Dynamic analysis of stochastic Cohen-Grossberg neural networks with time delays. (English) Zbl 1117.34080
The authors consider the following stochastic Cohen-Grossberg neural networks with time delays:
\[ dx(t)=-D(x(t))[C(x(t))-AF(x(t))-BF(x_{\tau }(t))]dt+\sigma (x(t))d\omega (t),\;t\geq 0, \tag{*} \]
where \(A=(a_{ij})_{n\times n},\;B=(b_{ij})_{n\times n},\;\sigma (x(t))=(\sigma _{ij}(x(t))_{n\times n},\) \(\varphi (t) =(\varphi _{1}(t),,,\varphi _{n}(t))^{T}\), \(D(x(t)) =\text{diag}(d_{1}(x_{1}(t)),,,d_{n}(x_{n}(t))),\) \(C(x(t)) =(c_{1}(x_{1}(t)),,,c_{n}(x_{n}(t)))\), \[ F(x(t)) =(f_{1}(x_{1}(t)),,,f_{n}(x_{n}(t))), \quad F(x_{\tau }(t)) =(f_{1}(x_{1}(t-\tau _{1})),,,f_{n}(x_{n}(t-\tau _{n}))). \]
By constructing a suitable Lyapunov functional and employing the semimartingale convergence theorem, they prove that if \(f_{i},\sigma _{ij}\) are globally Lipschitz and some conditions are satisfied, then the equilibrium point \(x^{\ast }\) to the system (*) is almost surely exponentially stable.
\[ dx(t)=-D(x(t))[C(x(t))-AF(x(t))-BF(x_{\tau }(t))]dt+\sigma (x(t))d\omega (t),\;t\geq 0, \tag{*} \]
where \(A=(a_{ij})_{n\times n},\;B=(b_{ij})_{n\times n},\;\sigma (x(t))=(\sigma _{ij}(x(t))_{n\times n},\) \(\varphi (t) =(\varphi _{1}(t),,,\varphi _{n}(t))^{T}\), \(D(x(t)) =\text{diag}(d_{1}(x_{1}(t)),,,d_{n}(x_{n}(t))),\) \(C(x(t)) =(c_{1}(x_{1}(t)),,,c_{n}(x_{n}(t)))\), \[ F(x(t)) =(f_{1}(x_{1}(t)),,,f_{n}(x_{n}(t))), \quad F(x_{\tau }(t)) =(f_{1}(x_{1}(t-\tau _{1})),,,f_{n}(x_{n}(t-\tau _{n}))). \]
By constructing a suitable Lyapunov functional and employing the semimartingale convergence theorem, they prove that if \(f_{i},\sigma _{ij}\) are globally Lipschitz and some conditions are satisfied, then the equilibrium point \(x^{\ast }\) to the system (*) is almost surely exponentially stable.
Reviewer: Takeshi Taniguchi (Kurume)
MSC:
| 34K50 | Stochastic functional-differential equations |
| 92B20 | Neural networks for/in biological studies, artificial life and related topics |
| 34K20 | Stability theory of functional-differential equations |
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