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Pure Mathematics 2024
网络上时滞相关脉冲非线性时滞耦合系统的积分输入到状态稳定性
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Abstract:
本文研究了网络上时滞相关脉冲的一般非线性时滞耦合系统的积分输入到状态稳定(iISS)性质。利用图论方法和Lyapunov-Krasovskii方法，在单个顶点系统iISS的Lyapunov函数的基础上，构造了整个网络iISS的Lyapunov函数，并推导出了网络上时滞相关脉冲的一般非线性时滞耦合系统存在iISS的充分条件。这些条件表明，如果每个节点上的连续时间系统都是iISS时，网络上时滞相关脉冲非线性时滞耦合系统在不稳定的脉冲出现的频率不太高的情况下仍能保证iISS的性质。
This paper investigates the integral-input-to-state stability (iISS) of general nonlinear delayed impulsive coupled systems on networks with delay-dependent impulses. With the assistance of graph theory and the Lyapunov-Krasovskii method, an iISS Lyapunov function for the total network is constructed based on the iISS Lyapunov functions of individual vertex systems, and sufficient conditions for iISS for general nonlinear delayed impulsive coupled systems on networks are derived. It is demonstrated that, when every continuous vertex system is iISS, the nonlinear delayed impulsive coupled systems on networks can still maintain iISS property provided the destabilizing impulses do not occur too frequently.
| [1] | Strogatz, S.H. (2001) Exploring Complex Networks. Nature, 410, 268-276. https://doi.org/10.1038/35065725 |
| [2] | Grillner, S. (2006) Biological Pattern Generation: The Cellular and Computational Logic of Networks in Motion. Neuron, 52, 751-766. https://doi.org/10.1016/j.neuron.2006.11.008 |
| [3] | Chandrasekar, A., Rakkiyappan, R. and Cao, J. (2015) Impulsive Synchronization of Markovian Jumping Randomly Coupled Neural Networks with Partly Unknown Transition Probabilities via Multiple Integral Approach. Neural Networks, 70, 27-38. https://doi.org/10.1016/j.neunet.2015.07.002 |
| [4] | Li, W., Pang, L., Su, H., et al. (2012) Global Stability for Discrete Cohen-Grossberg Neural Networks with Finite and Infinite Delays. Applied Mathematics Letters, 25, 2246-2251. https://doi.org/10.1016/j.aml.2012.06.011 |
| [5] | Wayman, J.A. and Varner, J.D. (2013) Biological Systems Modeling of Metabolic and Signaling Networks. Current Opinion in Chemical Engineering, 2, 365-372. https://doi.org/10.1016/j.coche.2013.09.001 |
| [6] | Sun, R. (2010) Global Stability of the Endemic Equilibrium of Multigroup SIR Models with Nonlinear Incidence. Computers & Mathematics with Applications, 60, 2286-2291. https://doi.org/10.1016/j.camwa.2010.08.020 |
| [7] | Kuniya, T. (2011) Global Stability Analysis with a Discretization Approach for an Age-Structured Multigroup SIR Epidemic Model. Nonlinear Analysis: Real World Applications, 12, 2640-2655. https://doi.org/10.1016/j.nonrwa.2011.03.011 |
| [8] | Sun, R. and Shi, J. (2011) Global Stability of Multigroup Epidemic Model with Group Mixing and Nonlinear Incidence Rates. Applied Mathematics and Computation, 218, 280-286. https://doi.org/10.1016/j.amc.2011.05.056 |
| [9] | Li, W., Su, H., Wei, D., et al. (2012) Global Stability of Coupled Nonlinear Systems with Markovian Switching. Communications in Nonlinear Science and Numerical Simulation, 17, 2609-2616. https://doi.org/10.1016/j.cnsns.2011.09.039 |
| [10] | Zhang, X., Li, W. and Wang, K. (2015) The Existence of Periodic Solutions for Coupled Systems on Networks with Time Delays. Neurocomputing, 152, 287-293. https://doi.org/10.1016/j.neucom.2014.10.067 |
| [11] | Li, C., Sun, J. and Sun, R. (2010) Stability Analysis of a Class of Stochastic Differential Delay Equations with Nonlinear Impulsive Effects. Journal of the Franklin Institute, 347, 1186-1198. https://doi.org/10.1016/j.jfranklin.2010.04.017 |
| [12] | Wu, Q., Zhou, J. and Xiang, L. (2011) Impulses-Induced Exponential Stability in Recurrent Delayed Neural Networks. Neurocomputing, 74, 3204-3211. https://doi.org/10.1016/j.neucom.2011.05.001 |
| [13] | Sontag, E.D. and Wang, Y. (1995) On Characterizations of the Input-to-State Stability Property. Systems & Control Letters, 24, 351-359. https://doi.org/10.1016/0167-6911(94)00050-6 |
| [14] | Grune, L. (2002) Input-to-State Dynamical Stability and Its Lyapunov Function Characterization. IEEE Transactions on Automatic Control, 47, 1499-1504. https://doi.org/10.1109/TAC.2002.802761 |
| [15] | Angeli, D. (2002) A Lyapunov Approach to Incremental Stability Properties. IEEE Transactions on Automatic Control, 47, 410-421. https://doi.org/10.1109/9.989067 |
| [16] | Arcak, M. and Teel, A. (2002) Input-to-State Stability for a Class of Lurie Systems. Automatica, 38, 1945-1949. https://doi.org/10.1016/S0005-1098(02)00100-0 |
