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SCIENCE CHINA Information Sciences, Volume 62, Issue 9: 192206(2019) https://doi.org/10.1007/s11432-018-9809-y

Halanay-type inequality with delayed impulses and its applications

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  • ReceivedDec 25, 2018
  • AcceptedJan 31, 2019
  • PublishedAug 1, 2019

Abstract

In this study, some properties of a novel Halanay-type inequality that simultaneously contains impulses and delayed impulses are investigated. Two concepts with respect to average impulsive gain are proposed to describe hybrid impulsive strength and hybrid delayed impulsive strength. Then, using the obtained results, two stability criteria are derived for the linear systems with impulses and delayed impulses. It is found that the stability of impulsive systems is robust with respect to delayed impulses of which the magnitude strength is relatively small. Whereas, if the impulse strength is small, the time-delayed impulses can also promote the stability of unstable systems. Two numerical examples are employed to illustrate the efficiency of our results.


References

[1] Matsumoto T, Kobayashi H, Togawa Y. Spatial versus temporal stability issues in image processing neuro chips.. IEEE Trans Neural Netw, 1992, 3: 540-569 CrossRef PubMed Google Scholar

[2] Bonnans J F, Shapiro A. Nondegeneracy and Quantitative Stability of Parameterized Optimization Problems with Multiple Solutions. SIAM J Optim, 1998, 8: 940-946 CrossRef Google Scholar

[3] Li Y, Li B, Liu Y. Set Stability and Stabilization of Switched Boolean Networks With State-Based Switching. IEEE Access, 2018, 6: 35624-35630 CrossRef Google Scholar

[4] Knochel R, Schunemann K. Noise and Transfer Properties of Harmonically Synchronized Oscillators. IEEE Trans Microwave Theor Techn, 1978, 26: 939-944 CrossRef ADS Google Scholar

[5] Stack J H, Whitney M, Rodems S M. A ubiquitin-based tagging system for controlled modulation of protein stability.. Nat Biotechnol, 2000, 18: 1298-1302 CrossRef PubMed Google Scholar

[6] Labovitz C, Malan G R, Jahanian F. Origins of internet routing instability. In: Proceedings of the 18th Annual Joint Conference of the IEEE Computer and Communications Societies, 1999. 218--226. Google Scholar

[7] Tang Y, Gao H, Zhang W. Leader-following consensus of a class of stochastic delayed multi-agent systems with partial mixed impulses. Automatica, 2015, 53: 346-354 CrossRef Google Scholar

[8] Li Y, Lou J, Wang Z. Synchronization of dynamical networks with nonlinearly coupling function under hybrid pinning impulsive controllers. J Franklin Institute, 2018, 355: 6520-6530 CrossRef Google Scholar

[9] Yang X, Lu J, Ho D W C. Synchronization of uncertain hybrid switching and impulsive complex networks. Appl Math Model, 2018, 59: 379-392 CrossRef Google Scholar

[10] Mayne D Q, Rawlings J B, Rao C V. Constrained model predictive control: Stability and optimality. Automatica, 2000, 36: 789-814 CrossRef Google Scholar

[11] Cao J, Li R. Fixed-time synchronization of delayed memristor-based recurrent neural networks. Sci China Inf Sci, 2017, 60: 032201 CrossRef Google Scholar

[12] Li Y. Impulsive Synchronization of Stochastic Neural Networks via Controlling Partial States. Neural Process Lett, 2017, 46: 59-69 CrossRef Google Scholar

[13] Yang X, Song Q, Cao J. Synchronization of Coupled Markovian Reaction-Diffusion Neural Networks With Proportional Delays Via Quantized Control.. IEEE Trans Neural Netw Learning Syst, 2019, 30: 951-958 CrossRef PubMed Google Scholar

[14] Halanay A. Differential Equations: Stability, Oscillations, Time Lags. Pittsburgh: Academic Press, 1966. Google Scholar

[15] Huang Z, Cao J, Raffoul Y N. Hilger-type impulsive differential inequality and its application to impulsive synchronization of delayed complex networks on time scales. Sci China Inf Sci, 2018, 61: 78201 CrossRef Google Scholar

[16] Brezis H, Lieb E H. Sobolev inequalities with remainder terms. J Funct Anal, 1985, 62: 73-86 CrossRef Google Scholar

