New discrete-time ZNN models and numerical algorithms derived from a new Zhang function for time-varying linear equations solving

Abstract

Different from the Zhang function (i.e., a type of error function) defined in the authors' previous work, a new Zhang function is proposed, designed and exploited to construct new Zhang neural network (ZNN) models in this paper. Moreover, two discrete-time ZNN models are developed and investigated to solve the problem of time-varying linear equations (TVLE). Such discrete-time ZNN models can exploit methodologically the time derivatives of time-varying coefficients and the theoretical inverse of the time-varying coefficient matrix. When the time-varying coefficient matrix is positive-definite and symmetric, the BFGS quasi-Newton method is introduced to eliminate the explicit matrix-inversion operation. Thus, two other discrete-time ZNN models combined with the BFGS quasi-Newton method (i.e., ZNN-BFGS) are proposed and investigated for TVLE solving. In addition, according to the criterion whether the time-derivative information of time-varying coefficients is explicitly known/used or not, these proposed discrete-time models are investigated in two categories: 1) the models with time-derivative information known (i.e., ZNN-K and ZNN-BFGS-K models); and 2) the models with time-derivative information unknown (i.e., ZNN-U and ZNN-BFGS-U models). Two illustrated examples verify the efficacy of these proposed models for TVLE solving.