This paper investigates the global polynomial synchronization (GPS) problem for quaternion-valued neural networks (QVNNs) featuring proportional delay, parameter uncertainty, and external disturbance. A combined approach of sliding mode control (SMC) and a non-separation strategy is adopted to achieve this goal. First, an integral-type sliding surface is designed for the system. Then, by constructing a delay-free Lyapunov functional and leveraging the properties of the quaternion vector norm and inequality techniques, sufficient conditions are derived to achieve GPS for the sliding mode dynamics. Furthermore, both a SMC law and an adaptive SMC law are designed, with a reachability analysis confirming that the system trajectories reach the predefined sliding surface in finite time. Finally, numerical examples with graphical analysis are provided to verify the obtained results, along with their application in color image pattern storage and retrieval.
Qu et al. (2026) studied this question.