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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
\[ \frac{dx_i(t)}{dt}=-a_ix_i(t)+\sum_{j=1}^m b_{ij}f_j(x_j(t))+c_i,~~t>t_0,~t\neq t_k, \]
\[ x(t_0+)=x_0\in\mathbb{R}^m, \]
\[ x_i(t_0+)-x_i(t_k-)=I_k(x_i(t_k-)),~~i=1,2,\dots,m,~~~k=1,2,\dots,~~t_k\rightarrow\infty, \]
and its discrete analog \[ y_i(n+1)=\frac{1}{1+a_ih}y_i(n)+\frac{h}{1+a_ih}\sum_{j=1}^m b_{ij}f_j(y_j(n)),~~i=1,2,\dots,m,~~ n>n_0, \]
\[ y_j(n_j+1)-y_j(n_j)=I_j(y_j(n_j)), ~~j=1,2,\dots. \] By constructing Lyapunov functions the author obtains explicit stability conditions for this equilibrium. A comparison with known results for nonimpulsive systems is presented.

MSC:

34A37 Ordinary differential equations with impulses
34D20 Stability of solutions to ordinary differential equations
39A12 Discrete version of topics in analysis
34C60 Qualitative investigation and simulation of ordinary differential equation models
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