Dynamic Approaches for Finding Least Squares Solution of Time-Varying Multi-Linear Systems

The primary focus of this study is to estimate least squares solutions for time-varying multi-linear systems (TVMLS) that involve time-varying (TV) perturbations on the right-hand side. To address this issue, we introduce two neural network models: the hybrid neural network (HNN) and a nonlinear enhancement of the HNN design that uses a modified weighted sign-bi-power (Mwsbp) activation function, referred to as MwsbpHNN. Theoretical findings demonstrate the convergence of both the HNN and MwsbpHNN dynamic models to the exact solutions of underlying TVMLS. Furthermore, we rigorously prove a fixed-time convergence of MwsbpHNN and provide upper bounds for the convergence time. Computational complexity of our theoretical framework is compared with current state-of-the-art methods for solving TVMLS, highlighting its efficiency. Numerical simulation further confirms the robustness of the proposed HNN and MwsbpHNN models. A model arising from discretized high-order Bellman equation is considered. Note to Practitioners—This research develops neural network models designed to compute least squares solutions of time-varying multi-linear systems (TVMLS) with time-varying perturbations on the right-hand side. A comprehensive convergence analysis reveals a predefined time convergence to a least squares solution of the TVMLS by the hybrid neural network (HNN) that is activated by the modified weighted sign-bi-power (Mwsbp) function. Additionally, a range of numerical simulations validate theoretical findings. We also provide rigorous theoretical and comparative simulation results to showcase the performance of the proposed models. Additionally, we analyze a model derived from a discretized high-order Bellman equation, serving as a fundamental theoretical framework in areas such as economics, engineering control theory, and reinforcement learning.