A direct analysis method to state bounding for high‐order quaternion Hopfield neural networks with mixed delays and bounded disturbances

Abstract

The main research in this paper is the state bounding problem of high‐order quaternion Hopfield neural networks (HOQHNNs) with time‐varying discrete delays, unbounded distributed delays, and bounded disturbances. In the research process, firstly, the considered HOQHNNs are decomposed into four equivalent real‐valued Hopfield neural networks (RHNNs) to solve the problem that quaternion multiplication is not commutative, and then, a sufficient condition related to time delays is given to ensure that the state trajectories that are contained in or globally exponentially convergent into a cuboid. Thereby, the corresponding reachable set estimation is obtained. Numerical examples are provided to illustrate validity of theoretical results. At present, the research on HOQHNNs with mixed time delays adopts Lyapunov–Krasovskii functional (LKF) method, which needs to not only choose the appropriate LKF but also take into account both conservatism and computational complexity. Besides, there is no literature to study the state bounding of HOQHNNs with mixed delays. The method suggested in the present text has three merits: (1) According to the definition of state bounding straightway, it has no use any LKF, which avoids massive calculations and solutions of high‐dimensional matrix inequalities; (2) it is available not only to HOQHNNs but also to RHNNs and complex‐valued Hopfield neural networks (CHNNs); and (3) make up for the blank of state bounding research of HOQHNNs.