Matrix Measure with Application in Quantized Synchronization Analysis of Complex Networks with Delayed Time via the General Intermittent Control

Author(s)
Qunli Zhang

ABSTRACT

This paper concerned with the quantized synchronization analysis problem. The scope of state vectors of dynamic systems, based on the matrix measure, is estimated. By using the general intermittent control, some simple yet generic criteria are derived ensuring the exponential stability of dynamic systems. Then, both the general intermittent networked controller and the quantized parameters can be designed, which guarantee that the nodes of the complex network are synchronized. Finally, simulation examples are given to illustrate the effectiveness and feasibility of the proposed method.

Cite this paper

Zhang, Q. (2013) Matrix Measure with Application in Quantized Synchronization Analysis of Complex Networks with Delayed Time via the General Intermittent Control.*Applied Mathematics*, **4**, 1417-1426. doi: 10.4236/am.2013.410192.

Zhang, Q. (2013) Matrix Measure with Application in Quantized Synchronization Analysis of Complex Networks with Delayed Time via the General Intermittent Control.

References

[1] H. Fujisaka and T. Yamada, “Stability Theory of Syn chronized Motion in a Coupled-Oscillator System,” Pro gress of Theoretical Physics, Vol. 69, No. 1, 1983, pp. 32-47. http://dx.doi.org/10.1143/PTP.69.32

[2] H. Fujisaka and T. Yamada, “A New Intermittency in Coupled Dynamical Systems,” Progress of Theoretical Physics, Vol. 74, No. 4, 1985, pp. 918-921. http://dx.doi.org/10.1143/PTP.74.918

[3] H. Fujisaka and T. Yamada, “Stability Theory of Syn chronized Motion in Coupled-Oscillator Systems IV,” Progress of Theoretical Physics, Vol. 75, No. 5, 1986, pp. 1087-1104. http://dx.doi.org/10.1143/

PTP.75.1087

[4] V. S. Afraimovich, N. N. Verichev and M. I. Rabinovich, “Stochastic Synchronization of Oscillations in Dissipative Systems,” Radiophysics and Quantum Electronics, Vol. 29, No. 9, 1986, pp. 747-751.

[5] L. M. Pecora and T. L. Carroll, “Synchronization in Chaotic Systems,” Physical Review Letters, Vol. 64, No. 8, 1990, pp. 821-824. http://dx.doi.org/10.1103/PhysRevLett.64.821

[6] W. H. Deng, J. H. Lu and C.-P. Li, “Stability of N-Dimensional Linear Systems with Multiple Delays and Application to Synchronization,” Journal of Systems Sci ence and Complexity, Vol. 19, No. 2, 2006, pp. 149-156.

[7] E. M. Elabbasy, H. N. Agiza and M. M. El-Dessoky, “Global Synchronization Criterion and Adaptive Syn chronization for New Chaotic System,” Chaos, Solitons and Fractals, Vol. 23, No. 4, 2005, pp. 1299-1309.

[8] Q. L. Zhang, J. Zhou and G. Zhang, “Stability Concerning Partial Variables for a Class of Time-Varying Systems and Its Applications in Chaos Synchronization,” Pro ceedings of the 24th Chinese Control Conference, South China University of Technology Press, Guangzhou, 2005, pp. 135-139.

[9] Q. L. Zhang and G. J. Jia, “Chaos Synchronization of Morse Oscillator via Backstepping Design,” Annals of Differential Equations, Vol. 22, No. 3, 2006, pp. 456-460.

[10] Q. L. Zhang, “Synchronization of Multi-Chaotic Systems via Ring Impulsive Control,” Control Theory & Applica tions, Vol. 27, No. 2, 2010, pp. 226-232.

[11] Z. Tang and J. W. Feng, “Adaptive Cluster Synchroni zation for Nondelayed and Delayed Coupling Complex Networks with Nonidentical Nodes,” Abstract and Applied Analysis, Vol. 2013, 2013, Article ID: 946243.

[12] B. Li and Q. K. Song, “Synchronization of Chaotic Delayed Fuzzy Neural Networks under Impulsive and Stochastic Perturbations,” Abstract and Applied Analysis, Vol. 2013, 2013, Article ID: 543549.

[13] M. Han, Y. Liu and J. Q. Lu, “Impulsive Control for the Synchronization of Chaotic Systems with Time Delay,” Abstract and Applied Analysis, Vol. 2013, 2013, Article ID: 647561.

