Complex-valued hybrid-order neural network: A novel noise-resilient solving strategy for dynamic generalized Lyapunov...

Lyapunov equation has been extensively investigated for its fundamental role in many research fields, such as image processing and stability analysis. While its standard form has been researched maturely, less attention has been focused on the much more general form, i.e., complex-valued dynamic generalized Lyapunov equation (CV-DGLE). In view of this, complex-valued hybrid-order neural network (CV-HBONN), a novel general neural solver, is designed for the first time to solve CV-DGLE. Unlike conventional approaches that transform the CV-DGLE into a standard form and solve it indirectly, a process that inherently involves time-consuming mathematical transformations, CV-HBONN tackles the equation directly. The proposed model integrates a fractional-power term, an integral term, and a cumulative hyper-power term, which respectively provide finite-time convergence, resistance to external noise, and accelerated error decay. In addition, nonlinear activation methods further improve the performance of the model. Rigorous theoretical analyses prove the convergence and noise-resistance properties of CV-HBONN. For comparison, representative ZNN models are extended to the complex domain as CV-HONN, CV-PIDZNN1, and CV-PIDZNN2. Numerical results show that CV-HBONN converges faster than these models, maintains residual errors at the 10^-3 level under constant noise, and produces smaller errors under random and linearly increasing noise.