Gradient-based neural networks for solving periodic Sylvester matrix equations

• Published in: Journal of the Franklin Institute Vol. 359; no. 18; pp. 10849 - 10866
• Main Authors: Lv, Lingling, Chen, Jinbo, Zhang, Lei, Zhang, Fengrui
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.12.2022
• ISSN: 0016-0032; 1879-2693
• DOI: 10.1016/j.jfranklin.2022.05.023

This paper considers neural network solutions of a category of matrix equation called periodic Sylvester matrix equation (PSME), which appear in the process of periodic system analysis and design. A linear gradient-based neural network (GNN) model aimed at solving the PSME is constructed, whose state is able to converge to the unknown matrix of the equation. In order to obtain a better convergence effect, the linear GNN model is extended to a nonlinear form through the intervention of appropriate activation functions, and its convergence is proved through theoretical derivation. Furthermore, the different convergence effects presented by the model with various activation functions are also explored and analyzed, for instance, the global exponential convergence and the global finite time convergence can be realized. Finally, the numerical examples are used to confirm the validity of the proposed GNN model for solving the PSME considered in this paper as well as the superiority in terms of the convergence effect presented by the model with different activation functions.