[14] S. J. Dan, S. X. Yang and W. Feng, “Lag Synchronization of Coupled Delayed Chaotic Neural Networks by Per iodically Intermittent Control,” Abstract and Applied Analysis, Vol. 2013, 2013, Article ID: 501461.

[15] C.-M. Lin, M.-H. Lin and R.-G. Yeh, “Synchronization of Unified Chaotic System via Adaptive Wavelet Cerebellar Model Articulation Controller,” Neural Computing and Applications, 2012, pp. 1-9. http://dx.doi.org/10.1007/s00521-012-1021-3

[16] J. D. Cao and L. L. Li, “Cluster Synchronization in an Array of Hybrid Coupled Neural Networks with Delay,” Neural Networks, Vol. 22, No. 4, 2009, pp. 335-342.

[17] L. L. Li and J. D. Cao, “Cluster Synchronization in an Array of Coupled Stochastic Delayed Neural Networks via Pinning Control,” Neurocomputing, Vol. 74, No. 5, 2011, pp. 846-856. http://dx.doi.org/

10.1016/j.neucom.2010.11.006

[18] H. Reza and P. Mass, “Delay-Range-Dependent Expo nential H∞ Synchronization of a Class of Delayed Neural Networks,” Chaos, Solitons and Fractals, Vol. 41, No. 3, 2009, pp. 1125-1135. http://dx.doi.org/10.1016/j.chaos.2008.04.051

[19] J. D. Cao, D. W. C. Ho and Y. Q. Yang, “Projective Syn chronization of a Class of Delayed Chaotic Systems via Impulsive Control,” Physics Letters A, Vol. 373, No. 35, 2009, pp. 3128-3133. http://dx.doi.org/10.1016/j.physleta.2009.06.056

[20] Z. Zahreddine, “Matrix Measure and Application to Stability of Matrices and Interval Dynamical Systems,” International Journal of Mathematics and Mathematical Sciences, Vol. 2003, No. 2, 2003, pp. 75-85. http://dx.doi.org/10.1155/S0161171203202295

[21] J. T. Sun, Y. P. Zhang, Y. Q. Liu and F. Q. Deng, “Exponential Stability of Interval Dynamical System with Multidelay,” Applied Mathematics and Mechanics, Vol. 23, No. 1, 2002, pp. 87-91.

[22] G. D. Zong, Y. Q. Wu and S. Y. Xu, “Stability Criteria for Switched Linear Systems with Time-Delay,” Control Theory & Applications, Vol. 25, No. 5, 2008, pp. 295-305.

[23] Y. H. Yuan, Q. L. Zhang and B. Chen, “Robust Fuzzy Control Based on Matrix Measure for Nonlinear Descriptor Systems with Time-Delay,” Control and Decision, Vol. 22, No. 2, 2007, pp. 174-178.

[24] L. M. Ding, J. B. Quan, Y. Zhang and Z. X. Li, “The Fixed Point Theorem and Asymptotic Stability of a Delay-Differential System,” Journal of Air Force Radar Academy, Vol. 23, No. 3, 2009, pp. 203-204.

[25] Y. X. Guo, “Matrix Measure and Uniform Ultimate Boundedness with Respect to Partial Variables for FDEs,” Journal of Wuhan University of Science and Engineering, Vol. 21, No. 4, 2008, pp. 15-19.

[26] J. J. Huang, C. D. Li and Q. Han, “Stabilization of Delayed Chaotic Neural Networks by Periodically Intermittent Control,” Circuits, Systems & Signal Processing, Vol. 28, No. 4, 2009, pp. 567-579. http://dx.doi.org/10.1007/s00034-009-9098-3

[27] J. Yu, H. J. Jiang and Z. D. Teng, “Synchronization of Nonlinear Systems with Delay via Periodically Intermit tent Control,” Journal of Xinjiang University (Natural Science Edition), Vol. 28, No. 3, 2010, pp. 310-315.

[28] Z. Y. Dong, Y. J. Wang, M. H. Bai and Z. Q. Zuo, “Ex ponential Synchronization of Uncertain Master-Slave Lur’e Systems via Intermittent Control,” Journal of Dy namics and Control, Vol. 7, No. 4, 2009, pp. 328-333.

