Delay-dependent exponential stability of recurrent neural networks with Markovian jumping parameters and proportional delays

This paper deals with the global exponential stability problem of a class of recurrent neural networks with Markovian jumping parameters and proportional delays. Here the proportional delay is unbounded time-varying, which is different from unbounded distributed delay. The nonlinear transformation $$z(t)=x({\text {e}}^{t})$$z(t)=x(et) transforms the recurrent neural networks with Markovian jumping parameters and proportional delays into the recurrent neural networks with Markovian jumping parameters, constant delays and variable coefficients. By constructing Lyapunov functional, a linear matrix inequality (LMI) approach is developed to establish a new delay-dependent global exponential stability sufficient condition in the mean square, which is related to the size of the proportional delay factor and can be easily checked by utilizing the numerically efficient MATLAB LMI toolbox, and no tuning of parameters is required. Two numerical examples and their simulations are given to illustrate the effectiveness of the obtained results.

[1]  L. Fox,et al.  On a Functional Differential Equation , 1971 .

[2]  Liqun Zhou.  Dissipativity of a class of cellular neural networks with proportional delays , 2013 .

[3]  Liqun Zhou,et al.  Global exponential stability of cellular neural networks with multi-proportional delays , 2015 .

[4]  Yonggui Kao,et al.  Delay-Dependent Robust Exponential Stability of Impulsive Markovian Jumping Reaction-Diffusion Cohen-Grossberg Neural Networks , 2012, Neural Processing Letters.

[5]  Rajendran Samidurai,et al.  Global asymptotic stability of BAM neural networks with mixed delays and impulses , 2009, Appl. Math. Comput..

[6]  Tingwen Huang,et al.  A mode-dependent approach to state estimation of recurrent neural networks with Markovian jumping parameters and mixed delays , 2013, Neural Networks.

[7]  Yan Liu,et al.  Global exponential stability of delayed fuzzy cellular neural networks with Markovian jumping parameters , 2011, Neural Computing and Applications.

[8]  Zidong Wang,et al.  Global exponential stability of generalized recurrent neural networks with discrete and distributed delays , 2006, Neural Networks.

[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]  Yonggui Kao,et al.  Global exponential stability analysis for cellular neural networks with variable coefficients and delays , 2008, Neural Computing and Applications.

[11]  Liqun Zhou,et al.  Delay-Dependent Exponential Stability of Cellular Neural Networks with Multi-Proportional Delays , 2012, Neural Processing Letters.

[12]  Yanyan Zhang,et al.  Global exponential stability of a class of impulsive recurrent neural networks with proportional delays via fixed point theory , 2016, J. Frankl. Inst..

[13]  Linshan Wang,et al.  Exponential stability of reaction–diffusion high-order Markovian jump Hopfield neural networks with time-varying delays , 2012 .

[14]  Wei Xing Zheng,et al.  Global Exponential Stability of Impulsive Neural Networks With Variable Delay: An LMI Approach , 2009, IEEE Transactions on Circuits and Systems I: Regular Papers.

[15]  Arieh Iserles,et al.  The asymptotic behaviour of certain difference equations with proportional delays , 1994 .

[16]  Lihong Huang,et al.  Periodic oscillation for a class of neural networks with variable coefficients , 2005 .

[17]  Zidong Wang,et al.  Exponential stability of delayed recurrent neural networks with Markovian jumping parameters , 2006 .

[18]  G. Derfel,et al.  Kato Problem for Functional-Differential Equations and Difference Schrödinger Operators , 1990 .

[19]  Pagavathigounder Balasubramaniam,et al.  On exponential stability results for fuzzy impulsive neural networks , 2010, Fuzzy Sets Syst..

[20]  R. Rakkiyappan,et al.  Delay-dependent stability criterion for a class of non-linear singular Markovian jump systems with mode-dependent interval time-varying delays , 2012 .

[21]  Liqun Zhou.  Global asymptotic stability of cellular neural networks with proportional delays , 2014 .

