Nonsmooth finite-time stabilization of neural networks with discontinuous activations.

*(English)*Zbl 1307.93353Summary: This paper is concerned with the finite-time stabilization for a class of Neural Networks (NNs) with discontinuous activations. The purpose of the addressed problem is to design a discontinuous controller to stabilize the states of such neural networks in finite time. Unlike the previous works, such stabilization objective will be realized for neural networks when the activations and controllers are both discontinuous. Based on the famous finite-time stability theorem of nonlinear systems and nonsmooth analysis in mathematics, sufficient conditions are established to ensure the finite-time stability of the dynamics of NNs. Then, the upper bound of the settling time for stabilization can be estimated in two forms due to two different methods of proof. Finally, two numerical examples are given to illustrate the effectiveness of the proposed design method.

##### MSC:

93D21 | Adaptive or robust stabilization |

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

93B51 | Design techniques (robust design, computer-aided design, etc.) |

93C10 | Nonlinear systems in control theory |

##### Keywords:

discontinuous activation; discontinuous controller; finite-time stabilization; neural network
Full Text:
DOI

##### References:

[1] | Aubin, J. P.; Cellina, A., Differential inclusions, (1984), Springer Berlin |

[2] | Aubin, J. P.; Frankowska, H., Set-valued analysis, (1990), Birkhäuser Boston |

[3] | Bhat, S., & Bernstein, D. (1997). Finite-time stability of homogeneous systems. In Proc. American control conference. June (pp. 2513-2514). |

[4] | Bhat, S.; Bernstein, D., Finite-time stability of continuous autonomous systems, SIAM Journal on Control and Optimization, 38, 3, 751-766, (2000) · Zbl 0945.34039 |

[5] | Chen, G.; Lewis, F. L.; Xie, L. H., Finite-time distributed consensus via binary control protocols, Automatica, 47, 9, 1962-1968, (2011) · Zbl 1226.93008 |

[6] | Clarke, F. H., Optimization and nonsmooth analysis, (1983), Wiley New York · Zbl 0582.49001 |

[7] | Cortés, J., Finite-time convergent gradient flows with applications to network consensus, Automatica, 42, 11, 1993-2000, (2006) · Zbl 1261.93058 |

[8] | Filippov, A. F., Differential equations with discontinuous right-hand side, (Mathematics and its applications, Soviet series, (1988), Kluwer Academic Publishers Boston) · Zbl 0138.32204 |

[9] | Forti, M.; Nistri, P., Global convergence of neural networks with discontinuous neuron activations, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 50, 11, 1421-1435, (2003) · Zbl 1368.34024 |

[10] | Haimo, V. T., Finite time controllers, SIAM Journal on Control and Optimization, 24, 4, 760-770, (1986) · Zbl 0603.93005 |

[11] | Hardy, G. H.; Littlewood, J. E.; Pólya, G., Inequalities, (1988), Cambridge University Press Cambridge · Zbl 0634.26008 |

[12] | Hong, Y.; Wang, J.; Cheng, D., Adaptive finite-time control of nonlinear systems with parametric uncertainty, IEEE Transactions on Automatic Control, 51, 5, 858-862, (2006) · Zbl 1366.93290 |

[13] | Hopfield, J. J.; Tank, D. W., Neurons with graded response have collective computational properties like those of two-state neurons, Proceedings of the National Academy of Sciences, 79, 3088-3092, (1984) · Zbl 1371.92015 |

[14] | Hopfield, J. J.; Tank, D. W., Computing with neural circuits: a model, Science, 233, 625-633, (1986) |

[15] | Huang, X.; Cao, J.; Ho, D. W.C., Existence and attractivity of almost periodic solution for recurrent neural networks with unbounded delays and variable coefficients, Nonlinear Dynamics, 45, 3, 337-351, (2006) · Zbl 1130.68084 |

[16] | Huang, H.; Feng, G.; Cao, J., Exponential synchronization of chaotic lur’e systems with delayed feedback control, Nonlinear Dynamics, 57, 3, 441-453, (2009) · Zbl 1176.70034 |

[17] | Hui, Q.; Haddad, W. M.; Bhat, S. P., Finite-time semistability and consensus for nonlinear dynamical networks, IEEE Transactions on Automatic Control, 53, 8, 1887-1900, (2008) · Zbl 1367.93434 |

[18] | Khoo, S. Y.; Xie, L. H.; Man, Z. H., Robust finite-time consensus tracking algorithm for multi robot systems, IEEE/ASME Transactions on Mechatronics, 14, 2, 219-228, (2009) |

[19] | Kwon, O. M.; Lee, S. M.; Park, J. H.; Cha, E. J., New approaches on stability criteria for neural networks with interval time-varying delays, Applied Mathematics and Computation, 218, 19, 9953-9964, (2012) · Zbl 1253.34066 |