[29] W. G. Xia and J. D. Cao, “Pinning Synchronization of Delayed Dynamical Networks via Periodically Intermit tent Control,” Chaos, Vol. 19, No. 1, 2009, Article ID: 013120. http://dx.doi.org/10.1063/

1.3071933

[30] P. Smolen, D. Baxter and J. Byrne, “Mathematical Mod eling of Gene Networks Review,” Neuron, Vol. 26, No. 3, 2000, pp. 567-580. http://dx.doi.org/10.1016/S0896-6273(00)81194-0

[31] Y. Wang, J. Hao and Z. Zuo, “A New Method for Exponential Synchronization of Chaotic Delayed Systems via Intermittent Control,” Physics Letters A, Vol. 374, No. 19-20, 2010, pp. 2024-2029. http://dx.doi.org/10.1016/j.physleta.2010.02.069

[32] T. Huang, C. Li, W. Yu and G. Chen, “Synchronization of Delayed Chaotic Systems with Parameter Mismatches by Using Intermittent Linear State Feedback,” Non linearity, Vol. 22, No. 3, 2009, pp. 569-584. http://dx.doi.org/10.1088/0951-7715/22/3/004

[33] H. Zhang, H. C. Yan, F. W. Yang and Q. J. Chen, “Quantized Control Design for Impulsive Fuzzy Net worked Systems,” IEEE Transactions on Fuzzy Systems, Vol. 19, No. 6, 2011, pp. 1153-1162. http://dx.doi.org/10.1109/TFUZZ.2011.2162525

[34] F. Fagnani and S. Zampieri, “Quantized Stabilization of Linear Systems: Complexity versus Performance,” IEEE Transactions on Automatic Control, Vol. 49, No. 9, 2004, pp. 1534-1548. http://

dx.doi.org/10.1109/TAC.2004.834111

[35] S. Sqartini, B. Schuller and A. Hussain, “Cognitive and Emotionan Infromation Processing for Human-Machine Interaction,” Cognitive Computation, Vol. 4, No. 4, 2012, pp. 383-387. http://dx.doi.org/

10.1007/s12559-012-9180-1

[36] J. B. Kim, J. S. Park and Y. H. Oh, “Speaker-Characterized Emotion Recognition Using Online and Iterative Speaker Adaptation,” Cognitive Computation, Vol. 4, No. 4, 2012, pp. 398-408. http://dx.doi.org/10.1007/s12559-012-9132-9

[37] L. I. Per lovsky and D. S. Levine, “The Drive for Creativity and the Escape from Creativity: Neurocognitive Mechanisms,” Cognitive Computation, Vol. 4, No. 3, 2012, pp. 292-305.

[38] Q. L. Zhang, “The Generalized Dahlquist Constant with Applications in Synchronization Analysis of Typical Neural Networks via General Intermittent Control,” Ad vances in Artificial Neural Systems, Vol. 2011, 2011, Article ID: 249136. http://dx.doi.org/10.1155/2011/249136

[39] G. T. Hui, B. N. Huang, Y. C. Wang and X. P. Meng, “Quantized Control Design for Coupled Dynamic Net works with Communication Constraints,” Cognitive Com putation, Vol. 5, No. 2, 2013, pp. 200-206.

[40] L. A. Montestruque and P. J. Antsaklis, “Static and Dynamic Quantization in Model-Based Networked Control Systems,” International Journal of Control, Vol. 80, No. 1, 2007, pp. 87-101. http://dx.doi.org/

10.1080/00207170600931663

[41] Q. Ye, H. B. Zhu and B. T. Cui, “Synchronization Analysis of Delayed Hybrid Dynamical Networks with Quantized Impulsive Effects,” Control Theory & Ap plications, Vol. 30, No. 1, 2013, pp. 61-68.

[42] W. Liu, Z. M. Wang and M. K. Ni, “Quantized Feedback Stabilization of Model-Based Networked Control Sys tems,” Control and Decision, Vol. 28, No. 2, 2013, pp. 285-288.

[43] R. Q. Lu, F. Wu and A. K. Xue, “A Reset Quantized-State Controller for Linear Systems,” Control Theory & Applications, Vol. 29, No. 4, 2012, pp. 507-512.

[44] X. X. Liao, “Theory and Application of Stability for Dynamical Systems,” National Defence Industry Press, Beijing, 2000, pp. 15-40.

[45] B. Shi, D. C. Zhang and M. J. Gai, “Theory and Applica tions of Differential Equations,” National Defense Indus try Press, Beijing, 2005, pp. 18-23.

