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 funct ions, guarantee the existence of a unique equilibrium under a synchron ous mode of operation as well as a class of asynchronous modes, For th e synchronous mode, these assumptions are also shown to imply local as ymptotic stability of the equilibrium, For the asynchronous mode of op eration, two results are derived, First, it Is shown that symmetry and stability of the interconnection matrix guarantee local asymptotic st ability of the equilibrium under a class of asynchronous modes-this is referred to as local absolute asymptotic stability, Second, it is sho wn that, under the standard assumptions, if the nonnegative matrix who se elements are the absolute values of the corresponding elements of t he 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 ap plications, robustness, and their relationship to earlier results.