Guaranteed cost control for exponential synchronization of cellular neural networks with mixed time-varying delays via hybrid feedback control.

*(English)*Zbl 1271.93064Summary: The problem of guaranteed cost control for exponential synchronization of cellular neural networks with interval non-differentiable and distributed time-varying delays via hybrid feedback control is considered. The interval time-varying delay function is not necessary to be differentiable. The construction of improved Lyapunov-Krasovskii functionals is based on Leibniz-Newton’s formula and methods of dealing with some integral terms. New delay-dependent sufficient conditions for the exponential synchronization of the error systems with memoryless hybrid feedback control are first established in terms of LMIs without introducing any free-weighting matrices. The optimal guaranteed cost control with linear error hybrid feedback is turned into the solvable problem of a set of LMIs. A numerical example is also given to illustrate the effectiveness of the proposed method.

##### MSC:

93B52 | Feedback control |

92B20 | Neural networks for/in biological studies, artificial life and related topics |

93D30 | Lyapunov and storage functions |

49N90 | Applications of optimal control and differential games |

##### Keywords:

cost control; cellular neural networks; hybrid feedback control; Lyapunov-Krasovskii functionals
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\textit{T. Botmart} and \textit{W. Weera}, Abstr. Appl. Anal. 2013, Article ID 175796, 12 p. (2013; Zbl 1271.93064)

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##### References:

[1] | Gupta, M. M.; Jin, L.; Homma, N., Static and Dynamic Neural Networks: From Fundamentalsto Advanced Theory, (2003), New York, NY, USA: Wiley, New York, NY, USA |

[2] | Pecora, L. M.; Carroll, T. L., Synchronization in chaotic systems, Physical Review Letters, 64, 8, 821-824, (1990) · Zbl 0938.37019 |

[3] | Liang, J.; Cao, J., Global asymptotic stability of bi-directional associative memory networks with distributed delays, Applied Mathematics and Computation, 152, 2, 415-424, (2004) · Zbl 1046.94020 |

[4] | Zhao, H., Global asymptotic stability of Hopfield neural network involving distributed delays, Neural Networks, 17, 1, 47-53, (2004) · Zbl 1082.68100 |

[5] | Balasubramaniam, P.; Vembarasan, V., Synchronization of recurrent neural networks with mixed time-delays via output coupling with delayed feedback, Nonlinear Dynamics, 70, 1, 677-691, (2012) · Zbl 1267.93138 |

[6] | Balasubramaniam, P.; Chandran, R.; Jeeva Sathya Theesar, S., Synchronization of chaotic nonlinear continuous neural networks with time-varying delay, Cognitive Neurodynamics, 5, 361-371, (2011) |

[7] | Thuan, M. V., Guaranteed cost control of neural networks with various activation functions and mixed time-varying delays in state and control, Differential Equations and Control Processes, 3, 18-29, (2011) · Zbl 1412.93040 |

[8] | Weera, W.; Niamsup, P., Exponential stabilization of neutral-type neural networks with interval nondifferentiable and distributed time-varying delays, Abstract and Applied Analysis, 2012, (2012) · Zbl 1237.93140 |

[9] | Phat, V. N.; Trinh, H., Exponential stabilization of neural networks with varous activation functions and mixed time-varying delays, IEEE Transactions on Neural Networks, 21, 1180-1184, (2010) |

[10] | Cai, J.; Wu, X.; Chen, S., Synchronization criteria for non-autonomous chaotic systems with sinusoidal state error feedback control, Physica Scripta, 75, 379-387, (2007) · Zbl 1130.93361 |

[11] | Botmart, T.; Niamsup, P., Adaptive control and synchronization of the perturbed Chua’s system, Mathematics and Computers in Simulation, 75, 1-2, 37-55, (2007) · Zbl 1115.37072 |

[12] | Jeeva Sathya Theesar, S.; Banerjee, S.; Balasubramaniam, P., Adaptive synchronization in noise perturbed chaotic systems, Physica Scripta, 85, 6, (2012) · Zbl 1282.34054 |

[13] | Cao, J.; Ho, D. W. C.; Yang, Y., Projective synchronization of a class of delayed chaotic systems via impulsive control, Physics Letters A, 373, 35, 3128-3133, (2009) · Zbl 1233.34017 |

[14] | Lu, J.; Ho, D. W. C.; Cao, J.; Kurths, J., Exponential synchronization of linearly coupled neural networks with impulsive disturbances, IEEE Transactions on Neural Networks, 22, 329-335, (2011) |

[15] | Li, G. H.; Zhou, S. P.; Yang, K., Generalized projective synchronization between two different chaotic systems using active backstepping control, PPhysics Letters A, 355, 326-330, (2006) |

[16] | Guo, H.; Zhong, S., Synchronization criteria of time-delay feedback control system with sector-bounded nonlinearity, Applied Mathematics and Computation, 191, 2, 550-559, (2007) · Zbl 1193.93144 |

[17] | Botmart, T.; Niamsup, P.; Liu, X., Synchronization of non-autonomous chaotic systems with time-varying delay via delayed feedback control, Communications in Nonlinear Science and Numerical Simulation, 17, 4, 1894-1907, (2012) · Zbl 1239.93043 |

[18] | Jeeva Sathya Theesar, S.; Banerjee, S.; Balasubramaniam, P., Synchronization of chaotic systems under sampled-data control, Nonlinear Dynamics, 70, 1977-1987, (2012) · Zbl 1268.93074 |

