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This work generalizes the projection scaling factor to a general constant matrix and proposes the matrix-projection synchronization (MPS) for fractional-order neural networks (FNNs) based on sliding mode control firstly. This kind of scaling factor is far more complex than the constant scaling factor, and it is highly variable and difficult to predict in the process of realizing the synchronization for the driving and response systems, which can ensure high security and strong confidentiality. Then, the fractional-order integral sliding surface and sliding mode controller for FNNs are designed. Furthermore, the criterion for realizing MPS is proved, and the reachability and stability of the synchronization error system are analyzed, so that the global MPS is realized for FNNs. Finally, a numerical application is given to demonstrate the feasibility of theory analysis. MPS is more general, so it is reduced to antisynchronization, complete synchronization, projective synchronization (PS), and modified PS when selecting different projective matrices. This work will enrich the synchronization theory of FNNs and provide a feasible method to study the MPS of other fractional-order dynamical models.

This paper is concerned with the global existence and exponential stability of solutions in H 4 for the compressible Navier-Stokes equations with the cylinder symmetry in R 3 when the initial total energy is sufficiently small. Moreover, the global existence and exponential stability of the classical solution can be also derived.

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In this paper, we study the initial layer problem of the Euler–Poisson in collisionless plasma that is rigorously proved by using the weighted energy method coupled with multiscaling asymptotic expansions. We apply a formal expansion for Debye length and derive the inner and the initial layer equations, which are used for deriving the error equations, and give out uniform estimate.

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