Global stability analysis of competitive neural networks with mixed time-varying delays and discontinuous neuron activations

In this paper, the global stability has been investigated for a novel class of competitive neural networks with mixed time delays and discontinuous activations. Without presuming the boundedness of activation functions, we draw two set of sufficient conditions ensuring the existence, uniqueness of the equilibrium, global exponential asymptotic stability of the solution and the associated output of the solution converging to the output equilibrium point in measure, by linear matrix inequality, M-matrix, Leray-Schauder alternative theorem in multivalued analysis, general Lyapunov method, topological degree theory of set-valued map. Meanwhile, we present a result on the global convergence in finite time of given neural networks. The results in the literature are generalized and significantly improved. Finally, one example and simulations are presented to demonstrate the effectiveness of the theoretical results.

[1]  Lihong Huang,et al.  Global dynamics of equilibrium point for delayed competitive neural networks with different time scales and discontinuous activations , 2014, Neurocomputing.

[2]  Jigui Jian,et al.  Stability in Lagrange sense for Cohen–Grossberg neural networks with time-varying delays and finite distributed delays , 2010 .

[3]  Huaiqin Wu.  Global stability analysis of a general class of discontinuous neural networks with linear growth activation functions , 2009, Inf. Sci..

[4]  Qiankun Song,et al.  Global exponential stability of the periodic solution of delayed Cohen-Grossberg neural networks with discontinuous activations , 2010, Neurocomputing.

[5]  Qing-Guo Wang,et al.  Delay-range-dependent stability for systems with time-varying delay , 2007, Autom..

[6]  Tianping Chen,et al.  Dynamical Behaviors of Delayed Neural Network Systems with Discontinuous Activation Functions , 2006 .

[7]  Xinzhi Liu,et al.  Global convergence of neural networks with mixed time-varying delays and discontinuous neuron activations , 2012, Inf. Sci..

[8]  Jinde Cao,et al.  Stability in delayed Cohen–Grossberg neural networks: LMI optimization approach , 2005 .

[9]  Anke Meyer-Bäse,et al.  Robust stability analysis of competitive neural networks with different time-scales under perturbations , 2007, Neurocomputing.

[10]  Sabri Arik,et al.  An analysis of exponential stability of delayed neural networks with time varying delays , 2004, Neural Networks.

[11]  M. Forti.  M-matrices and global convergence of discontinuous neural networks: Research Articles , 2007 .

[12]  Jinde Cao,et al.  A based-on LMI stability criterion for delayed recurrent neural networks , 2006 .

[13]  A Meyer-Base,et al.  Local and global stability analysis of an unsupervised competitive neural network. , 2008, IEEE transactions on neural networks.

[14]  J. Cao,et al.  Exponential stability of competitive neural networks with time-varying and distributed delays , 2008 .

[15]  Qingshan Liu,et al.  Two k-winners-take-all networks with discontinuous activation functions , 2008, Neural Networks.

[16]  M. Forti,et al.  Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-Lipschitz activations , 2006 .

[17]  Nachol Chaiyaratana,et al.  Thalassaemia classification by neural networks and genetic programming , 2007, Inf. Sci..

[18]  Sabri Arik,et al.  Global stability analysis of neural networks with multiple time varying delays , 2005, IEEE Transactions on Automatic Control.

[19]  Hongtao Lu,et al.  Global exponential stability of delayed competitive neural networks with different time scales , 2005, Neural Networks.

[20]  L. Pandolfi,et al.  On stability of cellular neural networks with delay , 1993 .

[21]  Lihong Huang,et al.  Existence and global asymptotic stability of periodic solution for discrete and distributed time-varying delayed neural networks with discontinuous activations , 2011, Neurocomputing.

[22]  Anke Meyer-Bäse,et al.  Stability Analysis of an Unsupervised Competitive Neural Network , 2006, The 2006 IEEE International Joint Conference on Neural Network Proceedings.

[23]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[24]  Huaiqin Wu,et al.  Stability analysis for periodic solution of BAM neural networks with discontinuous neuron activations and impulses , 2009 .

[25]  Zhenyuan Guo,et al.  Dynamical behavior of delayed Hopfield neural networks with discontinuous activations , 2009 .

