## $$p$$th moment exponential stability of stochastic recurrent neural networks with time-varying delays.(English)Zbl 1196.60125

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: \begin{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.\end{aligned} 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:
(1) $$\tau _{j}(t)$$ is a differentiable function with a constant $$\alpha >0$$ such that $$\frac{d}{dt}\tau _{j}(t)\leq \alpha <1$$,
(2) $$f_{j},g_{j}$$ satisfy the Lipschitz condition,
(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.
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.
This theorem is a generalization of the theorem on exponential stability proven by L. Wan and J. Sun [Phys. Lett., 342, No. 4, 331–340 (2005; Zbl 1222.93200)].
The authors also give two numerical examples.

### MSC:

 60H30 Applications of stochastic analysis (to PDEs, etc.) 68T05 Learning and adaptive systems in artificial intelligence 92B20 Neural networks for/in biological studies, artificial life and related topics

Zbl 1222.93200
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