Design, analysis and verification of recurrent neural dynamics for handling time-variant augmented Sylvester linear system

Augmented Sylvester linear system (ASLS) is one of the most important issues in various science and engineering fields. In this study, two recurrent neural dynamics (RND) methods in a continuous-time manner (termed as CTRND) and a discrete-time manner (termed as DTRND) are proposed for handling the continuous-form time-variant ASLS (CF-TV-ASLS) and discrete-form time-variant ASLS (DF-TV-ASLS), respectively. Specifically, first of all, aided with the Kronecker product and vectorization techniques, the CF-TV-ASLS is finally transformed into a continuous-form time-variant matrix-vector equation (CF-TV-MVE) by introducing an additional time-variant nonnegative variable. Analogously, the corresponding DF-TV-ASLS is transformed into a discrete-form time-variant matrix-vector equation (DF-TV-MVE). Whereafter, by exploiting the RND design formula, the CTRND method and DTRND method are proposed and investigated for solving obtained CF-TV-MVE and DF-TV-MVE, respectively. In addition, theoretical analyses about the convergence of CTRND method and DTRND method are presented. Finally, the instructive experiments, including a continuous-time example and a corresponding discrete-time one, substantiate the efficacy and superiority of the proposed CTRND method and DTRND method.