| [17] | Li, M.Y. and Shuai, Z. (2010) Global-Stability Problem for Coupled Systems of Differential Equations on Networks. Journal of Differential Equations, 248, 1-20. https://doi.org/10.1016/j.jde.2009.09.003 |
| [18] | Guo, H., Li, M.Y. and Shuai, Z. (2008) A Graph-Theoretic Approach to the Method of Global Lyapunov Functions. Proceedings of the American Mathematical Society, 136, 2793-2802. https://doi.org/10.1090/S0002-9939-08-09341-6 |
| [19] | Chen, H. and Sun, J. (2012) Stability Analysis for Coupled Systems with Time Delay on Networks. Physica A: Statistical Mechanics and its Applications, 391, 528-534. https://doi.org/10.1016/j.physa.2011.08.037 |
| [20] | Li, W., Su, H. and Wang, K. (2011) Global Stability Analysis for Stochastic Coupled Systems on Networks. Automatica, 47, 215-220. https://doi.org/10.1016/j.automatica.2010.10.041 |
| [21] | Su, H., Li, W. and Wang, K. (2012) Global Stability Analysis of Discrete-Time Coupled Systems on Networks and Its Applications. Chaos: An Interdisciplinary Journal of Nonlinear Science, 22, Article 033135. https://doi.org/10.1063/1.4748851 |
| [22] | Suo, J., Sun, J. and Zhang, Y. (2013) Stability Analysis for Impulsive Coupled Systems on Networks. Neurocomputing, 99, 172-177. https://doi.org/10.1016/j.neucom.2012.06.002 |
| [23] | Cao, J., Li, P. and Wang, W. (2006) Global Synchronization in Arrays of Delayed Neural Networks with Constant and Delayed Coupling. Physics Letters A, 353, 318-325. https://doi.org/10.1016/j.physleta.2005.12.092 |
| [24] | Amato, F., Ambrosino, R., Ariola, M., et al. (2009) Finite-Time Stability of Linear Time-Varying Systems with Jumps. Automatica, 45, 1354-1358. https://doi.org/10.1016/j.automatica.2008.12.016 |
| [25] | Sontag, E.D. (1989) Smooth Stabilization Implies Coprime Factorization. IEEE Transactions on Automatic Control, 34, 435-443. https://doi.org/10.1109/9.28018 |
| [26] | Sontag, E.D. (1998) Comments on Integral Variants of ISS. Systems & Control Letters, 34, 93-100. https://doi.org/10.1016/S0167-6911(98)00003-6 |
| [27] | Lazar, M., Heemels, W.P.M.H. and Teel, A.R. (2009) Lyapunov Functions, Stability and Input-to-State Stability Subtleties for Discrete-Time Discontinuous Systems. IEEE Transactions on Automatic Control, 54, 2421-2425. https://doi.org/10.1109/TAC.2009.2029297 |
| [28] | Liu, J., Liu, X. and Xie, W.C. (2011) Input-to-State Stability of Impulsive and Switching Hybrid Systems with Time-Delay. Automatica, 47, 899-908. https://doi.org/10.1016/j.automatica.2011.01.061 |
| [29] | Vu, L., Chatterjee, D. and Liberzon, D. (2007) Input-to-State Stability of Switched Systems and Switching Adaptive Control. Automatica, 43, 639-646. https://doi.org/10.1016/j.automatica.2006.10.007 |
| [30] | Ito, H. and Kellett, C.M. (2018) A Small-Gain Theorem in the Absence of Strong iISS. IEEE Transactions on Automatic Control, 64, 3897-3904. https://doi.org/10.1109/TAC.2018.2886955 |
| [31] | Angeli, D., Sontag, E.D. and Wang, Y. (2000) A Characterization of Integral Input-to-State Stability. IEEE Transactions on Automatic Control, 45, 1082-1097. https://doi.org/10.1109/9.863594 |
| [32] | Liberzon, D. (1999) ISS and Integral-ISS Disturbance Attenuation with Bounded Controls. Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, 7-10 December 1999, 2501-2506. https://doi.org/10.1109/CDC.1999.831303 |
| [33] | Hespanha, J.P., Liberzon, D. and Teel, A.R. (2008) Lyapunov Conditions for Input-to-State Stability of Impulsive Systems. Automatica, 44, 2735-2744. https://doi.org/10.1016/j.automatica.2008.03.021 |
| [34] | Chen, W.H. and Zheng, W.X. (2009) Input-to-State Stability and Integral Input-to-State Stability of Nonlinear Impulsive Systems with Delays. Automatica, 45, 1481-1488. https://doi.org/10.1016/j.automatica.2009.02.005 |
| [35] | Sun, X.M. and Wang, W. (2012) Integral Input-to-State Stability for Hybrid Delayed Systems with Unstable Continuous Dynamics. Automatica, 48, 2359-2364. https://doi.org/10.1016/j.automatica.2012.06.056 |
| [36] | Liu, X. and Zhang, K. (2019) Input-to-State Stability of Time-Delay Systems with Delay-Dependent Impulses. IEEE Transactions on Automatic Control, 65, 1676-1682. https://doi.org/10.1109/TAC.2019.2930239 |
| [37] | Zhang, K. (2020) Integral Input-to-State Stability of Nonlinear Time-Delay Systems with Delay-Dependent Impulse Effects. Nonlinear Analysis: Hybrid Systems, 37, Article 100907. https://doi.org/10.1016/j.nahs.2020.100907 |
