automatica.2015.10.002.

[60]

X. Li and J. Wu, Sufficient stability conditions of nonlinear differential systems under impulsive control with state-dependent delay, IEEE Trans. Automat. Control, 63 (2018), 306-311.  doi: 10.1109/TAC.2016.2639819.

[61]

X. Li, X. Yang and J. Cao, Event-triggered impulsive control for nonlinear delay systems, Automatica J. IFAC, 117 (2020), 108981, 7 pp. doi: 10.1016/j.automatica.2020.108981.

[62]

X. LiX. Yang and T. Huang, Persistence of delayed cooperative models: Impulsive control method, Appl. Math. Comput., 342 (2019), 130-146.  doi: 10.1016/j.amc.2018.09.003.

[63]

X. LiX. Yang and S. Song, Lyapunov conditions for finite-time stability of time-varying time-delay systems, Automatica J. IFAC, 103 (2019), 135-140.  doi: 10.1016/j.automatica.2019.01.031.

[64]

X. LiX. Zhang and S. Song, Effect of delayed impulses on input-to-state stability of nonlinear systems, Automatica J. IFAC, 76 (2017), 378-382.  doi: 10.1016/j.automatica.2016.08.009.

[65]

X. Li and Y. Zhao, Sliding mode control for linear impulsive systems with matched disturbances, IEEE Transactions on Automatic Control. doi: 10.1109/TAC.2021.3129735.

[66]

D. LinX. Li and D. O'Regan, $\mu$-stability of infinite delay functional differential systems with impulsive effects, Appl. Anal., 92 (2013), 15-26.  doi: 10.1080/00036811.2011.584185.

[67]

B. LiuD. J. Hill and Z. Sun, Stabilisation to input-to-state stability for continuous-time dynamical systems via event-triggered impulsive control with three levels of events, IET Control Theory Appl., 12 (2018), 1167-1179.  doi: 10.1049/iet-cta.2017.0820.

[68]

B. LiuX. LiuG. Chen and H. Wang, Robust impulsive synchronization of uncertain dynamical networks, IEEE Trans. Circuits Syst. I. Regul. Pap., 52 (2005), 1431-1441.  doi: 10.1109/TCSI.2005.851708.

[69]

B. Liu, Z. Sun, Y. Luo and Y. Zhong, Uniform synchronization for chaotic dynamical systems via event-triggered impulsive control, Phys. A, 531 (2019), 121725, 14 pp. doi: 10.1016/j.physa.2019.121725.

[70]

B. LiuB. Xu and T. Liu, Almost sure contraction for stochastic switched impulsive systems, IEEE Trans. Automat. Control, 66 (2021), 5393-5400.  doi: 10.1109/TAC.2020.3047554.

[71]

J. Liu and X. Li, Impulsive stabilization of high-order nonlinear retarded differential equations, Appl. Math., 58 (2013), 347-367.  doi: 10.1007/s10492-013-0017-3.

[72]

K. LiuA. Selivanov and E. Fridman, Survey on time-delay approach to networked control, Annu. Rev. Control, 48 (2019), 57-79.  doi: 10.1016/j.arcontrol.2019.06.005.

[73]

W. Liu, J. Sun, G. Wang and J. Chen, Quantized impulsive control of linear systems under bounded disturbances and DoS attacks, IEEE Transactions on Control of Network Systems. doi: 10.1109/TCNS.2021.3085759.

[74]

X. Liu, Practical stabilization of control systems with impulse effects, J. Math. Anal. Appl., 166 (1992), 563-576.  doi: 10.1016/0022-247X(92)90315-5.

[75]

X. Liu, Stability of impulsive control systems with time delay, Math. Comput. Modelling, 39 (2004), 511-519.  doi: 10.1016/S0895-7177(04)90522-5.

[76]

X. Liu and G. Ballinger, Uniform asymptotic stability of impulsive delay differential equations, Comput. Math. Appl., 41 (2001), 903-915.  doi: 10.1016/S0898-1221(00)00328-X.