[46] J. C. Kuang, “Applied Inequalities,” Shandong Science and Technology Press, Jinan, 2004, pp. 564-570.

[47] C. Liu, C. D. Li and S. K. Duan, “Stabilization of Oscillating Neural Networks with Time-Delay by Inter mittent Control,” International Journal of Control, Auto mation and Systems, Vol. 9, No. 6, 2011, pp. 1074-1079. http://dx.doi.org/10.1007/s12555-011-0607-3

[1] H. Fujisaka and T. Yamada, “Stability Theory of Syn chronized Motion in a Coupled-Oscillator System,” Pro gress of Theoretical Physics, Vol. 69, No. 1, 1983, pp. 32-47. http://dx.doi.org/10.1143/PTP.69.32

[2] H. Fujisaka and T. Yamada, “A New Intermittency in Coupled Dynamical Systems,” Progress of Theoretical Physics, Vol. 74, No. 4, 1985, pp. 918-921. http://dx.doi.org/10.1143/PTP.74.918

[3] H. Fujisaka and T. Yamada, “Stability Theory of Syn chronized Motion in Coupled-Oscillator Systems IV,” Progress of Theoretical Physics, Vol. 75, No. 5, 1986, pp. 1087-1104. http://dx.doi.org/10.1143/

PTP.75.1087

[4] V. S. Afraimovich, N. N. Verichev and M. I. Rabinovich, “Stochastic Synchronization of Oscillations in Dissipative Systems,” Radiophysics and Quantum Electronics, Vol. 29, No. 9, 1986, pp. 747-751.

[5] L. M. Pecora and T. L. Carroll, “Synchronization in Chaotic Systems,” Physical Review Letters, Vol. 64, No. 8, 1990, pp. 821-824. http://dx.doi.org/10.1103/PhysRevLett.64.821

[6] W. H. Deng, J. H. Lu and C.-P. Li, “Stability of N-Dimensional Linear Systems with Multiple Delays and Application to Synchronization,” Journal of Systems Sci ence and Complexity, Vol. 19, No. 2, 2006, pp. 149-156.

[7] E. M. Elabbasy, H. N. Agiza and M. M. El-Dessoky, “Global Synchronization Criterion and Adaptive Syn chronization for New Chaotic System,” Chaos, Solitons and Fractals, Vol. 23, No. 4, 2005, pp. 1299-1309.

[8] Q. L. Zhang, J. Zhou and G. Zhang, “Stability Concerning Partial Variables for a Class of Time-Varying Systems and Its Applications in Chaos Synchronization,” Pro ceedings of the 24th Chinese Control Conference, South China University of Technology Press, Guangzhou, 2005, pp. 135-139.

[9] Q. L. Zhang and G. J. Jia, “Chaos Synchronization of Morse Oscillator via Backstepping Design,” Annals of Differential Equations, Vol. 22, No. 3, 2006, pp. 456-460.

[10] Q. L. Zhang, “Synchronization of Multi-Chaotic Systems via Ring Impulsive Control,” Control Theory & Applica tions, Vol. 27, No. 2, 2010, pp. 226-232.

[11] Z. Tang and J. W. Feng, “Adaptive Cluster Synchroni zation for Nondelayed and Delayed Coupling Complex Networks with Nonidentical Nodes,” Abstract and Applied Analysis, Vol. 2013, 2013, Article ID: 946243.

[12] B. Li and Q. K. Song, “Synchronization of Chaotic Delayed Fuzzy Neural Networks under Impulsive and Stochastic Perturbations,” Abstract and Applied Analysis, Vol. 2013, 2013, Article ID: 543549.

[13] M. Han, Y. Liu and J. Q. Lu, “Impulsive Control for the Synchronization of Chaotic Systems with Time Delay,” Abstract and Applied Analysis, Vol. 2013, 2013, Article ID: 647561.

[14] S. J. Dan, S. X. Yang and W. Feng, “Lag Synchronization of Coupled Delayed Chaotic Neural Networks by Per iodically Intermittent Control,” Abstract and Applied Analysis, Vol. 2013, 2013, Article ID: 501461.

