# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
${L}_{r}$-synchronization and adaptive synchronization of a class of chaotic Lurie systems under perturbations. (${L}_{r}$-synchronization and adaptive synchronization of a class of chaotic Lur’e systems under perturbations.) (English) Zbl 1239.93050
Summary: The synchronous control of a class of disturbed chaotic Lur’e systems is probed in. The conception of ${L}_{r}$-synchronization of drive-respond systems is presented. Via Lyapunov function analysis and comparison principle, a ${L}_{r}$ synchronous controller of the drive-respond systems under perturbation is given and its robustness is also discussed. Barbalat’s lemma is further used to derive the adaptively synchronous controller for the unknown disturbance situation and the globally asymptotic synchronization is realized. All designed controllers are verified by the simulations and the given controllers are linear, which are convenient and can produce rapid convergence speed of the error systems.
##### MSC:
 93C15 Control systems governed by ODE 34H10 Chaos control (ODE) 93C73 Perturbations in control systems
##### References:
 [1] Li, X. D.; Ding, C. M.; Zhu, Q. X.: Synchronization of stochastic perturbed chaotic neural networks with mixed delays, Journal of the franklin institute 347, 1266-1280 (2010) · Zbl 1202.93057 · doi:10.1016/j.jfranklin.2010.06.001 [2] Grassi, G.: Propagation of projective synchronization in a series connection of chaotic systems, Journal of the franklin institute 347, 438-451 (2010) · Zbl 1185.93075 · doi:10.1016/j.jfranklin.2009.10.004 [3] Pecora, L. M.; Carroll, T. L.: Synchronization in chaotic systems, Physics review letters 64, 821-824 (1990) [4] Lazzouni, S. A.; Bowong, S.; Kakmeni, F. M. Moukam: An adaptive feedback control for chaos synchronization of nonlinear systems with different order, Communications in nonlinear science and numerical simulation 12, 568-583 (2007) [5] Fallahi, K.; Raoufi, R.; Khoshbin., H.: An application of Chen system for secure chaotic communication based on extended Kalman filter and multi-shift cipher algorithm, Communications in nonlinear science and numerical simulation 13, 763-781 (2008) · Zbl 1221.94046 · doi:10.1016/j.cnsns.2006.07.006 [6] Vincent, U. E.; Ucar, A.; Laoye, J. A.; Kareem, S. O.: Control and synchronization of chaos in RCL-shunted Josephson junction using backstepping design, Physica C 468, 374-382 (2008) [7] Yoshimura, K.; Muramatsu, J.; Davis, P.: Conditions for common-noise-induced synchronization in time-delay systems, Physica D 237, 3146-3152 (2008) · Zbl 1153.37360 · doi:10.1016/j.physd.2008.05.013 [8] Wu, X. G.; Li, J.; Chen., G. R.: Chaos in the fractional order unified system and its synchronization, Journal of the franklin institute 345, 392-401 (2008) · Zbl 1166.34030 · doi:10.1016/j.jfranklin.2007.11.003 [9] Oksasoglu, A.; Wang, Q. D.: Rank one chaos in a switch-controlled Chua’s circuit, Journal of the franklin institute 347, 1598-1622 (2010) · Zbl 1202.94239 · doi:10.1016/j.jfranklin.2010.06.006 [10] He, H. L.; Wang, Z. S.: Absolute stability of state feedback time-delay system, Lecture notes in control and information sciences 344, 469-473 (2006) · Zbl 1115.93079 [11] Wu, G. J.; He, H. L.: Absolute stability of perturbed lurie control systems by state feedback, Dynamics of continuous, discrete and impulsive systems, series B–applications and algorithms 6, 169-171 (2006) [12] Hao, F.: Absolute stability of uncertain discrete Lur’e systems and maximum admissible perturbed bounds, Journal of