Impulsive Control Of Discrete-Time Neural Networks Under Stochastic Perturbations

In the past few decades, due to the profound theoretical and practical importance,dynamical analysis of neural networks drew more and more attentions from researchersin various disciplines. It is a broad research field which has extensive applications andconnects tightly with computers, digital networks, embedded systems, and artificialintelligence, etc.In real world engineering, neural networks always experience abrupt state changeat certain moments. It is natural to assume these changes happen instantaneously,namely, in the form of impulses. In fact, impulsive phenomena always exist in theevolution processes of dynamical systems, such as in financial and biology models. It iswell known that impulses can bring about complex influence on the system dynamics:On the one hand, they may exist as perturbations, that is, they have disturbing effects onthe stability and can destabilize the system, even lead to oscillation or chaos. On theother hand, impulses probably serve as control power, which compensate the divergingtrend when the original component is unstable. Hence, there is a crucial need to studyhow impulses affect the dynamics, especially stability property, of the system.This paper studies the globally exponential stability of the delay-dependentdiscrete-time stochastic neural systems under the influence of impulses. Followingcreativities are obtained:①Introducing the Hopfield Neural Networks, and its discrete-time counterpart.Some novel stability criteria of such kind of neural system under stochasticperturbations are formulated in the form of LMI.②Destabilizing effect of impulses are considered, when impulses-free systemconverges to its equilibrium point, and impulses exist as a kind of perturbations, thefeasible interval of the impulsive jump operator that can preserve the stability propertyis derived.③Introducing the Lyapunov-Razumikhin theory, and presenting its application inthe dynamical analysis of neural networks. The stabilizing effects of impulses areinvestigated.④Studying the impulsive switching neural networks which include both stableand unstable subsystems. Some stability criteria are obtained.Finally, all of the above results have been illustrated by numerical examples. The proposed results can be used to design impulsive delayed neural networks.