[15] C.-M. Lin, M.-H. Lin and R.-G. Yeh, “Synchronization of Unified Chaotic System via Adaptive Wavelet Cerebellar Model Articulation Controller,” Neural Computing and Applications, 2012, pp. 1-9. http://dx.doi.org/10.1007/s00521-012-1021-3

[16] J. D. Cao and L. L. Li, “Cluster Synchronization in an Array of Hybrid Coupled Neural Networks with Delay,” Neural Networks, Vol. 22, No. 4, 2009, pp. 335-342.

[17] L. L. Li and J. D. Cao, “Cluster Synchronization in an Array of Coupled Stochastic Delayed Neural Networks via Pinning Control,” Neurocomputing, Vol. 74, No. 5, 2011, pp. 846-856. http://dx.doi.org/

10.1016/j.neucom.2010.11.006

[18] H. Reza and P. Mass, “Delay-Range-Dependent Expo nential H∞ Synchronization of a Class of Delayed Neural Networks,” Chaos, Solitons and Fractals, Vol. 41, No. 3, 2009, pp. 1125-1135. http://dx.doi.org/10.1016/j.chaos.2008.04.051

[19] J. D. Cao, D. W. C. Ho and Y. Q. Yang, “Projective Syn chronization of a Class of Delayed Chaotic Systems via Impulsive Control,” Physics Letters A, Vol. 373, No. 35, 2009, pp. 3128-3133. http://dx.doi.org/10.1016/j.physleta.2009.06.056

[20] Z. Zahreddine, “Matrix Measure and Application to Stability of Matrices and Interval Dynamical Systems,” International Journal of Mathematics and Mathematical Sciences, Vol. 2003, No. 2, 2003, pp. 75-85. http://dx.doi.org/10.1155/S0161171203202295

[21] J. T. Sun, Y. P. Zhang, Y. Q. Liu and F. Q. Deng, “Exponential Stability of Interval Dynamical System with Multidelay,” Applied Mathematics and Mechanics, Vol. 23, No. 1, 2002, pp. 87-91.

[22] G. D. Zong, Y. Q. Wu and S. Y. Xu, “Stability Criteria for Switched Linear Systems with Time-Delay,” Control Theory & Applications, Vol. 25, No. 5, 2008, pp. 295-305.

[23] Y. H. Yuan, Q. L. Zhang and B. Chen, “Robust Fuzzy Control Based on Matrix Measure for Nonlinear Descriptor Systems with Time-Delay,” Control and Decision, Vol. 22, No. 2, 2007, pp. 174-178.

[24] L. M. Ding, J. B. Quan, Y. Zhang and Z. X. Li, “The Fixed Point Theorem and Asymptotic Stability of a Delay-Differential System,” Journal of Air Force Radar Academy, Vol. 23, No. 3, 2009, pp. 203-204.

[25] Y. X. Guo, “Matrix Measure and Uniform Ultimate Boundedness with Respect to Partial Variables for FDEs,” Journal of Wuhan University of Science and Engineering, Vol. 21, No. 4, 2008, pp. 15-19.

[26] J. J. Huang, C. D. Li and Q. Han, “Stabilization of Delayed Chaotic Neural Networks by Periodically Intermittent Control,” Circuits, Systems & Signal Processing, Vol. 28, No. 4, 2009, pp. 567-579. http://dx.doi.org/10.1007/s00034-009-9098-3

[27] J. Yu, H. J. Jiang and Z. D. Teng, “Synchronization of Nonlinear Systems with Delay via Periodically Intermit tent Control,” Journal of Xinjiang University (Natural Science Edition), Vol. 28, No. 3, 2010, pp. 310-315.

[28] Z. Y. Dong, Y. J. Wang, M. H. Bai and Z. Q. Zuo, “Ex ponential Synchronization of Uncertain Master-Slave Lur’e Systems via Intermittent Control,” Journal of Dy namics and Control, Vol. 7, No. 4, 2009, pp. 328-333.

