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Exponential stability in Hopfield-type neural networks with impulses. (English) Zbl 1143.34031

The author considers networks described by \[ {dx_i(t)\over dt}=- a_ix_i(t)+ \sum^m_{j=1} b_{ij}f_i(x_j(t))+c_i,\quad t> t_0,\quad t\neq t_k, \]
\[ \Delta x_i|_{t= t_k}= x_i(t^+_k)- x_i(t^-_k)= I_k(x_i(t^-_k)),\qquad k= 1,2,3,\dots \] and \(t_0< t_1< t_2< t_3<\cdots< t_k\to \infty\) as \(k\to\infty\), \(\Delta t_k= t_k- t_{k-1}\geq \theta\), for \(k= 1,2,3,\dots\). The existence of an exponentially stable unique equilibrium state is proved.

MSC:

34D20 Stability of solutions to ordinary differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics
34A37 Ordinary differential equations with impulses
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