Spacial sampleddata control for Hxe2x88x9e output synchronization of directed coupled reactionxe2x80x93diffusion neural networks with mixed delays Abstract   This work investigates the      H    xe2x88x9e      output synchronization (HOS) of the directed coupled reactionxe2x80x93diffusion (Rxe2x80x93D) neural networks (NNs) with mixed delays. Firstly, a model of the directed state coupled Rxe2x80x93D NNs is introduced, which not only contains some discrete and distributed time delays, but also obeys a mixed Dirichletxe2x80x93Neumann boundary condition. Secondly, a spacial sampleddata controller is proposed to achieve the HOS of the considered networks. This type of controller can reduce the update rate in the process of control by measuring the state of networks at some fixed sampling points in the space region. Moreover, some criteria for the HOS are established by designing an appropriate Lyapunov functional, and some quantitative relations between diffusion coefficients, mixed delays, coupling strength and control parameters are given accurately by these criteria. Thirdly, the case of directed spatial diffusion coupled networks is also studied and, the following finding is obtained: the spatial diffusion coupling can suppress the HOS while the state coupling can promote it. Finally, one example is simulated as the verification of the theoretical results.

A new fixedtime stability theorem and its application to the fixedtime synchronization of neural networks Abstract   In this paper, we derive a new fixedtime stability theorem based on definite integral, variable substitution and some inequality techniques. The fixedtime stability criterion and the upper bound estimate formula for the settling time are different from those in the existing fixedtime stability theorems. Based on the new fixedtime stability theorem, the fixedtime synchronization of neural networks is investigated by designing feedback controller, and sufficient conditions are derived to guarantee the fixedtime synchronization of neural networks. To show the usability and superiority of the obtained theoretical results, we propose a secure communication scheme based on the fixedtime synchronization of neural networks. Numerical simulations illustrate that the new upper bound estimate formula for the settling time is much tighter than those in the existing fixedtime stability theorems. What is more, the plaintext signals can be recovered according to the new fixedtime stability theorem, while the plaintext signals can not be recovered according to the existing fixedtime stability theorems.