[26]  Walter Allegretto,et al.  Common Asymptotic Behavior of Solutions and Almost Periodicity for Discontinuous, Delayed, and Impulsive Neural Networks , 2010, IEEE Transactions on Neural Networks.

[27]  Tianping Chen,et al.  Dynamical Behaviors of Delayed Neural Network Systems with Discontinuous Activation Functions , 2006, Neural Computation.

[28]  M. Forti,et al.  Global convergence of neural networks with discontinuous neuron activations , 2003 .

[29]  Jinde Cao,et al.  Multistability of competitive neural networks with time-varying and distributed delays , 2009 .

[30]  Huaiqin Wu,et al.  Stability analysis for periodic solution of neural networks with discontinuous neuron activations , 2009 .

[31]  Lihong Huang,et al.  On the periodic dynamics of a class of time-varying delayed neural networks via differential inclusions , 2012, Neural Networks.

[32]  Zhao Lu,et al.  Adaptive feedback linearization control of chaotic systems via recurrent high-order neural networks , 2006, Inf. Sci..

[33]  Jinde Cao,et al.  Multistability of Second-Order Competitive Neural Networks With Nondecreasing Saturated Activation Functions , 2011, IEEE Transactions on Neural Networks.

[34]  Anke Meyer-Bäse,et al.  Singular Perturbation Analysis of Competitive Neural Networks with Different Time Scales , 1996, Neural Computation.

[35]  Abdesselam Bouzerdoum,et al.  Neural network for quadratic optimization with bound constraints , 1993, IEEE Trans. Neural Networks.

[36]  Francesca Maria Ceragioli,et al.  Discontinuous ordinary differential equations and stabilization , 2000 .

[37]  Vadim I. Utkin,et al.  Sliding Modes and their Application in Variable Structure Systems , 1978 .

[38]  D. Kelly,et al.  Stability in contractive nonlinear neural networks , 1990, IEEE Transactions on Biomedical Engineering.

[39]  Jinde Cao,et al.  On periodic solutions of neural networks via differential inclusions , 2009, Neural Networks.

[40]  Duccio Papini,et al.  Global exponential stability and global convergence in finite time of delayed neural networks with infinite gain , 2005, IEEE Transactions on Neural Networks.

[41]  Xiaofeng Liao,et al.  Delay-dependent asymptotic stability for neural networks with distributed delays , 2006 .

[42]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[43]  K. Deimling.  Fixed Point Theory , 2008 .

[44]  R. Westervelt,et al.  Stability of analog neural networks with delay. , 1989, Physical review. A, General physics.

[45]  Mauro Forti.  M-matrices and global convergence of discontinuous neural networks , 2007, Int. J. Circuit Theory Appl..

[46]  Hongtao Lu,et al.  Global Exponential Stability of Multitime Scale Competitive Neural Networks With Nonsmooth Functions , 2006, IEEE Transactions on Neural Networks.

[47]  Xiangnan Zhou,et al.  Existence and global asymptotic stability of periodic solutions for Hopfield neural networks with discontinuous activations , 2009 .

[48]  J. Hopfield.  Neurons withgraded response havecollective computational properties likethoseoftwo-state neurons , 1984 .

[49]  Jinde Cao,et al.  Exponential synchronization of the complex dynamical networks with a coupling delay and impulsive effects , 2010 .

[50]  Jinde Cao,et al.  Existence and global stability of equilibrium point for delayed competitive neural networks with discontinuous activation functions , 2012, Int. J. Syst. Sci..

[51]  Huaguang Zhang,et al.  Dynamical stability analysis of multiple equilibrium points in time-varying delayed recurrent neural networks with discontinuous activation functions , 2012, Neurocomputing.

[52]  Jean-Pierre Aubin,et al.  Viability theory , 1991 .

[53]  Tianping Chen,et al.  Almost Periodic Dynamics of a Class of Delayed Neural Networks with Discontinuous Activations , 2008, Neural Computation.

[54]  Everestus Eze.  Existence and Stability of Periodic Solutions... , 2017 .

[55]  J. Hale.  Theory of Functional Differential Equations , 1977 .

[56]  Tianping Chen,et al.  Dynamical behaviors of Cohen-Grossberg neural networks with discontinuous activation functions , 2005, Neural Networks.

[57]  Cui Bao-tong.  Global exponential stability of competitive neural networks with different time-scales , 2008 .