Stability of Sets Criteria for Impulsive Cohen-Grossberg Delayed Neural Networks with Reaction-Diffusion Terms

Impulsive Reaction-Diffusion Delayed Models in Biology: Integral Manifolds Approach

In this paper we study an impulsive delayed reaction-diffusion model applied in biology. The introduced model generalizes existing reaction-diffusion delayed epidemic models to the impulsive case. The integral manifolds notion has been introduced to the model under consideration. This notion extends the single state notion and has important applications in the study of multi-stable systems. By means of an extension of the Lyapunov method integral manifolds' existence, results are established. Based on the Lyapunov functions technique combined with a Poincarè-type inequality qualitative criteria related to boundedness, permanence, and stability of the integral manifolds are also presented. The application of the proposed impulsive control model is closely related to a most important problems in the mathematical biology-the problem of optimal control of epidemic models. The considered impulsive effects can be used by epidemiologists as a very effective therapy control strategy. In addition, since the integral manifolds approach is relevant in various contexts, our results can be applied in the qualitative investigations of many problems in the epidemiology of diverse interest.

Intermittent boundary stabilization of stochastic reaction-diffusion Cohen-Grossberg neural networks

Cohen-Grossberg neural networks (CGNNs) play an important role in many applications and the stabilization of this system has been well studied. This study considers the exponential stabilization for stochastic reaction-diffusion Cohen-Grossberg neural networks (SRDCGNNs) by means of an aperiodically intermittent boundary control. Both SRDCGNNs without and with time-delays are discussed. By employing the spatial integral functional method and Poincare's inequality, criteria are derived to ensure the controlled systems achieve mean square exponential stabilization. Based on these criteria, the effects of diffusion item, control gains, the minimum control proportion and time-delays on exponential stability are analyzed. Examples are given to illustrate the effectiveness of the obtained theoretical results.