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pth moment exponential stability of stochastic recurrent neural networks with time-varying delays. (English) Zbl 1196.60125

The authors discuss the pth moment exponential stability of zero solution to the generalized stochastically perturbed neural network model with time-varying defined by the state equation:

dx i (t)=-c i x i (t)+ j=1 n a ij f j (x j (t))+ j=1 n b ij g j (x j (t-τ j (t)))dt+ j=1 n σ ij (t,x j (t),x j (t-τ j (t)))dω j (t),i=1,2,,,,n,

where τ j (t)>0 is the transmission delay and ω(t) denotes an n-dimensional Brownian motion on a complete probability space. The following assumptions are given:

(1) τ j (t) is a differentiable function with a constant α>0 such that d dtτ j (t)α<1,

(2) f j ,g j satisfy the Lipschitz condition,

(3) f(0)=g(0)=σ(t,0,0)=0,

traceσ T (t,x,y)σ(t,x,y) i=1 n μ i x i 2 +ν i y i 2 ,(t,x,y)R×R n ×R n ·(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 pth moment exponential stability of zero solution. The main theorem is the following.

Theorem. If ρC -1 (MM 1 K+MM 2 K+NN 1 +NN 2 )1, then the zero solution to the above system is pth 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.

60H30Applications of stochastic analysis
68T05Learning and adaptive systems
92B20General theory of neural networks (mathematical biology)