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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.

\[ \frac{dx_i(t)}{dt}=-a_ix_i(t)+\sum_{j=1}^m b_{ij}f_j(x_j(t))+c_i,~~t>t_0,~t\neq t_k, \]

\[ x(t_0+)=x_0\in\mathbb{R}^m, \]

\[ x_i(t_0+)-x_i(t_k-)=I_k(x_i(t_k-)),~~i=1,2,\dots,m,~~~k=1,2,\dots,~~t_k\rightarrow\infty, \]

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.

Reviewer: Leonid Berezanski (Beer-Sheva)

### 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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\textit{K. Gopalsamy}, Appl. Math. Comput. 154, No. 3, 783--813 (2004; Zbl 1058.34008)

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