[29] W. G. Xia and J. D. Cao, “Pinning Synchronization of Delayed Dynamical Networks via Periodically Intermit tent Control,” Chaos, Vol. 19, No. 1, 2009, Article ID: 013120. http://dx.doi.org/10.1063/

1.3071933

[30] P. Smolen, D. Baxter and J. Byrne, “Mathematical Mod eling of Gene Networks Review,” Neuron, Vol. 26, No. 3, 2000, pp. 567-580. http://dx.doi.org/10.1016/S0896-6273(00)81194-0

[31] Y. Wang, J. Hao and Z. Zuo, “A New Method for Exponential Synchronization of Chaotic Delayed Systems via Intermittent Control,” Physics Letters A, Vol. 374, No. 19-20, 2010, pp. 2024-2029. http://dx.doi.org/10.1016/j.physleta.2010.02.069

[32] T. Huang, C. Li, W. Yu and G. Chen, “Synchronization of Delayed Chaotic Systems with Parameter Mismatches by Using Intermittent Linear State Feedback,” Non linearity, Vol. 22, No. 3, 2009, pp. 569-584. http://dx.doi.org/10.1088/0951-7715/22/3/004

[33] H. Zhang, H. C. Yan, F. W. Yang and Q. J. Chen, “Quantized Control Design for Impulsive Fuzzy Net worked Systems,” IEEE Transactions on Fuzzy Systems, Vol. 19, No. 6, 2011, pp. 1153-1162. http://dx.doi.org/10.1109/TFUZZ.2011.2162525

[34] F. Fagnani and S. Zampieri, “Quantized Stabilization of Linear Systems: Complexity versus Performance,” IEEE Transactions on Automatic Control, Vol. 49, No. 9, 2004, pp. 1534-1548. http://

dx.doi.org/10.1109/TAC.2004.834111

[35] S. Sqartini, B. Schuller and A. Hussain, “Cognitive and Emotionan Infromation Processing for Human-Machine Interaction,” Cognitive Computation, Vol. 4, No. 4, 2012, pp. 383-387. http://dx.doi.org/

10.1007/s12559-012-9180-1

[36] J. B. Kim, J. S. Park and Y. H. Oh, “Speaker-Characterized Emotion Recognition Using Online and Iterative Speaker Adaptation,” Cognitive Computation, Vol. 4, No. 4, 2012, pp. 398-408. http://dx.doi.org/10.1007/s12559-012-9132-9

[37] L. I. Per lovsky and D. S. Levine, “The Drive for Creativity and the Escape from Creativity: Neurocognitive Mechanisms,” Cognitive Computation, Vol. 4, No. 3, 2012, pp. 292-305.

[38] Q. L. Zhang, “The Generalized Dahlquist Constant with Applications in Synchronization Analysis of Typical Neural Networks via General Intermittent Control,” Ad vances in Artificial Neural Systems, Vol. 2011, 2011, Article ID: 249136. http://dx.doi.org/10.1155/2011/249136

[39] G. T. Hui, B. N. Huang, Y. C. Wang and X. P. Meng, “Quantized Control Design for Coupled Dynamic Net works with Communication Constraints,” Cognitive Com putation, Vol. 5, No. 2, 2013, pp. 200-206.

[40] L. A. Montestruque and P. J. Antsaklis, “Static and Dynamic Quantization in Model-Based Networked Control Systems,” International Journal of Control, Vol. 80, No. 1, 2007, pp. 87-101. http://dx.doi.org/

10.1080/00207170600931663

[41] Q. Ye, H. B. Zhu and B. T. Cui, “Synchronization Analysis of Delayed Hybrid Dynamical Networks with Quantized Impulsive Effects,” Control Theory & Ap plications, Vol. 30, No. 1, 2013, pp. 61-68.

[42] W. Liu, Z. M. Wang and M. K. Ni, “Quantized Feedback Stabilization of Model-Based Networked Control Sys tems,” Control and Decision, Vol. 28, No. 2, 2013, pp. 285-288.

[43] R. Q. Lu, F. Wu and A. K. Xue, “A Reset Quantized-State Controller for Linear Systems,” Control Theory & Applications, Vol. 29, No. 4, 2012, pp. 507-512.

[44] X. X. Liao, “Theory and Application of Stability for Dynamical Systems,” National Defence Industry Press, Beijing, 2000, pp. 15-40.

[45] B. Shi, D. C. Zhang and M. J. Gai, “Theory and Applica tions of Differential Equations,” National Defense Indus try Press, Beijing, 2005, pp. 18-23.

[46] J. C. Kuang, “Applied Inequalities,” Shandong Science and Technology Press, Jinan, 2004, pp. 564-570.

[47] C. Liu, C. D. Li and S. K. Duan, “Stabilization of Oscillating Neural Networks with Time-Delay by Inter mittent Control,” International Journal of Control, Auto mation and Systems, Vol. 9, No. 6, 2011, pp. 1074-1079. http://dx.doi.org/10.1007/s12555-011-0607-3