[19] | Chang, S. S. L.; Peng, T. K. C., Adaptive guaranteed cost control of systems with uncertain parameters, IEEE Transactions on Automatic Control, 17, 4, 474-483, (1972) · Zbl 0259.93018 |

[20] | Thuan, M. V.; Phat, V. N., Optimal guaranteed cost control of linear systems with mixed interval time-varying delayed state and control, Journal of Optimization Theory and Applications, 152, 2, 394-412, (2012) · Zbl 1237.49047 |

[21] | Tu, J.; He, H.; Xiong, P., Guaranteed cost synchronous control of time-varying delay cellular neural networks, Neural Computing and Applications, 22, 1, 103-110, (2013) |

[22] | Tu, J.; He, H., Guaranteed cost synchronization of chaotic cellular neural networks with time-varying delay, Neural Computation, 24, 1, 217-233, (2012) · Zbl 1237.92003 |

[23] | Chen, B.; Liu, X.; Tong, S., Guaranteed cost control of time-delay chaotic systems via memoryless state feedback, Chaos, Solitons & Fractals, 34, 5, 1683-1688, (2007) · Zbl 1152.93488 |

[24] | Lee, T. H.; Park, J. H.; Ji, D. H.; Kwon, O. M.; Lee, S. M., Guaranteed cost synchronization of a complex dynamical network via dynamic feedback control, Applied Mathematics and Computation, 218, 11, 6469-6481, (2012) · Zbl 1238.93070 |

[25] | Cheng, C. J.; Liao, T. L.; Yan, J. J.; Hwang, C. C., Exponential synchronization of a class of neural networks with time-varying delays, IEEE Transactions on Systems, Man, and Cybernetics B, 36, 209-215, (2006) |

[26] | Gao, X.; Zhong, S.; Gao, F., Exponential synchronization of neural networks with time-varying delays, Nonlinear Analysis. Theory, Methods & Applications, 71, 5-6, 2003-2011, (2009) · Zbl 1173.34349 |

[27] | Jeeva Sathya Theesar, S.; Chandran, R.; Balasubramaniam, P., Delay-dependent exponential synchronization criteria for chaotic neural networks with time-varying delays, Brazilian Journal of Physics, 42, 207-218, (2012) |

[28] | Sun, Y.; Cao, J.; Wang, Z., Exponential synchronization of stochastic perturbed chaotic delayed neural networks, Neurocomputing, 70, 13–15, 2477-2485, (2007) |

[29] | Li, T.; Fei, S.-M.; Zhu, Q.; Cong, S., Exponential synchronization of chaotic neural networks with mixed delays, Neurocomputing, 71, 13–15, 3005-3019, (2008) |

[30] | Li, T.; Fei, S.-M.; Zhang, K.-J., Synchronization control of recurrent neural networks with distributed delays, Physica A, 387, 4, 982-996, (2008) |

[31] | Karimi, H. R.; Gao, H., New delay-dependent exponential \(H_\infty\) synchronization for uncertain neural networks with mixed time delays, IEEE Transactions on Systems, Man, and Cybernetics B, 40, 173-185, (2010) |

[32] | Wu, Z.-G.; Shi, P.; Su, H.; Chu, J., Exponential synchronization of neural networks with discrete and distributed delays under time-varying sampling, IEEE Transactions on Neural Networks, 23, 1368-1376, (2012) |

[33] | Song, Q., Design of controller on synchronization of chaotic neural networks with mixed time-varying delays, Neurocomputing, 72, 13–15, 3288-3295, (2009) |

[34] | Botmart, T.; Niamsup, P.; Phat, V. N., Delay-dependent exponential stabilization for uncertain linear systems with interval non-differentiable time-varying delays, Applied Mathematics and Computation, 217, 21, 8236-8247, (2011) · Zbl 1241.34080 |

[35] | Kwon, O. M.; Park, J. H., Exponential stability analysis for uncertain neural networks with interval time-varying delays, Applied Mathematics and Computation, 212, 2, 530-541, (2009) · Zbl 1179.34080 |

[36] | Tian, J.; Zhou, X., Improved asymptotic stability criteria for neural for networks with interval time-varying delay, Expert Systems with Applications, 37, 7521-7525, (2010) |

[37] | Wang, D.; Wang, W., Delay-dependent robust exponential stabilization for uncertain systems with interval time-varying delays, Journal of Control Theory and Applications, 7, 3, 257-263, (2009) |

[38] | Park, J. H., Further result on asymptotic stability criterion of cellular neural networks with time-varying discrete and distributed delays, Applied Mathematics and Computation, 182, 2, 1661-1666, (2006) · Zbl 1154.92302 |

[39] | Park, J. H., On global stability criterion of neural networks with continuously distributed delays, Chaos, Solitons and Fractals, 37, 2, 444-449, (2008) · Zbl 1141.93054 |

[40] | Park, J. H., An analysis of global robust stability of uncertain cellular neural networks with discrete and distributed delays, Chaos, Solitons and Fractals, 73, 2789-2792, (2010) |

[41] | Park, J. H.; Cho, H. J., A delay-dependent asymptotic stability criterion of cellular neural networks with time-varying discrete and distributed delays, Chaos, Solitons and Fractals, 33, 2, 436-442, (2007) · Zbl 1142.34379 |

[42] | Gu, K.; Kharitonov, V. L.; Chen, J., Stability of Time-Delay System, (2003), Boston, Mass, USA: BirkhĂ¤auser, Boston, Mass, USA |

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