In this article, the finite-time synchronization (FTSYN) of a class of quaternion-valued neural networks (QVNNs) with discrete and distributed time delays is studied. Furthermore, the FTSYN and fixed-time synchronization (FIXSYN) of the QVNNs without time delay are investigated. Different from the existing results, which used decomposition techniques, by introducing an improved one-norm, we use a direct analytical method to study the synchronization problems. Incidentally, several properties of one-norm of the quaternion are analyzed, and then, three effective controllers are proposed to synchronize the drive and response QVNNs within a finite time or fixed time. Moreover, efficient criteria are proposed to guarantee that the synchronization of QVNNs with or without mixed time delays can be realized within a finite and fixed time interval, respectively. In addition, the settling times are reckoned. Compared with the existing work, our advantages are mainly reflected in the simpler Lyapunov analytical process and more general activation function. Finally, the validity and practicability of the conclusions are illustrated via four numerical examples.

This paper investigates the stability of a class of quaternion-valued neural networks (QVNNs) with discrete and distributed delays. By decomposing the QVNN and forming an equivalent real-valued vector-matrix differential equation (RVVDE), based on the Lyapunov theory and some matrix inequalities, some sufficient conditions are derived to ensure the existence and uniqueness, the globally exponential stability and the globally power stability of the equilibrium of RVVDE. These conditions also apply to the QVNN. Two numerical examples are given to show the advantage and the effectiveness of the main results.

This study is concerned with the problem of finite-time state estimation for T-S fuzzy stochastic jumping neural networks, where the communication links between the stochastic jumping neural networks and its estimator are imperfect. By introducing the fuzzy technique, both the nonlinearities and the stochastic disturbances are represented by T-S model. Stochastic variables subject to the Bernoulli white sequences are employed to determine the nonlinearities occurring in different sector bounds. Some sufficient conditions for the existence of the state estimator are given in terms of linear matrix inequalities, whose effectiveness are illustrated with the aid of simulation results.