Multistability of Fuzzy Neural Networks With a General Class of Activation Functions and State-Dependent Switching Rules
Shiqin Ou, Zhenyuan Guo, Jun Wang
Abstract
This paper addresses the multistability of switched fuzzy neural networks with a general class of activation functions under state-dependent switching. The existence, stability, and attraction basins of equilibria are analyzed via state-space decomposition based on Brouwer fixed point theorem and M-matrix properties. It is shown that there exist <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$5^{k_{1}}3^{k_{2}}$</tex-math></inline-formula> equilibria, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$3^{k_{1}}2^{k_{2}}$</tex-math></inline-formula> of them are locally exponentially stable under four sets of sufficient conditions for an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -neuron switched network, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k_{1}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k_{2}$</tex-math></inline-formula> are nonnegative integers such that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$0< k_{1}+k_{2}\leq n$</tex-math></inline-formula> . The results reveal that the switched fuzzy neural networks have much more equilibria than conventional fuzzy neural networks. Four numerical examples with simulation results are discussed to substantiate the theoretical results.