Stochastic Synchronization of Reaction-Diffusion Neural Networks under General Impulsive Controller with Mixed Delays

and Applied Analysis 3 global exponential stability and synchronization of delayed reaction-diffusion neural networks under hybrid state feedback control and impulsive control. However, to the authors knowledge, impulsive control has not been considered in the literature to realize synchronization of reaction-diffusion neural networks. Moreover, the impulsive controllers in 3, 9, 36–40 were nondelayed. Recently, in 41 , global exponential stability of fuzzy reactiondiffusion cellular neural networks with time-varying discrete delays and unbounded distributed delays and impulsive perturbations were studied. Nevertheless, to the best of our knowledge, there are no results on stability or synchronization of reaction-diffusion neural networks with time-varying discrete delays and distributed delays under impulsive controller which has multiple time-varying delays, let alone impulsive controller with distributed delays. If these delays are considered in impulsive controller, the analysis methods used in 3, 9, 26, 36–40 are not applicable anymore. Considering the fact that both discrete delays and distributed delays are unavoidable in practice, it is of great importance to consider delayed impulsive control to synchronize-delayed neural neural networks. Being motivated by the above discussions, this paper aims to study the global exponential derive-response synchronization of reaction-diffusion neural networks with multiple time-varying discrete delays and unbounded distributed delays via general impulsive control. The general impulsive controller is assumed to be nonlinear and has multiple timevarying discrete and distributed delays. Since time delays are always vary and unavoidable in practical operation, the general impulsive controller is essentially important and more practical than existing nondelayed impulsive controller. Stochastic perturbations in the response system are also considered. By using a novel integral inequality in 35 , the problem of distributed delays with not-equal-to-1 delay kernel can be solved by matrix method. By utilizing the novel integral inequality, the properties of random variables and Lyapunov functional method, sufficient conditions guaranteeing the considered drive-response systems to realize synchronization in mean square are derived through strict mathematical proof. The proof process and the results are very simple. Finally, numerical simulations are given to show the effectiveness of the theoretical results. The rest of this paper is organized as follows. In Section 2, the considered model of coupled reaction-diffusion neural networks with delays is presented. Some necessary assumptions, definitions, and lemmas are also given in this section. In Section 3, synchronization for the proposed model is studied. Then, in Section 4, simulation examples are presented to show the effectiveness of the theoretical results. Finally, Section 5 provides some conclusions. Notations. In the sequel, if not explicitly stated, matrices are assumed to have compatible dimensions. N denotes the set of positive integers. In denotes the n × n identity matrix. R n denotes the Euclidean space, and Rn×m is the set of all n × m real matrix. λmax A and λmin A mean the largest and smallest eigenvalues of matrix A, respectively, ‖A‖ √ λmax ATA , where T denotes transposition. C diag c1, c2, . . . , cn means C is a diagonal matrix. Moreover, let S,F, {Ft}t≥0, P be a complete probability space with filtration {Ft}t≥0 satisfying the usual conditions i.e., the filtration contains all P -null sets and is right continuous . Denote by LPF0 −∞, 0 ;R the family of all F0-measurable C −∞, 0 ;R valued random variables ξ {ξ s : s ≤ 0} such that sups≤0E ‖ξ s ‖ < ∞, where E{·} stands formathematical expectation operator with respect to the given probabilitymeasure P . Sometimes, the arguments of a function or a matrix will be omitted in the analysis when no confusion can arise. 4 Abstract and Applied Analysis 2. Preliminaries Consider a delayed neural network with reaction-diffusion terms which is described as follows: