Finite time stabilization of delayed neural networks.

*(English)*Zbl 1396.93101Summary: In this paper, the problem of finite time stabilization for a class of Delayed Neural Networks (DNNs) is investigated. The general conditions on the feedback control law are provided to ensure the finite time stabilization of DNNs. Then, some specific conditions are derived by designing two different controllers which include the delay-dependent and delay-independent ones. In addition, the upper bound of the settling time for stabilization is estimated. Under fixed control strength, discussions of the extremum of settling time functional are made and a switched controller is designed to optimize the settling time. Finally, numerical simulations are carried out to demonstrate the effectiveness of the obtained results.

##### MSC:

93D15 | Stabilization of systems by feedback |

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

93C55 | Discrete-time control/observation systems |

93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |

68T05 | Learning and adaptive systems in artificial intelligence |

93B52 | Feedback control |

##### Keywords:

finite time stabilization; delayed neural networks; delay-dependent controller; delay-independent controller; switched controller; settling time
Full Text:
DOI

##### References:

[1] | Berman, A.; Plemmons, R. J., Nonnegative matrices in the mathematical science, (1979), Academic New York |

[2] | Bhat, S. P.; Bernstein, D. S., Finite-time stability of continuous autonomous systems, SIAM Journal on Control and Optimization, 38, 751-766, (2000) · Zbl 0945.34039 |

[3] | Cao, J.; Wang, J., Global exponential stability and periodicity of recurrent neural networks with time delays, IEEE Transactions on Circuits and Systems I: Regular Papers, 52, 920-931, (2005) · Zbl 1374.34279 |

[4] | Chen, X.; Huang, L.; Guo, Z., Finite time stability of periodic solution for Hopfield neural networks with discontinuous activations, Neurocomputing, 103, 43-49, (2013) |

[5] | Chen, W. H.; Zhong, J.; Jiang, Z.; Lu, X., Periodically intermittent stabilization of delayed neural networks based on piecewise Lyapunov functions/functionals, Circuits, Systems, and Signal Processing, 33, 3757-3782, (2014) · Zbl 1342.93096 |

[6] | Chua, L. O.; Yang, L., Celluar neural networks: applications, IEEE Transactions on Circuits and Systems, 35, 1273-1290, (1988) |

[7] | Cohen, M. A.; Grossberg, S., Absolute stability of global pattern formation and parallel memory storage by competitive neural networks, IEEE Transactions on Systems, Man and Cybernetics, 13, 815-826, (1983) · Zbl 0553.92009 |

[8] | Efimov, D.; Polyakov, A.; Fridman, E.; Perruquetti, E.; Richard, J. P., Comments on finite-time stability of time-delay systems, Automatica, 50, 1944-1947, (2014) · Zbl 1296.93150 |

[9] | Faydasicok, O.; Arik, S., Robust stability analysis of a class of neural networks with discrete time delays, Neural Networks, 29-30, 52-59, (2012) · Zbl 1245.93111 |

[10] | Forti, M.; Nistri, P.; Papini, D., Global exponential stability and global convergence in finite time of delayed neural networks with infinite gain, IEEE Transactions on Neural Networks and Learning Systems, 16, 1449-1463, (2005) |

[11] | Forti, M.; Tesi, A., New conditions for global stability of neural networks with application to linear and quadratic programming problems, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 42, 354-366, (1995) · Zbl 0849.68105 |

[12] | Guan, Z. H.; Zhang, H., Stabilization of complex network with hybrid impulsive and switching control, Chaos, Solitons & Fractals, 37, 1372-1382, (2008) · Zbl 1142.93423 |

[13] | Guo, Z. Y.; Wang, J.; Yan, Z., Global exponential dissipativity and stabilization of memristor-based recurrent neural networks with time-varying delays, Neural Networks, 48, 158-172, (2013) · Zbl 1297.93129 |

[14] | Hardy, G.; Littlewood, J.; Polya, G., Inequalities, (1988), Cambridge University Press Cambridge · Zbl 0634.26008 |

[15] | He, Y.; Wu, M.; She, J. H., Delay-dependent exponential stability of delayed neural networks with time-varying delay, IEEE Transactions on Circuits Systems II: Express Briefs, 53, 553-557, (2006) |

[16] | Hong, Y.; Jiang, Z. P., Finite-time stabilization of nonlinear systems with parametric and dynamic uncertainties, IEEE Transactions on Automatic Control, 51, 1950-1956, (2006) · Zbl 1366.93577 |

[17] | Huang, H.; Huang, T.; Chen, X.; Qian, C., Exponential stabilization of delayed recurrent neural networks: A state estimation based approach, Neural Networks, 48, 153-157, (2013) · Zbl 1297.93153 |

[18] | Huang, T.; Li, C.; Duan, S.; Starzyk, J. A., Robust exponential stability of uncertain delayed neural networks with stochastic perturbation and impulse effects, IEEE Transactions on Neural Networks and Learning Systems, 23, 866-875, (2012) |

[19] | Huang, J.; Li, C.; Han, Q., Stabilization of delayed chaotic neural networks by periodically intermittent control, Circuits, Systems and Signal Processing, 28, 567-579, (2009) · Zbl 1170.93370 |

[20] | Huang, J.; Li, C.; Huang, T.; He, X., Finite-time lag synchronization of delayed neural networks, Neurocomputing, 139, 145-149, (2014) |

[21] | Hu, C.; Yu, J.; Jiang, H., Finite-time synchronization of delayed neural networks with Cohen-Grossberg type based on delayed feedback control, Neurocomputing, 143, 90-96, (2014) |

