Stability and stabilization of quaternion-valued neural networks with uncertain time-delayed impulses: Direct quaternion method

Abstract

In this paper, the exponential stability of impulsive quaternion-valued neural networks (IQVNNs) with time delays is investigated. Both the differential equation and impulses are affected by time-varying delays which may be nonidentical. Uncertain parameters in impulses are also considered. Firstly, the existence and uniqueness of the equilibrium of the IQVNNs are discussed with the help of homeomorphic mapping theorem. Then, by utilizing Lyapunov functions and a newly developed inequality, several sufficient criteria are established in the form of quaternion-valued linear matrix inequalities (LMIs). It should be noted that, compared to most of existing results which are obtained by decomposing quaternion-valued neural networks (QVNNs) into four real-valued neural networks (RVNNs) or two complex-valued neural networks (CVNNs), our results are easier to be verified since the quaternion-valued LMIs exhibit lower dimensions. Then, based on obtained results, one impulsive controller is designed to realize the stabilization of delayed QVNNs. Finally, two numerical examples are presented to verify the effectiveness and merits of the theoretical analysis.