Study On Stability And Synchronization For Delayed Neural Networks

Delayed neural networks exhibiting rich dynamical behaviors are a class of complex large scale nonlinear dynamical systems. Time delay is commonly encountered in real systems, and its existence is frequently a source of instability and poor performance. In recent years, the stability of delayed neural networks has been extensively studied because of their applications in many areas such as pattern recognition, associative memory, combinatorial optimization and so on. Meanwhile, the synchronization of the delayed chaotic neural networks has attracted worldwide attention. This thesis mainly focuses on the existence and global stability of the equilibrium point and synchronization for the delayed neural networks based on the Lyapunov stability theory and matrix theory. The main work and research results of this thesis can be briefly described as follows:(1) The global robust stability of a class of uncertain neural networks with both multiple time-varying discrete delays and distributed delays is studied. By using the Lyapunov method, several sufficient conditions guaranteeing the global robust stability of the equilibrium point are presented. The parameter uncertainties which can be commonly encountered because of the inaccuracies and changes in the environment of the model are considered. All proposed sufficient conditions are represented in terms of linear matrix inequalities (LMIs), which can be easily verified by using the Matlab LMI Toolbox. Moreover, the obtained results are superior to the existing ones in the previous literatures.(2) The problem of robust exponential stability for a class of interval Cohen-Grossberg neural networks with time-varying delays is discussed. Without assuming the boundedness and differentiability of the activation functions and any symmetry of interconnection matrices, some sufficient conditions for the existence, uniqueness, and global robust exponential stability of the equilibrium point are derived. Some comparisons between the results presented in this paper and the previous results admit that our results are the improvement and extension of the existed ones. The validity and performance of the new results are further illustrated by two simulation examples.(3) The global stability of two classes of delayed Markovian jumping neural networks is investigated. Firstly, without assuming the differentiability and monotonicity of the activation functions and any symmetry of interconnection matrices, some sufficient conditions are proposed for the robust stability of the delayed bidirectional associative memory neural networks with Markovian jumping parameters by constructing suitable Lyapunov functional. Secondly, by utilizing the linear matrix inequality technique, the global exponential stability of the delayed Markovian jumping recurrent neural networks is analyzed. The proposed conditions here are extended to the uncertain cases, which are shown to be the improvement and extension of the existing ones.(4) The stability of two classes of discrete-time neural networks with delays is analyzed. Firstly, we study the global robust exponential stability for a class of discrete-time interval bidirectional associative memory neural networks with time-varying delays. By employing the Lyapunov functional, a new sufficient criterion is proposed for the global robust exponential stability of discrete-time BAM neural networks which contain uncertain parameters with their values being bounded. Secondly, a class of uncertain Markovian jump discrete-time recurrent neural networks with time delays is investigated. The uncertainty is assumed to be of norm bounded form. By using the Lyapunov stability theory, some sufficient criteria are proposed for the robust stochastic stability of the Markovian jump discrete-time recurrent neural networks with constant or mode-dependent time delays. The proposed LMI-based results are computationally efficient and improve the previous results.(5) The synchronization of the chaotic delayed neural networks is investigated. Firstly, based on drive-response synchronization principle, the synchronization control for a class of chaotic delayed neural networks with parameter uncertainty is studied. By using Lyapunov stability theory and linear matrix inequality approach, some sufficient conditions are obtained to guarantee the global synchronization of two chaotic neural networks and a procedure to construct a synchronization controller is presented. Secondly, the global synchronization in an array of linearly coupled neural networks with constant and delayed coupling is discussed. By a simple combination of adaptive control and linear feedback with the updated laws, some sufficient conditions are derived for the global synchronization of the coupled neural networks. The coupling configuration matrix of the coupled system is assumed to be asymmetric and the delayed coupling is considered. It is shown that our results are less restrictive. Numerical simulations are presented to demonstrate the effectiveness of the theoretical results.