[77]

X. Liu and G. Ballinger, On boundedness of solutoins for impulsive systems in terms of two measures, Nonlinear World, 4 (1997), 417-434. 

[78]

X. Liu and G. Ballinger, Existence and continuability of solutions for differential equations with delays and state-dependent impulses, Nonlinear Anal., 51 (2002), 633-647.  doi: 10.1016/S0362-546X(01)00847-1.

[79]

X. Liu and K. Rohlf, Impulsive control of a Lotka-Volterra system, IMA J. Math. Control Inform., 15 (1998), 269-284.  doi: 10.1093/imamci/15.3.269.

[80]

X. Liu and K. Zhang, Synchronization of linear dynamical networks on time scales: Pinning control via delayed impulses, Automatica J. IFAC, 72 (2016), 147-152.  doi: 10.1016/j.automatica.2016.06.001.

[81]

X. Liu and K. Zhang, Input-to-state stability of time-delay systems with delay-dependent impulses, IEEE Trans. Automat. Control, 65 (2020), 1676-1682.  doi: 10.1109/TAC.2019.2930239.

[82]

X. LiuK. Zhang and W.-C. Xie, Consensus seeking in multi-agent systems via hybrid protocols with impulse delays, Nonlinear Anal. Hybrid Syst., 25 (2017), 90-98.  doi: 10.1016/j.nahs.2017.03.002.

[83]

Y. LiuS. Zhao and J. Lu, A new fuzzy impulsive control of chaotic systems based on T–S fuzzy model, IEEE Transactions on Fuzzy Systems, 19 (2011), 393-398.  doi: 10.1109/TFUZZ.2010.2090162.

[84]

Z.-W. LiuG. WenX. YuZ.-H. Guan and T. Huang, Delayed impulsive control for consensus of multiagent systems with switching communication graphs, IEEE Transactions on Cybernetics, 50 (2020), 3045-3055.  doi: 10.1109/TCYB.2019.2926115.

[85]

J. LuD. W. C. Ho and J. Cao, A unified synchronization criterion for impulsive dynamical networks, Automatica J. IFAC, 46 (2010), 1215-1221.  doi: 10.1016/j.automatica.2010.04.005.

[86]

J. LuD. W. C. HoJ. Cao and J. Kurths, Exponential synchronization of linearly coupled neural networks with impulsive disturbances, IEEE Transactions on Neural Networks, 22 (2011), 329-336.  doi: 10.1109/TNN.2010.2101081.

[87]

S. Luo, F. Deng and W.-H. Chen, Stability and stabilization of linear impulsive systems with large impulse-delays: A stabilizing delay perspective, Automatica J. IFAC, 127 (2021), 109533, 7 pp. doi: 10.1016/j.automatica.2021.109533.

[88]

X. LvJ. CaoX. LiM. Abdel-Aty and U. A. Al-Juboori, Synchronization analysis for complex dynamical networks with coupling delay via event-triggered delayed impulsive control, IEEE Transactions on Cybernetics, 51 (2021), 5269-5278.  doi: 10.1109/TCYB.2020.2974315.

[89]

X. Lv and X. Li, Finite time stability and controller design for nonlinear impulsive sampled-data systems with applications, ISA Transactions, 70 (2017), 30-36.  doi: 10.1016/j.isatra.2017.07.025.

[90]

J. L. Mancilla-Aguilar, H. Haimovich and P. Feketa, Uniform stability of nonlinear time-varying impulsive systems with eventually uniformly bounded impulse frequency, Nonlinear Anal. Hybrid Syst., 38 (2020), 100933, 16 pp. doi: 10.1016/j.nahs.2020.100933.

[91]

P. NaghshtabriziJ. P. Hespanha and A. R. Teel, Exponential stability of impulsive systems with application to uncertain sampled-data systems, Systems Control Lett., 57 (2008), 378-385.  doi: 10.1016/j.sysconle.2007.10.009.

