Exponential stability of artificial neural networks with distributed delays and large impulses.

*(English)*Zbl 1154.34042The authors deal with an equation for an artificial neural network subject to delays and impulse effects. Both delays and impulses can be destabilizing factors of the system. In the first part of the paper, the authors state several results on the existence and uniqueness of the equilibrium state. In the second part of the paper, exponential stability of the equilibrium is investigated. The initial equation is conveniently modified to a new impulsive delayed equation for which is enough to establish the global exponential stability of the trivial equilibrium.

In each part of the article the authors bring out important improvements to the theory of impulsive delayed equations. We mention the contributions concerning the exponential stability of the equilibrium. The authors employ the method of Lyapunov functions combined with the technique of Halanay inequalities and obtain good sufficient conditions governing the network parameters and the magnitude of the impulses. Unlike other results, the magnitude of an impulse is allowed to be exponentially proportional to the size of the corresponding inter-impulse interval, for instance. Some computer simulations are presented.

In each part of the article the authors bring out important improvements to the theory of impulsive delayed equations. We mention the contributions concerning the exponential stability of the equilibrium. The authors employ the method of Lyapunov functions combined with the technique of Halanay inequalities and obtain good sufficient conditions governing the network parameters and the magnitude of the impulses. Unlike other results, the magnitude of an impulse is allowed to be exponentially proportional to the size of the corresponding inter-impulse interval, for instance. Some computer simulations are presented.

Reviewer: Marcia Federson (São Paulo)

##### MSC:

34K45 | Functional-differential equations with impulses |

34K20 | Stability theory of functional-differential equations |

92B20 | Neural networks for/in biological studies, artificial life and related topics |

##### Keywords:

neural networks; distributed delays; impulsive state displacement; Lyapunov functions; Hananay inequalities; exponential stability
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\textit{S. Mohamad} et al., Nonlinear Anal., Real World Appl. 9, No. 3, 872--888 (2008; Zbl 1154.34042)

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