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Zbl 1078.34052
Liao, Xiaofeng; Li, Chuandong
An LMI approach to asymptotical stability of multi-delayed neural networks. (English)
[J] Physica D 200, No. 1-2, 139-155 (2005). ISSN 0167-2789

The authors consider the following neural network with multiple delays $$ \frac{du(t)}{dt}=-Au(t)+W^{(0)}g(u(t))+\sum_{k=1}^{r}W^{(k)}g(u(t-\tau _{k}))+I, $$ where $u(t)=[u_{1}(t),\dots ,u_{n}(t)]^{T}$ is the neuron state vector, $A=\text{diag}(a_{1},\dots ,a_{n})$ is a positive diagonal matrix, $W^{(k)}=(w_{ij}^{(k)}) _{n\times n},$ $k=0,\dots ,r$, are the interconnection matrices, $g(u)=[g_{1}(u_{1}), \dots ,g_{n}(u_{n})]^{T}$ denotes the neuron activation with $g(0)=0$ and $I=[I_{1},\dots,I_{n}]^{T}$ is a constant input vector, while $\tau _{k}>0,$ $k=1,\dots ,r,$ being the delay parameters. The authors present sufficient conditions for the origin to be asymptotically stable by constructing a Lyapunov-Krasovskii functional for the cases where the time delays are constants or are time-varying, respectively. The main results are generalizations of some results reported in the literature. Three examples are given in this paper.
[Takeshi Taniguchi (Kurume)]

MSC 2000:: *34K20 Stability theory of functional-differential equations
92B20 General theory of neural networks

Keywords: Neural networks; Global asymptotic stability; Lyapunov-Krasovskii functional; Linear matrix inequality; Multiple time delays

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