[92]

S. G. Nersesov and W. M. Haddad, Finite-time stabilization of nonlinear impulsive dynamical systems, Nonlinear Anal. Hybrid Syst., 2 (2008), 832-845.  doi: 10.1016/j.nahs.2007.12.001.

[93]

S.-I. Niculescu, Delay Effects on Stability: A Robust Control Approach, vol. 269, Springer-Verlag London, Ltd., London, 2001.

[94]

S. Peng and F. Deng, New criteria on $p$th moment input-to-state stability of impulsive stochastic delayed differential systems, IEEE Trans. Automat. Control, 62 (2017), 3573-3579.  doi: 10.1109/TAC.2017.2660066.

[95]

R. RakkiyappanP. Balasubramaniam and J. Cao, Global exponential stability results for neutral-type impulsive neural networks, Nonlinear Anal. Real World Appl., 11 (2010), 122-130.  doi: 10.1016/j.nonrwa.2008.10.050.

[96]

W. Ren and J. Xiong, Vector-Lyapunov-function-based input-to-state stability of stochastic impulsive switched time-delay systems, IEEE Trans. Automat. Control, 64 (2019), 654-669. 

[97]

W. Ren and J. Xiong, Stability analysis of impulsive switched time-delay systems with state-dependent impulses, IEEE Trans. Automat. Control, 64 (2019), 3928-3935.  doi: 10.1109/TAC.2018.2890768.

[98]

W. Ren and J. Xiong, Stability analysis of stochastic impulsive switched systems with deterministic state-dependent impulses and switches, SIAM J. Control Optim., 59 (2021), 2068-2092.  doi: 10.1137/20M1353460.

[99]

H. RíosL. Hetel and D. Efimov, Nonlinear impulsive systems: 2D stability analysis approach, Automatica J. IFAC, 80 (2017), 32-40.  doi: 10.1016/j.automatica.2017.01.010.

[100]

H. RíosL. Hetel and D. Efimov, Robust output-feedback control for uncertain linear sampled-data systems: A 2D impulsive system approach, Nonlinear Anal. Hybrid Syst., 32 (2019), 177-201.  doi: 10.1016/j.nahs.2018.11.005.

[101]

J. Shen and J. Li, Existence and global attractivity of positive periodic solutions for impulsive predator–prey model with dispersion and time delays, Nonlinear Anal. Real World Appl., 10 (2009), 227-243.  doi: 10.1016/j.nonrwa.2007.08.026.

[102]

Q. Song and J. Zhang, Global exponential stability of impulsive Cohen–Grossberg neural network with time-varying delays, Nonlinear Anal. Real World Appl., 9 (2008), 500-510.  doi: 10.1016/j.nonrwa.2006.11.015.

[103]

G. StamovE. Gospodinova and I. Stamova, Practical exponential stability with respect to $h-$manifolds of discontinuous delayed cohen–grossberg neural networks with variable impulsive perturbations, Mathematical Modelling and Control, 1 (2021), 26-34.  doi: 10.3934/mmc.2021003.

[104]

G. T. Stamov and I. M. Stamova, Almost periodic solutions for impulsive neural networks with delay, Applied Mathematical Modelling, 31 (2007), 1263-1270.  doi: 10.1016/j.apm.2006.04.008.

[105]

J. SunQ.-L. Han and X. Jiang, Impulsive control of time-delay systems using delayed impulse and its application to impulsive master–slave synchronization, Phys. Lett. A, 372 (2008), 6375-6380.  doi: 10.1016/j.physleta.2008.08.067.

[106]

X. TanJ. Cao and X. Li, Consensus of leader-following multiagent systems: A distributed event-triggered impulsive control strategy, IEEE Transactions on Cybernetics, 49 (2019), 792-801.  doi: 10.1109/TCYB.2017.2786474.

[107]

Y. TangH. GaoW. Zhang and J. Kurths, Leader-following consensus of a class of stochastic delayed multi-agent systems with partial mixed impulses, Automatica J. IFAC, 53 (2015), 346-354.  doi: 10.1016/j.automatica.2015.01.008.

