Global exponential stability of nonautonomous neural network models with continuous distributed delays

Abstract

For a family of non-autonomous differential equations with distributed delays, we give sufficient conditions for the global exponential stability of an equilibrium point. This family includes most of the delayed models of neural networks of Hopfield type, with time-varying coefficients and distributed delays. For these models, we establish sufficient conditions for their global exponential stability. The existence and global exponential stability of a periodic solution is also addressed. A comparison of results shows that these results are general, news, and add something new to some earlier publications.

Global exponential stability of nonautonomous neural network models with unbounded delays

“…for which its global exponential stability was studied in [9]. Trivially, hypotheses (A1) and (A2) hold with a i = a i = 1 and D i (t) = 0 respectively, and we have the next result.…”

“…Remark 3.2. We note that Corollary 3.5 improves the exponential stability criterion in [9], because here we have a model with unbounded delays.…”

“…In 1989, Marcus and Westervelt [22] introduced for the first time a discrete delay in the Hopfield model (1.2), and they observed that the delay can destabilize the system. In fact, the delays can affect the dynamic behavior of neural network models [3,22] and, for this reason, stability of delayed neural network models has been investigated extensively (see [1,2,4,6,7,9,11,18,19,20,23,25,26,27,28,29,30], and the references therein). Another relevant fact to take into account is that the neuron charging time, the interconnection weights, and the external inputs often change as time proceeds.…”

“…For neural network models with time-varying coefficients, many authors derive sufficient conditions ensuring that all solutions converge exponentially to zero or to an equilibrium point [9,19,30]. Other authors assume periodic, or almost periodic, coefficient functions and derive sufficient conditions ensuring the existence of a periodic, or almost periodic, solution and its global exponential stability [7,18,20,26,27].…”

Global asymptotic stability of nonautonomous Cohen–Grossberg neural network models with infinite delays

“…In 1989, Marcus and Westervelt [18] introduced for the first time a discrete delay in the Hopfield model (1.2), and they observed that the delay can destabilize the system. In fact, the delays can affect the dynamic behavior of neural network models [1] and, for this reason, stability of delayed neural networks has been investigated extensively ( [2,3,7,8,13,14,16,19,20,23,25], and the references therein). Another relevant fact to take into account is that the neuron charging time, the interconnection weights, and the external inputs often change as time proceeds.…”