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On Popov-type stability criteria for neural networks. (English) Zbl 0971.93067
There 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.
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
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