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.