Mittag–Leffler and asymptotic adaptive projective synchronization of fractional inertial neural networks in quaternion field

Abstract

This paper investigates the problems of Mittag–Leffler projective synchronization (MLPS) and asymptotic adaptive projective synchronization (AAPS) in fractional quaternion-valued inertial neural networks (FQVINNs) subject to parametric uncertainties. To facilitate the analysis, the original FQVINN model is reformulated into an equivalent fractional system through an appropriate variable transformation. Two synchronization strategies are proposed: a quaternion-valued feedback controller is designed to realize MLPS, while a fractional adaptive controller is developed to achieve AAPS. By employing tools from fractional differential inequality theory and Lyapunov stability analysis, sufficient conditions for the synchronization of FQVINNs are rigorously established. The effectiveness of the proposed control schemes is demonstrated through a numerical example, which confirms the theoretical predictions and highlights the practical applicability of the methods.