[17] Frank R L, Lieb E H, Seiringer R. Hardy-Lieb-Thirring inequalities for fractional Schroedinger operators. J Amer Math Soc, 2008, 21: 925-950 CrossRef ADS Google Scholar

[18] Li X, Cao J. An Impulsive Delay Inequality Involving Unbounded Time-Varying Delay and Applications. IEEE Trans Automat Contr, 2017, 62: 3618-3625 CrossRef Google Scholar

[19] Zhang W, Tang Y, Wong W K. Stochastic stability of delayed neural networks with local impulsive effects.. IEEE Trans Neural Netw Learning Syst, 2015, 26: 2336-2345 CrossRef PubMed Google Scholar

[20] Peng S, Deng F. New Criteria on $p$th Moment Input-to-State Stability of Impulsive Stochastic Delayed Differential Systems. IEEE Trans Automat Contr, 2017, 62: 3573-3579 CrossRef Google Scholar

[21] Yang R, Wu B, Liu Y. A Halanay-type inequality approach to the stability analysis of discrete-time neural networks with delays. Appl Math Computation, 2015, 265: 696-707 CrossRef Google Scholar

[22] Hien L V, Phat V N, Trinh H. New generalized Halanay inequalities with applications to stability of nonlinear non-autonomous time-delay systems. NOnlinear Dyn, 2015, 82: 563-575 CrossRef Google Scholar

[23] Song Y, Shen Y, Yin Q. New discrete Halanay-type inequalities and applications. Appl Math Lett, 2013, 26: 258-263 CrossRef Google Scholar

[24] He B B, Zhou H C, Chen Y Q. Asymptotical stability of fractional order systems with time delay via an integral inequality. IET Control Theor Appl, 2018, 12: 1748-1754 CrossRef Google Scholar

[25] Li X, Wu J. Sufficient Stability Conditions of Nonlinear Differential Systems Under Impulsive Control With State-Dependent Delay. IEEE Trans Automat Contr, 2018, 63: 306-311 CrossRef Google Scholar

[26] Liu X. Stability of impulsive control systems with time delay. Math Comput Model, 2004, 39: 511-519 CrossRef Google Scholar

[27] Li H, Li C, Huang T. Fixed-time stabilization of impulsive Cohen-Grossberg BAM neural networks.. Neural Networks, 2018, 98: 203-211 CrossRef PubMed Google Scholar

[28] Aouiti C, Assali E A, Cao J D. Stability analysis for a class of impulsive competitive neural networks with leakage time-varying delays. Sci China Technol Sci, 2018, 61: 1384-1403 CrossRef Google Scholar

[29] Li X, Ho D W C, Cao J. Finite-time stability and settling-time estimation of nonlinear impulsive systems. Automatica, 2019, 99: 361-368 CrossRef Google Scholar

[30] Yang Z, Xu D. Stability Analysis and Design of Impulsive Control Systems With Time Delay. IEEE Trans Automat Contr, 2007, 52: 1448-1454 CrossRef Google Scholar

[31] Wu Q, Zhou J, Xiang L. Impulses-induced exponential stability in recurrent delayed neural networks. Neurocomputing, 2011, 74: 3204-3211 CrossRef Google Scholar

[32] Yang X, Yang Z. Synchronization of TS fuzzy complex dynamical networks with time-varying impulsive delays and stochastic effects. Fuzzy Sets Syst, 2014, 235: 25-43 CrossRef Google Scholar

[33] Zhang L, Yang X, Xu C. Exponential synchronization of complex-valued complex networks with time-varying delays and stochastic perturbations via time-delayed impulsive control. Appl Math Computation, 2017, 306: 22-30 CrossRef Google Scholar

[34] Li X, Song S. Stabilization of Delay Systems: Delay-Dependent Impulsive Control. IEEE Trans Automat Contr, 2017, 62: 406-411 CrossRef Google Scholar

[35] Yang H, Wang X, Zhong S. Synchronization of nonlinear complex dynamical systems via delayed impulsive distributed control. Appl Math Computation, 2018, 320: 75-85 CrossRef Google Scholar

[36] Lu J, Ho D W C, Cao J. A unified synchronization criterion for impulsive dynamical networks. Automatica, 2010, 46: 1215-1221 CrossRef Google Scholar

[37] Wang N, Li X, Lu J. Unified synchronization criteria in an array of coupled neural networks with hybrid impulses.. Neural Networks, 2018, 101: 25-32 CrossRef PubMed Google Scholar

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