Stability Of Hopfield Neural Networks

This paper focuses on retarded Hopfield neural networks and studies the exponential stability of them. The retardant is characteristic of the neural networks, which makes the study more practical. The reason why neural networks are continuously paid much attention is because they have the abilities of nonlinear reflection, adjustment, and remembrance. They are also significant because of their method for dealing with parallel information and their high tolerance for any mistake. When people use neural networks, they can always expect fast global convergence, wide reflection and differentiation, and less cost. Therefore, it is very useful to try to decrease the additive requirements and apply the flexibility of neural networks and their according machines to the programs. We studied this in two ways briefly. One is to study continuous-time Hopfield networks. They imitate the output constants from the neurons with the parallel connection of the resistance Ri and the capacitance C_i. Moreover, connector T_ij imitates the character of neurons connection, and the calculative enlarger imitates the nonlinear of neurons. Given u_i is the ith input of neurons, v_i is the ith output of them accordingly. With the connection of n neurons, the differential equation of the neural networks is shown as follows: where g_i is the one-dimension nonlinear retarded function of neurons. In the proof, this paper studied the nonlinear retarded differential equation, used disproof, parameters alteration and the technique of inequality analysis. Consequently, we got the theorems about the k ?global exponential stability of the continuous-time retarded Hopfield neural networks. Seeing the requirements of the proof, we can easily find that the presupposition of (T_i)_jn×n symmetry has been cancelled. Meanwhile, the conclusions are made by more than one approximation, which makes some progress for solving the problems of the nonlinear neural networks and the unstable control systems. It is very useful for any coming study of...