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Stability of artificial neural networks with impulses. (English) Zbl 1058.34008

The author studies the existence and uniqueness of the positive equilibrium of the following impulsive problem

dx i (t) dt=-a i x i (t)+ j=1 m b ij f j (x j (t))+c i ,t>t 0 ,tt k ,
x(t 0 +)=x 0 m ,
x i (t 0 +)-x i (t k -)=I k (x i (t k -)),i=1,2,,m,k=1,2,,t k ,

and its discrete analog

y i (n+1)=1 1+a i hy i (n)+h 1+a i h j=1 m b ij f j (y j (n)),i=1,2,,m,n>n 0 ,
y j (n j +1)-y j (n j )=I j (y j (n j )),j=1,2,·

By constructing Lyapunov functions the author obtains explicit stability conditions for this equilibrium. A comparison with known results for nonimpulsive systems is presented.

34A37Differential equations with impulses
34D20Stability of ODE
39A12Discrete version of topics in analysis
34C60Qualitative investigation and simulation of models (ODE)