the franklin institute 347, 1511-1525 (2010) · Zbl 1202.93102 · doi:10.1016/j.jfranklin.2010.07.003 [13] Lee, S. M.; Park, Ju H.: Delay-dependent criteria for absolute stability of uncertain time-delayed Lur’e dynamical systems, Journal of the franklin institute 347, 146-153 (2010) [14] Park, Ju H.; Ji, D. H.; Won, S. C.; Lee, S. M.; Choi., S. J.: H$\infty$ control of Lur’e systems with sector and slope restricted nonlinearities, Physics letters A 373, 3734-3740 (2009) · Zbl 1233.93048 · doi:10.1016/j.physleta.2009.08.018 [15] Yin, C.; Zhong, S. M.; Chen, W. F.: On delay-dependent robust stability of a class of uncertain mixed neutral and Lur’e dynamical systems with interval time-varying delays, Journal of the franklin institute 347, 1623-1642 (2010) [16] Liao, X. X.; Luo, H. G.; Zhao, X. Q.: Absolute stability of lurie control systems with circulative feedback and its applications in chaotic synchronization, Progress in natural science 16, 543-554 (2006) [17] Xu, H. L.; Ding, X. J.; Chen, H. P.: Impulsive synchronization of uncertain chaotic lurie system, Journal of huazhong university of science and technology (Nature science edition) 36, No. 1, 115-117 (2008) [18] Sun, J. T.; Wu, Q. D.: Impulsive control for the stabilization and synchronization of Lur’e systems, Applied mathematics and mechanics 25, No. 3, 291-296 (2004) · Zbl 1151.93365 · doi:10.1007/BF02437335 [19] Dzhunusov, I. A.; Fradkov, A. L.: Adaptive synchronization of a network of interconnected nonlinear Lur’e systems, Automation and remote control 70, 1190-1205 (2009) · Zbl 1181.93048 · doi:10.1134/S0005117909070108 [20] Lee, S. M.; Choi, S. J.; Ji, D. H.; Park, Ju H.; Won, S. C.: Synchronization for chaotic Lur’e systems with sector restricted nonlinearities via delayed feedback control, Nonlinear dynamics 59, 277-288 (2010) · Zbl 1183.70073 · doi:10.1007/s11071-009-9537-5 [21] Xia, Y. H.; Yang, Z. J.; Han, M. A.: Lag synchronization of unknown chaotic delayed Yang–Yang-type fuzzy neural networks with noise perturbation based on adaptive control and parameter identification, IEEE transactions on neural networks 20, 1165-1180 (2009) [22] Koofigar, H. R.; Hosseinnia, S.; Sheikholeslam, F.: Robust adaptive synchronization of uncertain unified chaotic systems, Nonlinear dynamics 59, 477-483 (2010) · Zbl 1183.70072 · doi:10.1007/s11071-009-9554-4 [23] Buscarino, A.; Fortuna, L.; Frasca, M.: Experimental robust synchronization of hyperchaotic circuits, Physica D 238, 1917-1922 (2009) · Zbl 1179.37047 · doi:10.1016/j.physd.2009.06.021 [24] Ge, Z. M.; Yang, C. H.: Pragmatical generalized synchronization of chaotic systems with uncertain parameters by adaptive control, Physica D 231, 87-94 (2007) · Zbl 1167.34357 · doi:10.1016/j.physd.2007.03.019 [25] Ji, D. H.; Park, Ju H.; Yoo, W. J.; Won, S. C.; Lee, S. M.: Synchronization criterion for Lur’e type complex dynamical networks with time-varying delay, Physics letters A 374, 1218-1227 (2010) [26] He, H. L.; Tu, J. J.; Xiong, P.: Chaos synchronization between lurie systems based on estimation of Lipschitz constant, Systems engineering and electronics 33, 600-602 (2011) [27] Cheng, C. K.; Kuo, H. H.; Hou, Y. Y.; Hwang, C. C.; Liao, T. L.: Robust chaos synchronization of noise-perturbed chaotic systems with multiple time-delays, Physica A 387, 3093-3102 (2008) [28] He, H. L.; Tu, J. J.; Xiong, P.: Global asymptotical synchronization of a class of chaotic lurie systems, Journal of huazhong university of science and technology (Natural science edition) 38, No. 2, 38-40 (2010) · Zbl 1224.93094