Step-width theoretics and numerics of four-point general DTZN model for future minimization using Jury stability criterion.

Future minimization, or say, discrete-time time-varying minimization, can be successfully solved by discrete-time Zhang neuronet (DTZN) models. A DTZN model is obtained through the discretization of a continuous-time Zhang neuronet (CTZN) model. However, most of the existing numerical differentiation formulas cannot be applied to discretizing a CTZN model for obtaining an effective DTZN model. Zhang et al discretization (ZeaD) formulas are the ones that can successfully address the problem about discretization. Recently, the four-point general ZeaD formula has been proposed (Hu et al., 2018), which can be applied to obtaining the four-point general DTZN model for future minimization. This paper focuses on researching the step-width-related stability of the four-point general DTZN model for future minimization. The step-width is a crucial parameter in the DTZN model. If the step-width is out of its corresponding interval, DTZN model does not converge in terms of residual error and fails in solving future minimization problem. By using Jury stability criterion, the analytic solution of step-width interval is proposed corresponding to the four-point general DTZN model. In addition, the optimal value of step-width is investigated, and the analytic solution of the optimal value is proposed as well. When step-width is equal to the optimal value, DTZN model has the fastest convergence rate to steady state with respect to residual error, and is farthest from instable state, thus having the best stability. The theoretical analyses are consistent with previous special-case investigations, and their correctness is confirmed by numerical experiments and results (i.e., numerics).