Existence and Stability of a Unique Equilibrium in Continuous-Valued Discrete-Time Asynchronous Hopfield Neural Networks

It is shown that the assumption of D-stability of the interconnection matrix, together with the standard assumptions on the activation functions, guarantee the existence of a unique equilibrium under a synchronous mode of operation as well as a class of asynchronous modes. For the synchronous mode, these assumptions are also shown to imply local asymptotic stability of the equilibrium. For the asynchronous mode of operation, two results are derived. First, it is shown that symmetry and stability of the interconnection matrix guarantee local asymptotic stability of the equilibrium under a class of asynchronous modes-this is referred to as local absolute asymptotic stability. Second, it is shown that, under the standard assumptions, if the nonnegative matrix whose elements are the absolute values of the corresponding elements of the interconnection matrix is stable, then the equilibrium is globally absolutely asymptotically stable under a class of asynchronous modes. The results obtained are discussed from the points of view of their applications, robustness, and their relationship to earlier results.