Global Well-Posedness and Dynamical Behavior of Delayed Reaction-Diffusion BAM Neural Networks Driven by Wiener Processes

Abstract

This paper studies the global existence and uniqueness as well as asymptotic behavior of the reaction-diffusion bidirectional associative memory neural networks with S-type distributed delays and infinite dimensional Wiener processes. The conspicuous characteristics of this system are neurons in one layer interacting with neurons in another layer and the noise which disturbed this system has both time and spatial structure. First, several inequalities are proposed and proved for the preparation of future study. Then, the system is coped in the framework of semigroup theory and functional space. Furthermore, the local existence and uniqueness of mild solution for this system is proven by using the contraction mapping principle coupled with many functional inequalities such as Young inequality, Burkholder–Davis–Gundy inequality, Poincare inequality. The global well-posedness are proven through a prior estimate from constructing appropriate Lyapunov–Krasovskii functional. Moreover, the existence of equilibrium is solved by using the topological degree theory and homotopy invariance. At last, the globally exponential stability of the equilibrium in the mean square sense is studied by constructing appropriate vector Lyapunov–Krasovskii functional and using an improved inequality proposed by us. The criteria of stability are given in the form of matrix form. It is easy to verify them in the computer and they will have a wider application. We give an example to examine the availability of our result, and the code is performed in Matlab. The approach used in this paper can also be extended to other systems.