On delay-dependent robust exponential stability of stochastic neural networks with mixed time delays and Markovian switching

Abstract This paper deals with the global exponential stability analysis problem for a general class of uncertain stochastic neural networks with mixed time delays and Markovian switching. The mixed time delays under consideration comprise both the discrete time-varying delays and the distributed time-delays. The main purpose of this paper is to establish easily verifiable conditions under which the delayed stochastic neural network is robustly exponentially stable in the mean square in the presence of parameters uncertainties, mixed time delays, and Markovian switching. By employing new Lyapunov–Krasovskii functionals and conducting stochastic analysis, a linear matrix inequality (LMI) approach is developed to derive the criteria for the robust exponential stability, which can be readily checked by using some standard numerical packages such as the Matlab LMI Toolbox. The criteria derived are dependent on both the discrete time delay and distributed time delay, and, are therefore, less conservative. A simple example is provided to demonstrate the effectiveness and applicability of the proposed testing criteria.

[1]  H. Unbehauen,et al.  Robust H2/H∞-state estimation for discrete-time systems with error variance constraints , 1997, IEEE Trans. Autom. Control..

[2]  R. Khasminskii.  Stochastic Stability of Differential Equations , 1980 .

[3]  James Lam,et al.  Stochastic stability analysis of fuzzy hopfield neural networks with time-varying delays , 2005, IEEE Transactions on Circuits and Systems II: Express Briefs.

[4]  Jinde Cao,et al.  Stability in Cohen–Grossberg-type bidirectional associative memory neural networks with time-varying delays , 2006 .

[5]  Heinz Unbehauen,et al.  Robust H2/H∞-state estimation for systems with error variance constraints: the continuous-time case , 1999, IEEE Trans. Autom. Control..

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

[7]  Guanrong Chen,et al.  New criteria for synchronization stability of general complex dynamical networks with coupling delays , 2006 .

[8]  Huijun Gao,et al.  Parameter-dependent robust stability of uncertain time-delay systems , 2007 .

[9]  Maozhen Li,et al.  Stability analysis for stochastic Cohen-Grossberg neural networks with mixed time delays , 2006, IEEE Transactions on Neural Networks.

[10]  Peng Shi,et al.  New global asymptotic stability criterion for neural networks with discrete and distributed delays , 2007 .

[11]  Huijun Gao,et al.  New Results on Stability of Discrete-Time Systems With Time-Varying State Delay , 2007, IEEE Transactions on Automatic Control.

[12]  H. Chizeck,et al.  Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control , 1990 .

[13]  Hongyong Zhao,et al.  Global exponential stability and periodicity of cellular neural networks with variable delays , 2005 .

[14]  Jinde Cao,et al.  Periodic solutions and its exponential stability of reaction–diffusion recurrent neural networks with continuously distributed delays , 2006 .

[15]  Jinde Cao,et al.  Global robust stability of delayed recurrent neural networks , 2004 .

[16]  Yan Shi,et al.  Resilient observer-based control of uncertain time-delay systems , 2007 .

[17]  Peter Tiño,et al.  Markovian architectural bias of recurrent neural networks , 2004, IEEE Transactions on Neural Networks.

[18]  Wei Wang,et al.  TWO DESIGN SCHEMES FOR ROBUST ADAPTIVE CONTROL OF A CLASS OF LINEAR UNCERTAIN NEUTRAL DELAY SYSTEMS , 2007 .

[19]  Mike Casey,et al.  The Dynamics of Discrete-Time Computation, with Application to Recurrent Neural Networks and Finite State Machine Extraction , 1996, Neural Computation.

[20]  S. Arik.  Stability analysis of delayed neural networks , 2000 .

[21]  Q. Song,et al.  Global exponential stability of BAM neural networks with distributed delays and reaction–diffusion terms , 2005 .

[22]  Hongyong Zhao,et al.  Existence of periodic oscillatory solution of reaction-diffusion neural networks with delays [rapid communication] , 2005 .

[23]  Yuanqing Xia,et al.  On designing of sliding-mode control for stochastic jump systems , 2006, IEEE Transactions on Automatic Control.

[24]  A. Skorokhod.  Asymptotic Methods in the Theory of Stochastic Differential Equations , 2008 .

[25]  Tianping Chen,et al.  Robust global exponential stability of Cohen-Grossberg neural networks with time delays , 2004, IEEE Transactions on Neural Networks.

[26]  K. Gu.  An integral inequality in the stability problem of time-delay systems , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[27]  Zidong Wang,et al.  On global asymptotic stability of neural networks with discrete and distributed delays , 2005 .

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

[29]  E. Boukas,et al.  On Stochastic Stabilization of Discrete‐Time Markovian Jump Systems with Delay in State , 2003 .

[30]  H. B. Huang,et al.  Robust stability analysis of switched Hopfield neural networks with time-varying delay under uncertainty , 2005 .

[31]  Wenlian Lu,et al.  Dynamical Behaviors of a Large Class of General Delayed Neural Networks , 2005, Neural Computation.