[108]

Y. Tang, X. Wu, P. Shi and F. Qian, Input-to-state stability for nonlinear systems with stochastic impulses, Automatica J. IFAC, 113 (2020), 108766, 12 pp. doi: 10.1016/j.automatica.2019.108766.

[109]

Y. TangX. XingH. R. KarimiL. Kocarev and J. Kurths, Tracking control of networked multi-agent systems under new characterizations of impulses and its applications in robotic systems, IEEE Transactions on Industrial Electronics, 63 (2016), 1299-1307.  doi: 10.1109/TIE.2015.2453412.

[110]

L. Wang and X. Li, $\mu$-stability of impulsive differential systems with unbounded time-varying delays and nonlinear perturbations, Math. Methods Appl. Sci., 36 (2013), 1140-1446.  doi: 10.1002/mma.2696.

[111]

X. WangC. LiT. Huang and X. Pan, Impulsive control and synchronization of nonlinear system with impulse time window, Nonlinear Dynam., 78 (2014), 2837-2845.  doi: 10.1007/s11071-014-1629-1.

[112]

Y. Wang and J. Lu, Some recent results of analysis and control for impulsive systems, Commun. Nonlinear Sci. Numer. Simul., 80 (2020), 104862, 15 pp. doi: 10.1016/j.cnsns.2019.104862.

[113]

Y. WangJ. LuX. Li and J. Liang, Synchronization of coupled neural networks under mixed impulsive effects: A novel delay inequality approach, Neural Networks, 127 (2020), 38-46.  doi: 10.1016/j.neunet.2020.04.002.

[114]

Y. Wang, J. Lu and Y. Lou, Halanay-type inequality with delayed impulses and its applications, Sci. China Inf. Sci., 62 (2019), 192206, 10 pp. doi: 10.1007/s11432-018-9809-y.

[115]

T. WeiX. Xie and X. Li, Persistence and periodicity of survival red blood cells model with time-varying delays and impulses, Mathematical Modelling and Control, 1 (2021), 12-25.  doi: 10.3934/mmc.2021002.

[116]

S. WuX. SunX. Li and H. Wang, On controllability and observability of impulsive control systems with delayed impulses, Math. Comput. Simulation, 171 (2020), 65-78.  doi: 10.1016/j.matcom.2019.03.013.

[117]

X. WuP. ShiY. Tang and W. Zhang, Input-to-state stability of nonlinear stochastic time-varying systems with impulsive effects, Internat. J. Robust Nonlinear Control, 27 (2017), 1792-1809.  doi: 10.1002/rnc.3637.

[118]

X. WuY. Tang and W. Zhang, Input-to-state stability of impulsive stochastic delayed systems under linear assumptions, Automatica J. IFAC, 66 (2016), 195-204.  doi: 10.1016/j.automatica.2016.01.002.

[119]

X. Wu, L. Yan, W. Zhang and Y. Tang, Exponential stability of stochastic differential delay systems with delayed impulse effects, J. Math. Phys., 52 (2011), 092702, 14 pp. doi: 10.1063/1.3638037.

[120]

D. Xu and Z. Yang, Impulsive delay differential inequality and stability of neural networks, J. Math. Anal. Appl., 305 (2005), 107-120.  doi: 10.1016/j.jmaa.2004.10.040.

[121]

D. XuZ. Yang and Z. Yang, Exponential stability of nonlinear impulsive neutral differential equations with delays, Nonlinear Anal., 67 (2007), 1426-1439.  doi: 10.1016/j.na.2006.07.043.

[122]

F. XuL. DongD. WangX. Li and R. Rakkiyappan, Globally exponential stability of nonlinear impulsive switched systems, Math. Notes, 97 (2015), 803-810.  doi: 10.1134/S0001434615050156.

[123]

Z. XuX. Li and P. Duan, Synchronization of complex networks with time-varying delay of unknown bound via delayed impulsive control, Neural Networks, 125 (2020), 224-232.  doi: 10.1016/j.neunet.2020.02.003.

