Direct approach-based synchronization of fully quaternion-valued neural networks with inertial term and time-varying delay

In this article, the global exponential and asymptotic synchronization for a type of quaternion-valued inertial delayed neural networks (QIDNNs) are explored by introducing a direct analysis approach. Above all, without reducing the order or decomposing the models of QIDNNs, a compact quaternion-valued control scheme is developed to ensure synchronization. In addition, two new Lyapunov functionals are constructed directly and several compact criteria represented by matrix inequalities are derived by means of the theory of quaternion and inequality technique. Furthermore, a quaternion-valued adaptive control law is proposed to automatically determine the control costs, and the synchronization is guaranteed by the aid of Barbalat Lemma. Lastly, the derived results are affirmed via a numerical example.