A critical analysis on global convergence of Hopfield-type neural networks

This paper deals with the global convergence and stability of the Hopfield-type neural networks under the critical condition that M₁(Γ)=L^-1DΓ-(ΓW+W^TΓ)/(2) is nonnegative for any diagonal matrix Γ, where W is the weight matrix of the network, L=diag{L₁,L₂,...,L_N} with L_i being the Lipschitz constant of g_i and G(u)=(g₁(u₁),g₂(u₂),...,g_N(u_N))^T is the activation mapping of the network. Many stability results have been obtained for the Hopfield-type neural networks in the noncritical case that M₁(Γ) is positive definite for some positive definite diagonal matrix Γ. However, very few results are available on the global convergence and stability of the networks in the critical case. In this paper, by exploring two intrinsic features of the activation mapping, two generic global convergence results are established in the critical case for the Hopfield-type neural networks, which extend most of the previously known globally asymptotic stability criteria to the critical case. The results obtained discriminate the critical dynamics of the networks, and can be applied directly to a group of Hopfield-type neural network models. An example has also been presented to demonstrate both theoretical importance and practical significance of the critical results obtained.