Abstract
This paper further investigates the stability of the n-dimensional linear systems with multiple delays. Using Laplace transform, we introduce a definition of characteristic equation for the n-dimensional linear systems with multiple delays. Moreover, one sufficient condition is attained for the Lyapunov globally asymptotical stability of the general multi-delay linear systems. In particular, our result shows that some uncommensurate linear delays systems have the similar stability criterion as that of the commensurate linear delays systems. This result also generalizes that of Chen and Moore (2002). Finally, this theorem is applied to chaos synchronization of the multi-delay coupled Chua’s systems.
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References
Y. Q. Chen and K. L. Moore, Analytical stability bound for delayed second-order systems with repeating poles using Lambert function W, Automatica, 2002, 38(5): 891–895.
Y. Q. Chen and K. L. Moore, Analytical stability bound for a class of delayed fractional-order dynamics systems, Nonlinear Dynamics, 2002, 29(1–4): 191–200.
M. De Sousa Vieira and A. J. Lichtenberg, Controlling chaos using nonlinear feedback with delay, Phys. Rev. E, 1996, 54(2): 1200–1207.
R. He and P. G. Vaidya, Time delayed chaotic systems and their synchronization, Phys. Rev. E, 1999, 59(4): 4048–4051.
J. W. Ryu, W. H. Kye, S. Y. Lee, M. W. Kim, M. H. Choi, S. Rim, Y. J. Park, and C. M. Kim, Effects of time-delayed feedback on chaotic oscillators, Phys. Rev. E, 2004, 70(3): 036220.
W. H. Deng, Y. J. Wu, and C. P. Li, Stability analysis of differential equations with time-dependent delay, Int. J. Bifurcation and Chaos, 2006, 16(2): 465–472.
J. Lü and G. Chen, A time-varying complex dynamical network models and its controlled synchronization criteria, IEEE Trans. Auto. Contr., 2005, 50(6): 841–846.
J. Zhou, J. A. Lu, and J. Lü, Adaptive synchronization of an uncertain complex dynamical network, IEEE Trans. Auto. Contr., 2006, 51(4): 652–656.
S. -I. Niculescu and W. Michiels, Stabilizing a chain of integrators using multiple delays, IEEE Trans. Automat. Contr., 2004, 49(5): 802–807.
J. Lü, F. L. Han, X. H. Yu, and G. Chen, Generating 3-D multi-scroll chaotic attractors: A hysteresis series switching method, Automatica, 2004, 40(10): 1677–1687.
J. Lü and G. Chen, A new chaotic attractor coined, Int. J. Bifurcation and Chaos, 2002, 12(3): 659–661.
W. H. Deng and C. P. Li, Synchronization of chaotic fractional Chen system, Journal of Physical Society of Japan, 2005, 74(6): 1645–1648.
W. H. Deng and C. P. Li, Chaos synchronization of the fractional Lü system, Physica A, 2005, 353: 61–72.
C. P. Li, W. H. Deng, and D. Xu, Chaos synchronization of the Chua system with a fractional order, Physica A, 2006, 360(2): 171–185.
S. I. Niculescu, Delay Effects on Stability: A Robust Control Approach, Springer-Verlag, Heidelberg, Germany, 2001.
K. Gu, V. Kharitonov, and J. Chen, Stability of Time-Delay Systems, Birkhauser, Boston, 2003.
K. Gu and S. I. Niculescu, Survey on recent results in the stability and control of time-delay systems, J. Dynam. Syst. Measur. Contr., 2003, 125: 158–165.
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This work was supported by the National Natural Science Foundation of China under Grants 60304017, 20336040, and 60221301, the Scientific Research Startup Special Foundation on Excellent PhD Thesis and Presidential Award of Chinese Academy of Sciences, and the Tianyuan Foundation under Grant A0324651.
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Deng, W., Lü, J. & Li, C. Stability of N-Dimensional Linear Systems with Multiple Delays and Application to Synchronization. Jrl Syst Sci & Complex 19, 149–156 (2006). https://doi.org/10.1007/s11424-006-0149-6
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DOI: https://doi.org/10.1007/s11424-006-0149-6