[20] | Li, C.; Liao, X.; Huang, T., Exponential stabilization of chaotic systems with delay by periodically intermittent control, Chaos, 17, 013103, (2007) · Zbl 1159.93353 |

[21] | Liang, J.; Wang, Z.; Liu, X., Exponential synchronization of stochastic delayed discrete-time complex networks, Nonlinear Dynamics, 53, 1, 153-165, (2008) · Zbl 1172.92002 |

[22] | Liu, X.; Cao, J., On periodic solutions of neural networks via differential inclusions, Neural Networks, 22, 4, 329-334, (2009) · Zbl 1335.93058 |

[23] | Liu, X.; Cao, J., Local synchronization of one-to-one coupled neural networks with discontinuous activations, Cognitive Neurodynamics, 5, 13-20, (2011) |

[24] | Liu, X.; Chen, T.; Cao, J.; Lu, W., Dissipativity and quasi-synchronization for neural networks with discontinuous activations and parameter mismatches, Neural Networks, 24, 10, 1013-1021, (2011) · Zbl 1264.93048 |

[25] | Liu, Y.; Wang, Z.; Liu, X., Stability criteria for periodic neural networks with discrete and distributed delays, Nonlinear Dynamics, 49, 1, 93-103, (2007) · Zbl 1176.92006 |

[26] | Liu, Y.; Wang, Z.; Liu, X., On delay-dependent robust exponential stability of stochastic neural networks with mixed time delays and Markovian switching, Nonlinear Dynamics, 54, 3, 199-212, (2008) · Zbl 1169.92003 |

[27] | Lu, J.; Cao, J., Adaptive synchronization of uncertain dynamical networks with delayed coupling, Nonlinear Dynamics, 53, 1, 107-115, (2008) · Zbl 1182.92007 |

[28] | Lu, W.; Chen, T., Dynamical behaviors of delayed neural networks systems with discontinuous activation functions, Neural Computation, 18, 683-708, (2006) · Zbl 1094.68625 |

[29] | Lu, W.; Chen, T., Almost periodic dynamics of a class of delayed neural networks with discontinuous activations, Neural Computation, 20, 1065-1090, (2008) · Zbl 1146.68422 |

[30] | Lu, J.; Ho, D. W.C.; Wang, Z., Pinning stabilization of linearly coupled stochastic neural networks via minimum number of controllers, IEEE Transactions on Neural Networks, 20, 10, 1617-1629, (2009) |

[31] | Lu, J.; Ho, D. W.C.; Wu, L., Exponential stabilization of switched stochastic dynamical networks, Nonlinearity, 22, 889-911, (2009) · Zbl 1158.93413 |

[32] | Ott, E.; Grebogi, G.; Yorke, J. A., Controlling chaos, Physical Review Letters, 64, 1196-1199, (1990) · Zbl 0964.37501 |

[33] | Park, M. J.; Kwon, O. M.; Lee, S. M.; Park, J. H.; Cha, E. J., Synchronization criteria for coupled stochastic neural networks with time-varying delays and leakage delay, Journal of the Franklin Institute, 349, 5, 1699-1720, (2012) · Zbl 1254.93012 |

[34] | Shen, J.; Cao, J., Finite-time synchronization of coupled neural networks via discontinuous controllers, Cognitive Neurodynamics, 5, 4, 373-385, (2011) |

[35] | Sontag, E. D., A “universal” construction of artsteins theorem on nonlinear stabilization, Systems & Control Letters, 13, 117-123, (1989) · Zbl 0684.93063 |

[36] | Wang, G.; Cao, J.; Liang, J., Exponential stability in the mean square for stochastic neural networks with mixed time-delays and Markovian jumping parameters, Nonlinear Dynamics, 57, 1, 209-218, (2009) · Zbl 1176.92007 |

[37] | Wang, L.; Xiao, F., Finite-time consensus problems for networks of dynamic agents, IEEE Transactions on Automatic Control, 55, 4, 950-955, (2010) · Zbl 1368.93391 |

[38] | Wu, Z.; Park, J. H.; Su, H.; Chu, J., New results on exponential passivity of neural networks with time-varying delays, Nonlinear Analysis: Real World Applications, 13, 4, 1593-1599, (2012) · Zbl 1257.34055 |

[39] | Xiao, F.; Wang, L.; Chen, J.; Gao, Y. P., Finite-time formation control for multi-agent systems, Automatica, 45, 11, 2605-2611, (2009) · Zbl 1180.93006 |

[40] | Yu, S., Ma, Z., Zhu, Q., & Wu, D. (2006). Nonsmooth finite-time control of uncertain affine planar systems. In Proceedings of the 6th world congress on intelligent control and automation (pp. 21-23). |

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