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Zbl 1196.60125
Sun, Yonghui; Cao, Jinde
$p$th moment exponential stability of stochastic recurrent neural networks with time-varying delays. (English)
[J] Nonlinear Anal., Real World Appl. 8, No. 4, 1171-1185 (2007). ISSN 1468-1218

The authors discuss the $p$th moment exponential stability of zero solution to the generalized stochastically perturbed neural network model with time-varying defined by the state equation: $$\aligned dx_{i}(t) =\left[ -c_{i}x_{i}(t)+\sum_{j=1}^{n}a_{ij}f_{j}(x_{j}(t))+ \sum_{j=1}^{n}b_{ij}g_{j}(x_{j}(t-\tau _{j}(t)))\right] dt \\ \left. +\sum_{j=1}^{n}\sigma _{ij}(t,x_{j}(t),x_{j}(t-\tau _{j}(t)))d\omega _{j}(t),\ i=1,2,,,,n,\right.\endaligned$$ where $\tau _{j}(t)>0$ is the transmission delay and $\omega (t)$ denotes an $n-$dimensional Brownian motion on a complete probability space. The following assumptions are given: \par (1) $\tau _{j}(t)$ is a differentiable function with a constant $\alpha >0$ such that $\frac{d}{dt}\tau _{j}(t)\leq \alpha <1$, \par (2) \ $f_{j},g_{j}$ satisfy the Lipschitz condition,\par (3) $f(0)=g(0)=\sigma (t,0,0)=0$, $$\text{trace}\left[ \sigma ^{T}(t,x,y)\sigma (t,x,y)\right] \leq \sum_{i=1}^{n}\left( \mu _{i}x_{i}^{2}+\nu _{i}y_{i}^{2}\right) ,\ \ (t,x,y)\in R\times R^{n}\times R^{n}.\tag{4}$$ It is known that under the above conditions there exists a unique global solution to the above state equation. Thus, the authors discuss the $p$th moment exponential stability of zero solution. The main theorem is the following. \par Theorem. If $\rho \left[ C^{-1}(MM_{1}K+MM_{2}K+NN_{1}+NN_{2})\right] \leq 1,\ $ then the zero solution to the above system is $p$th moment exponentially stable. \par This theorem is a generalization of the theorem on exponential stability proven by {\it L. Wan} and {\it J. Sun} [Phys. Lett., 342, No. 4, 331--340 (2005; Zbl 1222.93200)]. \par The authors also give two numerical examples.
[Takeshi Taniguchi (Kurume)]

MSC 2000:: *60H30 Appl. of stochastic analysis
68T05 Learning and adaptive systems
92B20 General theory of neural networks

Keywords: stochastic recurrent neural networks; $p$th moment exponential stability; time-varying delays; Burkholder-Davis-Gundy inequality

Citations: Zbl 1222.93200

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