Dynamical behaviors of Cohen-Grossberg neural networks with mixed time delays and discontinuous activations

In this paper, we investigate the dynamical behaviors of a novel class of Cohen-Grossberg neural networks with not only mixed time delays, i.e., time varying delays and distributed delays, but also discontinuous activations, which may be unbounded or nonmonotonic. Some sufficient conditions are derived to ensure 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 using the Leray-Schauder alternative theorem in multivalued analysis, matrix theory and generalized Lyapunov-like approach. Some recent results in the literature are generalized and significantly improved. One example is provided to demonstrate the effectiveness of the theoretical results.

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

[2]  Lihong Huang,et al.  Global convergence of periodic solution of neural networks with discontinuous activation functions , 2009 .

[3]  Tianping Chen,et al.  Delay-independent stability analysis of Cohen-Grossberg neural networks , 2003 .

[4]  Neyir Ozcan,et al.  Global robust stability analysis of neural networks with multiple time delays , 2006, IEEE Transactions on Circuits and Systems I: Regular Papers.

[5]  Lin Wang,et al.  Exponential stability of Cohen-Grossberg neural networks , 2002, Neural Networks.

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

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

[8]  Yingwei Li,et al.  Global stability analysis in Cohen-Grossberg neural networks with delays and inverse Hölder neuron activation functions , 2010, Inf. Sci..

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

[10]  Tianping Chen,et al.  Nonnegative periodic dynamics of delayed Cohen-Grossberg neural networks with discontinuous activations , 2010, Neurocomputing.

[11]  P. Hartman.  Ordinary Differential Equations , 1965 .

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

[13]  Y. Chen.  Global asymptotic stability of delayed Cohen-Grossberg neural networks , 2006, IEEE Transactions on Circuits and Systems I: Regular Papers.

[14]  Lihong Huang,et al.  Dynamics of a class of Cohen–Grossberg neural networks with time-varying delays , 2007 .

[15]  Arkadi Nemirovski,et al.  Lmi Control Toolbox For Use With Matlab , 2014 .

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

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

[18]  Lihong Huang,et al.  Stability analysis of Cohen–Grossberg neural networks with discontinuous neuron activations , 2010 .

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

[20]  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.

[21]  Jinde Cao,et al.  Stability analysis of Cohen-Grossberg neural network with both time-varying and continuously distributed delays , 2006 .

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

[23]  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.

[24]  M. Forti,et al.  Global Convergence of Neural Networks With , 2003 .

[25]  Jinde Cao,et al.  Boundedness and stability for Cohen–Grossberg neural network with time-varying delays☆ , 2004 .

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

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

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

[29]  T. Mexia,et al.  Author ' s personal copy , 2009 .

[30]  J J Hopfield,et al.  Neurons with graded response have collective computational properties like those of two-state neurons. , 1984, Proceedings of the National Academy of Sciences of the United States of America.

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

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

[33]  Jiafu Wang,et al.  Almost periodicity for a class of delayed Cohen–Grossberg neural networks with discontinuous activations , 2012 .

[34]  Stephen Grossberg,et al.  Nonlinear neural networks: Principles, mechanisms, and architectures , 1988, Neural Networks.

[35]  R. Schwarzenberger.  ORDINARY DIFFERENTIAL EQUATIONS , 1982 .