[22] | Hu, C.; Yu, J.; Jiang, H.; Teng, Z., Exponential stabilization and synchronization of neural networks with time-varying delays via periodically intermittent control, Nonlinearity, 23, 2369-2391, (2010) · Zbl 1197.92005 |

[23] | Karafyllis, I., Finite-time global stabilization by means of time-varying distributed delay feedback, SIAM Journal on Control and Optimization, 45, 320-342, (2006) · Zbl 1132.93036 |

[24] | Liu, Y., Global finite-time stabilization via time-varying feedback for uncertain nonlinear systems, SIAM Journal on Control and Optimization, 52, 1886-1913, (2014) · Zbl 1295.93067 |

[25] | Liu, X.; Ho, D. W.C.; Yu, W.; Cao, J., A new switching design to finite-time stabilization of nonlinear systems with applications to neural networks, Neural Networks, 57, 94-102, (2014) · Zbl 1323.93064 |

[26] | Liu, X.; Jiang, N.; Cao, J.; Wang, S.; Wang, Z., Finite-time stochastic stabilization for BAM neural networks with uncertainties, Journal of the Franklin Institute, 350, 2109-2123, (2013) · Zbl 1293.93176 |

[27] | Liu, X.; Park, J. H.; Jiang, N.; Cao, J., Nonsmooth finite-time stabilization of neural networks with discontinuous activations, Neural Networks, 52, 25-32, (2014) · Zbl 1307.93353 |

[28] | Liu, Y.; Wang, Z.; Liu, X., Global exponential stability of generalized recurrent neural networks with discrete and distributed delays, Neural Networks, 19, 667-675, (2006) · Zbl 1102.68569 |

[29] | Lu, H., Chaotic attractors in delayed neural networks, Physics Letters A, 298, 109-116, (2002) · Zbl 0995.92004 |

[30] | Moulay, E.; Dambrine, M.; Yeganefar, N.; Perruquetti, W., Finite-time stability and stabilization of time-delay systems, Systems & Control Letters, 57, 561-566, (2008) · Zbl 1140.93447 |

[31] | Moulay, E.; Perruquetti, W., Finite time stability of differential inclusions, IMA Journal of Mathematical Control and Information, 22, 465-475, (2005) · Zbl 1115.93351 |

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

[33] | Shen, J.; Cao, J., Finite-time synchronization of coupled neural networks via discontinuous controllers, Cognitive Neurodynamics, 5, 373-385, (2011) |

[34] | Shen, H.; Park, J. H.; Wu, Z. G., Finite-time synchronization control for uncertain Markov jump neural networks with input constraints, Nonlinear Dynamics, 77, 1709-1720, (2014) · Zbl 1331.92019 |

[35] | Shen, Y.; Wang, J., Almost sure exponential stability of recurrent neural networks with Markovian switching, IEEE Transactions on Neural Networks, 20, 840-855, (2009) |

[36] | Shen, Y.; Wang, J., Robustness analysis of global exponential stability of recurrent neural networks in the presence of time delays and random disturbances, IEEE Transactions on Neural Networks and Learning Systems, 23, 87-96, (2012) |

[37] | Sun, L.; Feng, G.; Wang, Y., Finite-time stabilization and H\(\infty\) control for a class of nonlinear Hamiltonian descriptor systems with application to affine nonlinear descriptor systems, Automatica, 50, 2090-2097, (2014) · Zbl 1297.93132 |

[38] | Wang, Z.; Liu, Y.; Li, M.; Liu, X., Stability analysis for stochastic Cohen-Grossberg neural networks with mixed time delays, IEEE Transactions on Neural Networks and Learning Systems, 17, 814-820, (2006) |

[39] | Wang, Z.; Liu, Y.; Liu, X., On global asymptotic stability of neural networks with discrete and distributed delays, Physics Letters A, 345, 299-308, (2005) · Zbl 1345.92017 |

[40] | Wang, L.; Shen, Y., New results on passivity analysis of memristor-based neural networks with time-varying delays, Neurocomputing, 144, 208-214, (2014) |

[41] | Wang, L.; Shen, Y.; Yin, Q.; Zhang, G., Adaptive synchronization of memristor-based neural networks with time-varying delays, IEEE Transactions on Neural Networks and Learning Systems, 2014, (2014) |

[42] | Wang, L.; Xiao, F., Finite-time consensus problems for networks of dynamic agents, IEEE Transactions on Automatic Control, 55, 950-955, (2010) · Zbl 1368.93391 |

[43] | Wen, S.; Huang, T.; Zeng, Z.; Chen, Y.; Li, P., Circuit design and exponential stabilization of memristive neural networks, Neural Networks, 63, 48-56, (2015) · Zbl 1323.93065 |

[44] | Wu, R.; Lu, Y.; Chen, L., Finite-time stability of fractional delayed neural networks, Neurocomputing, 149, 700-707, (2015) |

[45] | Wu, A. L.; Zeng, Z. G., Exponential stabilization of memristive neural networks with time delays, IEEE Transactions on Neural Networks and Learning Systems, 23, 1919-1929, (2012) |

[46] | Yang, R.; Wang, Y., Finite-time stability and stabilization of a class of nonlinear time-delay systems, SIAM Journal on Control and Optimization, 50, 3113-3131, (2012) · Zbl 1279.34087 |

[47] | Zeng, Z.; Zheng, W. X., Multistability of neural networks with time-varying delays and concave-convex characteristics, IEEE Transactions on Neural Networks and Learning Systems, 23, 293-305, (2012) |

[48] | Zhang, G. D.; Shen, Y., Exponential stabilization of memristor-based chaotic neural networks with time-varying delays via intermittent control, IEEE Transactions on Neural Networks and Learning Systems, 26, 1431-1441, (2015) |

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