[124]

Z. Xu, X. Li and V. Stojanovic, Exponential stability of nonlinear state-dependent delayed impulsive systems with applications, Nonlinear Anal. Hybrid Syst., 42 (2021), 101088, 12 pp. doi: 10.1016/j.nahs.2021.101088.

[125]

T. Yang, Impulsive Control Theory, vol. 272, Springer-Verlag, Berlin, 2001.

[126]

X. YangJ. Cao and J. Lu, Stochastic synchronization of complex networks with nonidentical nodes via hybrid adaptive and impulsive control, IEEE Trans. Circuits Syst. I. Regul. Pap., 59 (2012), 371-384.  doi: 10.1109/TCSI.2011.2163969.

[127]

X. YangJ. Cao and J. Lu, Synchronization of delayed complex dynamical networks with impulsive and stochastic effects, Nonlinear Anal. Real World Appl., 12 (2011), 2252-2266.  doi: 10.1016/j.nonrwa.2011.01.007.

[128]

X. YangJ. Cao and J. Qiu, Pth moment exponential stochastic synchronization of coupled memristor-based neural networks with mixed delays via delayed impulsive control, Neural Networks, 65 (2015), 80-91. 

[129]

X. Yang and X. Li, Finite-time stability of nonlinear impulsive systems with applications to neural networks, IEEE Transactions on Neural Networks and Learning Systems. doi: 10.1109/TNNLS.2021.3093418.

[130]

X. YangX. LiQ. Xi and P. Duan, Review of stability and stabilization for impulsive delayed systems, Math. Biosci. Eng., 15 (2018), 1495-1515.  doi: 10.3934/mbe.2018069.

[131]

X. YangC. LiQ. SongH. Li and J. Huang, Effects of state-dependent impulses on robust exponential stability of quaternion-valued neural networks under parametric uncertainty, IEEE Trans. Neural Netw. Learn. Syst., 30 (2019), 2197-2211.  doi: 10.1109/TNNLS.2018.2877152.

[132]

X. Yang and J. Lu, Finite-time synchronization of coupled networks with markovian topology and impulsive effects, IEEE Trans. Automat. Control, 61 (2016), 2256-2261.  doi: 10.1109/TAC.2015.2484328.

[133]

X. YangJ. LuD. W. C. Ho and Q. Song, Synchronization of uncertain hybrid switching and impulsive complex networks, Appl. Math. Model., 59 (2018), 379-392.  doi: 10.1016/j.apm.2018.01.046.

[134]

X. YangZ. Yang and X. Nie, Exponential synchronization of discontinuous chaotic systems via delayed impulsive control and its application to secure communication, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 1529-1543.  doi: 10.1016/j.cnsns.2013.09.012.

[135]

Z. Yang and D. Xu, Stability analysis of delay neural networks with impulsive effects, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 13 (2006), 563-573. 

[136]

Z. Yang and D. Xu, Stability analysis and design of impulsive control systems with time delay, IEEE Trans. Automat. Control, 52 (2007), 1448-1454.  doi: 10.1109/TAC.2007.902748.

[137]

H. ZhangZ.-H. Guan and G. Feng, Reliable dissipative control for stochastic impulsive systems, Automatica J. IFAC, 44 (2008), 1004-1010.  doi: 10.1016/j.automatica.2007.08.018.

[138]

H. ZhangT. MaG.-B. Huang and Z. Wang, Robust global exponential synchronization of uncertain chaotic delayed neural networks via dual-stage impulsive control, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 40 (2010), 831-844.  doi: 10.1109/TSMCB.2009.2030506.

[139]

K. Zhang and E. Braverman, Time-delay systems with delayed impulses: A unified criterion on asymptotic stability, Automatica J. IFAC, 125 (2021), 109470, 8 pp. doi: 10.1016/j.automatica.2020.109470.

[140]

L. ZhangX. YangC. Xu and J. Feng, Exponential synchronization of complex-valued complex networks with time-varying delays and stochastic perturbations via time-delayed impulsive control, Appl. Math. Comput., 306 (2017), 22-30.  doi: 10.1016/j.amc.2017.02.004.

