Stability Analysis And Adaptive Synchronization Of Stochastic Neural Networks In Martingale Methods

In the past about decade, stochastic neural networks have attracted a lot of attention in various fields of all the sciences and humanities. Stochastic neural networks are becoming increasingly ubiquitous in the real word, such as global optimization processing, signal processing, pattern recognition. Therefore, the research of stochastic neural networks is of great significance for us. On other hand, the problem of synchronization for stochastic neural networks is a common phenomenon in engineering and nature field.In studying of a class of stochastic neural networks, the stability analysis and adaptive synchronization are the hot problems of system dynamics and control theory at present, it not only has theoretical significance, but also has real applications. This dissertation aims to study the asymptotic stability in mean square, the exponential stability in mean square and the adaptive synchronization in pth moment for stochastic neural networks based on M-matrix approach. By applying the convergence theorem of martingale and the martingale properties of Brownian motion for stochastic neural networks with Markovian switching, several sufficient conditions to ensure the asymptotic stability in mean square, the exponential stability in mean square, the adaptive synchronization and the parameters identification scheme for drive system are derived. The main work and innovation are as follows:1. First, after introducing the significance and the motive of the stochastic neural networks stability and adaptive synchronization, then, we gave out the innovations and the main contents of this study. 2. Second, the main contents of the stochastic neural networks stability and adaptive synchronization were summarized in this letter. We introduced the definition and characteristics of stochastic neural networks, stochastic neural networks stability and adaptive synchronization features, and focused on research situation of the stability and adaptive synchronization.3. Globally exponential stability of stochastic neutral-type delayed neural networks with impulsive perturbations and Markovian switchingThe problem of globally exponential stability of stochastic neutral-type delayed neural networks with impulsive perturbations and Markovian switching is studied in this paper. By the method of the linear matrix inequality (LMI) technique, some novel sufficient conditions are derived to guarantee globally exponential stability in the mean square. It has been showed that the stability condition improves and generalizes some existing ones in the literature. Particularly, the impulsive perturbations terms are taken into account in the models. Therefore, the proposed results in this letter are more general than those reported in existing results. Meanwhile, the proposed model consists of impulsive perturbations, so these proposed criteria are universal and representative. By using Lyapunov-Krasovskii method and stochastic analysis approach, a sufficient condition to ensure globally exponential stability for the stochastic neutral-type delayed neural networks with impulsive perturbations and Markovian switching is derived. Finally, a numerical example is given to illustrate the effectiveness of the stability result.4. Adaptive almost surely asymptotically synchronization for stochastic delay neural networks with Markovian switchingThe adaptive feedback control scheme and the M-matrix approach are studied to the problem of the almost surely asymptotically synchronization for stochastic discrete time-varying delays neural networks with Markovian switching and random noise in detail. By utilizing the M-matrix approach and adaptive feedback technique, a rigorous, simple, and systematic synchronization based Automatic identification scheme proposed to solve the problem addressed. In reality, the method using in this paper simple to implement. Some appropriate parameters analysis and update laws which are found via the adaptive feedback control techniques can adjust the synchronization speed. On the other hand, almost surely asymptotically synchronization for stochastic discrete time-varying delays neural networks with Markovian switching and random noise is presented. Based on Lyapunov-Krasovskii and M-matrix approach, the activation functions are assumed to be neither differentiable, nor monotonic, nor bounded, sufficient conditions have been developed to ensure the almost surely asymptotically synchronization for the error system, and thus the drive system can synchronize with the response system. Finally, an illustrative numerical example to demonstrate the effectiveness of the M-matrix-based synchronization condition derived in this paper.5. Adaptive exponential synchronization in pth moment for stochastic delayed neural networks with Markovian switching and parameter estimationA general model of an array of Markovian jumping stochastic neural networks with parameters identification scheme and mixed time-varying delays is proposed. By utilizing the M-matrix approach and the adaptive feedback control scheme, the problem of adaptive exponential synchronization in pth moment for delay-dependent neural networks with Markovian switching and stochastic disturbances described in terms of the Brownian motion is analyzed. The time-varying delays network switches from one mode to another according to a Markovian chain with known transition probability and adjust parameters identification speed and the synchronization speed by regulating the adaptive feedback control gain. On the other hand, based on M-matrix approach and free-weighting matrices, some new stability criteria are derived to ensure the adaptive exponential synchronization in pth moment for delay-dependent neural networks in terms of linear matrix inequality (LMI) by using the Lyapunov-Krasovskii functional approach and Ito formula. Numerical examples are presented to show the effectiveness of the obtained approach finally.6. Adaptive exponential synchronization in pth moment of neutral-type delays neural networks with Markovian switching and parameter estimation The analysis issue for the problem of adaptive exponential synchronization in pth moment for Markovian switching neutral-type neural networks with parameters identification scheme and time-varying delays is considered. In this case, the new method is developed based on Lyapunov-Krasovskii and M-matrix approach to take place of traditional criteria. The proposed method is shown to be simple yet effective for analyzing the stability of adaptive exponential synchronization in pth moment of the neutral-type delays neural networks systems. Especially, when researching on the stability of neural network with Markovian switching, it can avoid losing much necessary information during the finite modes switching from one to another and will release mathematically complex which involved in time-varying, stochastic noise perturbations and parameter estimation for investigating the stability of the neural network system. In addition, the analysis issue of stability criteria is establish by using M-matrix approach for testing whether the neutral-type delayed neural networks is adaptive exponentially synchronization in pth moment. The numerical example is provided finally to illustrate the usefulness of the derived M-matrix-based synchronization conditions.