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Exponential stability of nonlinear time-delay systems with delayed impulse effects. (English) Zbl 1233.93080
Summary: The problem of exponential stability for nonlinear time-delay systems with delayed impulses is addressed in this paper. Lyapunov-based sufficient conditions for exponential stability are derived, respectively, for two kinds of delayed impulses (i.e., destabilizing delayed impulses and stabilizing delayed impulses). It is shown that if a nonlinear impulsive time-delay system without impulse input delays is exponentially stable, then under some conditions, its stability is robust with respect to small impulse input delays. Moreover, it is also shown that for a stable nonlinear impulsive time-delay system, if the magnitude of the delayed impulses is sufficiently small, then under some conditions, the delayed impulses do not destroy the stability irrespective of the sizes of the impulse input delays. The efficiency of the proposed results is illustrated by three numerical examples.
##### MSC:
 93D20 Asymptotic stability of control systems 93C10 Nonlinear control systems 93D09 Robust stability of control systems 93C15 Control systems governed by ODE
##### Keywords:
impulsive systems; delayed impulses; exponential stability
##### References:
 [1] Anokhin, A.; Berezansky, L.; Braverman, E.: Exponential stability of linear delay impulsive differential equations, Journal of mathematical analysis and applications 193, No. 3, 923-941 (1995) · Zbl 0837.34076 · doi:10.1006/jmaa.1995.1275 [2] Bainov, D. D.; Simeonov, P. S.: Systems with impulse effect: stability, theory and applications, (1989) · Zbl 0683.34032 [3] Ballinger, G.; Liu, X.: Existence and uniqueness results for impulsive delay differential equations, Dynamics of continuous, discrete and impulsive systems 5, No. 1–4, 579-591 (1999) · Zbl 0955.34068 [4] Bernstein, D. S.: Matrix mathematics: theory, facts, and formulas with application to linear systems theory, (2005) [5] Chen, W. -H.; Wang, J. -G.; Tang, Y. -J.; Lu, X.: Robust H$\infty$ control of uncertain linear impulsive stochastic systems, International journal of robust and nonlinear control 18, No. 13, 1348-1371 (2008) [6] Chen, W. -H.; Zheng, W. X.: Robust stability and H$\infty$-control of uncertain impulsive systems with time-delay, Automatica 45, No. 1, 109-117 (2009) · Zbl 1154.93406 · doi:10.1016/j.automatica.2008.05.020 [7] Chen, W. -H.; Zheng, W. X.: Input-to-state stability and integral input-to-state stability of nonlinear impulsive systems with delays, Automatica 45, No. 6, 1481-1488 (2009) · Zbl 1166.93370 · doi:10.1016/j.automatica.2009.02.005 [8] Chen, W. -H.; Zheng, W. X.: Global exponential stability of impulsive neural networks with variable delay: an LMI approach, IEEE transactions on circuits and systems I. Regular papers 56, No. 6, 1248-1259 (2009) [9] Guan, Z. -H.: Decentralized stabilization for impulsive large scale systems with delays, Dynamics of continuous, discrete and impulsive systems 6, No. 3, 367-379 (1999) · Zbl 0934.93009 [10] Hespanha, J. P.; Liberzon, D.; Teel, A. R.: Lyapunov conditions for input-to-state stability of impulsive systems, Automatica 44, No. 11, 2735-2744 (2008) · Zbl 1152.93050 · doi:10.1016/j.automatica.2008.03.021 [11] Ho, D. W. C.; Liang, J.; Lam, J.: Global exponential stability of impulsive high-order BAM neural networks with time-varying delays, Neural networks 19, No. 10, 1581-1590 (2006) · Zbl 1178.68417 · doi:10.1016/j.neunet.2006.02.006 [12] Khadra, A.; Liu, X.; Shen, X. S.: Analyzing the robustness of impulsive synchronization coupled by linear delayed impulses, IEEE transactions on automatic control 4, No. 4, 923-928 (2009) [13] Li, Z. G.; Soh, C. B.; Xu, X. H.: Stability of impulsive differential systems, Journal of mathematical analysis and applications 216, No. 2, 644-653 (1997) · Zbl 0899.34010 · doi:10.1006/jmaa.1997.5695 [14] Li, Z. G.; Wen, C. Y.; Soh, Y. C.: Analysis and design of impulsive control systems, IEEE transactions on automatic control 46, No. 6, 894-897 (2001) · Zbl 1001.93068 · doi:10.1109/9.928590 [15] Liu, B.: Stability of solutions for stochastic impulsive systems via comparison approach, IEEE transactions on automatic control 53, No. 9, 2128-2133 (2008) [16] Liu, X.; Ballinger, G.: Uniform asymptotic stability of impulsive delay differential equations, Computers mathematics with applications 41, No. 7–8, 903-915 (2001) [17] Liu, X.; Shen, X. S.; Zhang, Y.; Wang, Q.: Stability criteria for impulsive systems with time delay and unstable system matrices, IEEE transactions on circuits and systems I. Regular papers 54, No. 10, 2288-2298 (2007) [18] Lu, H. T.: Chaotic attractors in delayed neural networks, Physics letters A 298, No. 2–3, 109-116 (2002) · Zbl 0995.92004 · doi:10.1016/S0375-9601(02)00538-8 [19] Naghshtabrizi, P.; Hespanha, J. P.; Teel, A. R.: Exponential stability of impulsive systems with application to uncertain sampled-data systems, Systems control letters 57, No. 5, 378-385 (2008) · Zbl 1140.93036 · doi:10.1016/j.sysconle.2007.10.009 [20] Van De Wouw, N.; Naghshtabrizi, P.; Cloosterman, M. B. G.; Hespanha, J. P.: Tracking control for sampled-data systems with uncertain time-varying sampling intervals and delays, International journal of robust and nonlinear control 20, No. 4, 387-411 (2010) [21] Wang, Q.; Liu, X.: Exponential stability for impulsive delay differential equations by razumikhin method, Journal of mathematical analysis and applications 309, No. 2, 462-473 (2005) · Zbl 1084.34066 · doi:10.1016/j.jmaa.2004.09.016 [22] Yang, T.: Impulsive systems and control: theory and applications, (2001) [23] Yang, Z.; Xu, D.: Stability analysis and design of impulsive control systems with time delay, IEEE transactions on automatic control 52, No. 8, 1448-1454 (2007)