Adaptive Gradient Neural Networks for Solving the Time-Varying Sylvester Equation

This paper develops several new dynamical designs, based on the gradient neural network (GNN), from the perspective of control theory to solve the time-varying Sylvester equation (TVSE). We start with an adaptive gradient neural network called AGNN-S. Then, based on Lyapunov theory, we propose three improved models: ACGNN, AIGNN, and ABGNN. Among them, the ABGNN model stands out for its fastest convergence speed and strongest robustness to noise. Theoretical analysis confirms that all proposed models solve the TVSE effectively, with ACGNN, AIGNN, and ABGNN converging faster than AGNN-S. Theoretical analysis shows that using certain nonlinear activation functions can further boost convergence speed. A robustness analysis indicates that the AGNN-S, ACGNN, and ABGNN models maintain stable convergence even in the presence of differentiation or model-implementation errors. Comparisons with the classical zeroing neural network (ZNN) and other six state-of-the-art GNN- and ZNN-type models demonstrate the superior accuracy and efficiency of our approaches, especially the ABGNN model. Numerical experiments notably highlight ABGNN’s advantages over other models under certain parameter setting. Finally, we showcase applications in time-varying quadratic programming and robotic arm trajectory tracking to verify the practical value of the models. Note to Practitioners—This paper targets practitioners seeking efficient and robust neural network-based solvers for time-varying matrix equations, particularly the time-varying Sylvester equation, which commonly arises in adaptive control, signal processing, and dynamic system modeling. Classical methods, such as the GNN, may suffer from limited solution accuracy. The proposed models, referred to as AGNN-S, ACGNN, AIGNN, and ABGNN, are designed for rapid convergence and strong noise resistance under time-varying conditions. Among them, ABGNN exhibits the fastest convergence and strongest robustness, making it particularly suitable for deployment in real-world engineering systems with uncertain or fluctuating dynamics for the goal of good performance. These models are relevant for hardware implementation where stability and speed are critical. Additionally, the adaptability of these neural networks extends to broader applications, such as solving constrained optimization problems and robot mobile arm tracking.