On Popov-type stability criteria for neural networks.

*(English)*Zbl 0971.93067There is no general solution to the stability of neural networks problem. The point is that neural networks are strongly nonlinear dynamical systems with a potentiality for qualitatively different kinds of instabilities. The presented note concerns the case of Hopfield-type neural networks. In this case some improvement of stability criteria for continuous-time neural networks is presented. But the conditions are limited to nonlinear transfer functions which are bounded and slope restricted. In such a case the more general Lyapunov function, even quite different in comparison with the usual energy function, is constructed by means of solving Lur’e-type equations. Then the existence of solutions for such equations is ensured by a Popov-type frequency domain inequality. In other words, the neural networks’ stability analysis is in this case embedded into the framework of the qualitative theory of systems with several equilibria.

Reviewer: Ladislav Andrey (Praha)

##### MSC:

93D20 | Asymptotic stability in control theory |

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

93D30 | Lyapunov and storage functions |

93D10 | Popov-type stability of feedback systems |