[22]  Qiankun Song,et al.  Exponential stability of recurrent neural networks with both time-varying delays and general activation functions via LMI approach , 2008, Neurocomputing.

[23]  Ting Wang,et al.  Delay-Derivative-Dependent Stability for Delayed Neural Networks With Unbound Distributed Delay , 2010, IEEE Transactions on Neural Networks.

[24]  Cheng-Zhong Xu,et al.  A robust packet scheduling algorithm for proportional delay differentiation services , 2004, IEEE Global Telecommunications Conference, 2004. GLOBECOM '04..

[25]  Le Van Hien,et al.  Finite-time stability of a class of non-autonomous neural networks with heterogeneous proportional delays , 2015, Appl. Math. Comput..

[26]  Liqun Zhou,et al.  Global exponential periodicity and stability of recurrent neural networks with multi-proportional delays. , 2016, ISA transactions.

[27]  Zidong Wang,et al.  Stability and Synchronization of Discrete-Time Markovian Jumping Neural Networks With Mixed Mode-Dependent Time Delays , 2009, IEEE Transactions on Neural Networks.

[28]  Ewing L. Lusk,et al.  Experiments with resolution-based theorem-proving algorithms , 1982 .

[29]  Pagavathigounder Balasubramaniam,et al.  Stochastic stability of uncertain fuzzy recurrent neural networks with Markovian jumping parameters , 2011, Int. J. Comput. Math..

[30]  Zidong Wang,et al.  A delay-dependent LMI approach to dynamics analysis of discrete-time recurrent neural networks with time-varying delays , 2007 .

[31]  Shouming Zhong,et al.  Delay-dependent exponential stability for Markovian jumping stochastic Cohen-Grossberg neural networks with p-Laplace diffusion and partially known transition rates via a differential inequality , 2013, Advances in Difference Equations.

[32]  Pagavathigounder Balasubramaniam,et al.  Synchronization of recurrent neural networks with mixed time-delays via output coupling with delayed feedback , 2012, Nonlinear Dynamics.

[33]  Ke Wang,et al.  Stability in distribution of neutral stochastic functional differential equations with Markovian switching , 2012 .

[34]  Yunkang Liu,et al.  Asymptotic behaviour of functional-differential equations with proportional time delays , 1996, European Journal of Applied Mathematics.

[35]  Jinde Cao,et al.  Matrix measure based stability criteria for high-order neural networks with proportional delay , 2015, Neurocomputing.

[36]  Quanxin Zhu,et al.  Stochastically asymptotic stability of delayed recurrent neural networks with both Markovian jump parameters and nonlinear disturbances , 2010, J. Frankl. Inst..

[37]  Linshan Wang,et al.  Stochastic exponential stability of the delayed reaction–diffusion recurrent neural networks with Markovian jumping parameters , 2008 .

[38]  Daniel W. C. Ho,et al.  State estimation for delayed neural networks , 2005, IEEE Transactions on Neural Networks.

[39]  Yixian Yang,et al.  Asymptotic stability of cellular neural networks with multiple proportional delays , 2014, Appl. Math. Comput..

[40]  Liqun Zhou,et al.  Novel global exponential stability criteria for hybrid BAM neural networks with proportional delays , 2015, Neurocomputing.

[41]  Liqun Zhou.  Delay-Dependent Exponential Synchronization of Recurrent Neural Networks with Multiple Proportional Delays , 2014, Neural Processing Letters.

[42]  Krishnan Balachandran,et al.  Global exponential stability of neutral-type impulsive neural networks with discrete and distributed delays , 2010 .

[43]  Rajendran Samidurai,et al.  Dissipativity of discrete-time BAM stochastic neural networks with Markovian switching and impulses , 2013, J. Frankl. Inst..

[44]  Manchun Tan,et al.  Global Asymptotic Stability of Fuzzy Cellular Neural Networks with Unbounded Distributed Delays , 2010, Neural Processing Letters.