Novel stability criteria for impulsive delayed reaction–diffusion Cohen–Grossberg neural networks via Hardy–Poincarè inequality

Abstract This work is devoted to the investigation of stability theory for impulsive delayed reaction–diffusion Cohen–Grossberg neural networks with Dirichlet boundary condition. By means of Hardy–Poincare inequality and Gronwall–Bellman-type impulsive integral inequality, we summarize some new and concise sufficient conditions ensuring global exponential stability of the equilibrium point. The presented stability criteria show that not only reaction–diffusion coefficients but also regional features as well as the first eigenvalue of the Dirichlet Laplacian will impact the stability. In conclusion, two examples are illustrated to demonstrate the effectiveness of our obtained results.

[1]  Kelin Li,et al.  Exponential stability of impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms , 2008, Neurocomputing.

[2]  Xinzhi Liu.  Stability results for impulsive differential systems with applications to population growth models , 1994 .

[3]  Kwok-Wo Wong,et al.  Criteria for exponential stability of Cohen-Grossberg neural networks , 2004, Neural Networks.

[4]  Kelin Li,et al.  Stability analysis of impulsive fuzzy cellular neural networks with distributed delays and reaction-diffusion terms , 2009 .

[5]  V. Lakshmikantham,et al.  Theory of Impulsive Differential Equations , 1989, Series in Modern Applied Mathematics.

[6]  Xinzhi Liu,et al.  Stability criteria for impulsive reaction-diffusion Cohen-Grossberg neural networks with time-varying delays , 2010, Math. Comput. Model..

[7]  Xinzhi Liu,et al.  Impulsive Stabilization of High-Order Hopfield-Type Neural Networks With Time-Varying Delays , 2008, IEEE Transactions on Neural Networks.

[8]  Jianlong Qiu,et al.  Exponential stability of impulsive neural networks with time-varying delays and reaction-diffusion terms , 2007, Neurocomputing.

[9]  P. Balasubramaniam,et al.  Robust stability of uncertain Markovian jumping stochastic Cohen-Grossberg type bam neural networks with time-varying delays and reaction diffusion terms , 2011, Neural Parallel Sci. Comput..

[10]  Jinde Cao,et al.  Global exponential stability and periodicity of recurrent neural networks with time delays , 2005, IEEE Transactions on Circuits and Systems I: Regular Papers.

[11]  Haim Brezis,et al.  Blow-up solutions of some nonlinear elliptic problems , 1997 .

[12]  Xinhua Zhang,et al.  Delay-dependent exponential stability for impulsive Cohen–Grossberg neural networks with time-varying delays and reaction–diffusion terms , 2011 .

[13]  S. Arik,et al.  Global stability analysis of Cohen–Grossberg neural networks with time varying delays , 2005 .

[14]  Shouming Zhong,et al.  Dynamical behaviors of impulsive reaction-diffusion Cohen-Grossberg neural network with delays , 2010, Neurocomputing.

[15]  P. Balasubramaniam,et al.  Exponential stability of stochastic reaction-diffusion uncertain fuzzy neural networks with mixed delays and Markovian jumping parameters , 2012, Expert Syst. Appl..

[16]  Q. Song,et al.  Global exponential stability of impulsive Cohen-Grossberg neural network with time-varying delays , 2008 .

[17]  Daoyi Xu,et al.  Impulsive effects on stability of Cohen-Grossberg neural networks with variable delays , 2006, Appl. Math. Comput..

[18]  Jiye Zhang,et al.  Global exponential stability of Cohen–Grossberg neural networks with variable delays☆ , 2005 .

[19]  Chuandong Li,et al.  Stability of Cohen-Grossberg neural networks with unbounded distributed delays , 2007 .

[20]  P. Balasubramaniam,et al.  Global asymptotic stability of stochastic BAM neural networks with distributed delays and reaction-diffusion terms , 2010, J. Comput. Appl. Math..

[21]  Pagavathigounder Balasubramaniam,et al.  Robust stability of uncertain fuzzy cellular neural networks with time-varying delays and reaction diffusion terms , 2010, Neurocomputing.

[22]  Daoyi Xu,et al.  Global exponential stability of impulsive fuzzy cellular neural networks with mixed delays and reaction-diffusion terms , 2009 .

[23]  Zuoan Li,et al.  Stability analysis of impulsive Cohen-Grossberg neural networks with distributed delays and reaction-diffusion terms , 2009 .

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

[25]  Stephen Grossberg,et al.  Absolute stability of global pattern formation and parallel memory storage by competitive neural networks , 1983, IEEE Transactions on Systems, Man, and Cybernetics.

[26]  J. Ruan,et al.  Global dynamic analysis of general Cohen–Grossberg neural networks with impulse , 2007 .

[27]  J. Ruan,et al.  Global stability analysis of impulsive Cohen–Grossberg neural networks with delay , 2005 .

[28]  Wei Zhu.  Global exponential stability of impulsive reaction-diffusion equation with variable delays , 2008, Appl. Math. Comput..