[141]

W. ZhangY. TangJ.-A. Fang and X. Wu, Stability of delayed neural networks with time-varying impulses, Neural Networks, 36 (2012), 59-63.  doi: 10.1016/j.neunet.2012.08.014.

[142]

X. Zhang and C. Li, Finite-time stability of nonlinear systems with state-dependent delayed impulses, Nonlinear Dynamics, 102 (2020), 197-210.  doi: 10.1007/s11071-020-05953-4.

[143]

X. ZhangC. Li and H. Li, Finite-time stabilization of nonlinear systems via impulsive control with state-dependent delay, J. Franklin Inst., 359 (2022), 1196-1214.  doi: 10.1016/j.jfranklin.2021.11.013.

[144]

Y. Zhang and J. Sun, Stability of impulsive neural networks with time delays, Physics Letters A, 348 (2005), 44-50.  doi: 10.1016/j.physleta.2005.08.030.

[145]

Y. ZhangJ. Sun and G. Feng, Impulsive control of discrete systems with time delay, IEEE Trans. Automat. Control, 54 (2009), 871-875.  doi: 10.1109/TAC.2008.2010968.

[146]

Y. Zhao, X. Li and J. Cao, Global exponential stability for impulsive systems with infinite distributed delay based on flexible impulse frequency, Appl. Math. Comput., 386 (2020), 125467, 10 pp. doi: 10.1016/j.amc.2020.125467.

[147]

Y. ZhouH. Zhang and Z. Zeng, Quasi-synchronization of delayed memristive neural networks via a hybrid impulsive control, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 51 (2021), 1954-1965.  doi: 10.1109/TSMC.2019.2911366.

[148]

C. Zhu, X. Li and J. Cao, Finite-time ${H}_\infty$ dynamic output feedback control for nonlinear impulsive switched systems, Nonlinear Anal. Hybrid Syst., 39 (2021), 100975, 13 pp. doi: 10.1016/j.nahs.2020.100975.

[149]

W. ZhuD. WangL. Liu and G. Feng, Event-based impulsive control of continuous-time dynamic systems and its application to synchronization of memristive neural networks, IEEE Trans. Neural Netw. Learn. Syst., 29 (2018), 3599-3609.  doi: 10.1109/TNNLS.2017.2731865.

Figure 1.  Outline of this paper
Figure 2.  Trajectories of system (5) with $ a = 1 $ and $ b = 0.5 $
Figure 3.  Trajectories of system (5) with $ a = -1 $ and $ b = 1.5 $
Table 1.  Summary of stability results of impulsive systems with DIs
Type of DI Stability Constrains Main results
Imp control Imp disturbance Hybrid
TDDI UAS Condi (6) Th 3.2 Th 3.3 n/a
Uniform ES Condi (8) Cor 1 Cor 1 Th 3.4
Uniform ISS Condi (11) Th 3.6 Cor 2 Th 3.5
FTS/FTCS Condi (13) Th 3.9/3.10 Th 3.7/3.8 Th 3.11
SDDI US/UAS Condi (6) Th 3.12 Th 3.13 n/a
Local ES Condi (14) Th 3.14 Th 3.14 n/a
FTS/FTCS Condi (15) Th 3.16 Th 3.15 n/a
Type of DI Stability Constrains Main results
Imp control Imp disturbance Hybrid
TDDI UAS Condi (6) Th 3.2 Th 3.3 n/a
Uniform ES Condi (8) Cor 1 Cor 1 Th 3.4
Uniform ISS Condi (11) Th 3.6 Cor 2 Th 3.5
FTS/FTCS Condi (13) Th 3.9/3.10 Th 3.7/3.8 Th 3.11
SDDI US/UAS Condi (6) Th 3.12 Th 3.13 n/a
Local ES Condi (14) Th 3.14 Th 3.14 n/a
FTS/FTCS Condi (15) Th 3.16 Th 3.15 n/a
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