Global Stability Of Clifford-valued Recurrent Neural Networks With Time Delays

Clifford algebra (geometric algebra) was introduced by William K. Clifford (1845 1879). It has been applied to various fields such as neural computing, computer and robot vision, image and signal processing, control problems and other areas due to its practical and powerful framework for the representation and solution of geometrical problem. Recently, as an extension of real value models, Clifford neural networks has become an active research field. Neural networks for function approximation require feature enhancement, rotation and dilatation operations, such as back propagation (BP) neural networks. These operations are limited by the Euclidean metric in real-valued neural networks and can be carried out more efficiently in Clifford-valued neural networks due to the Clifford algebra coordinate-free framework which allows to pro-cess patterns between layers and make the metric possible only due to the promising projective split.In practical applications, the neural network system is often required to be stable. So it’s necessary and important to study the stability for neural networks. Through the study of the past few decades, the research of the real-valued neural networks is mature and have gained remarkable achievements. Meanwhile, the research of complex-valued neural networks has achieved gratifying success in recent years. Therefore, it’s urgent to study the stability analysis for Clifford-valued recurrent neural networks. However, there are a little paper to study the stability for Clifford-valued recurrent neural networks. To the best of our knowledge, the stability problem for Clifford-valued systems with time delays has still not been solved. This paper investigate the stability for Clifford-valued recurrent neural networks based on the Lyapunov stability theory, the M-matrix theory and linear matrix inequality (LMI) method. In chapter two, the existence and uniqueness for the equilibrium and the global asymptotic stability are obtained based on the M-matrix theory. In chapter three, we first explore the existence and uniqueness for the equilibrium of delayed Clifford-valued recurrent neural networks (RNNs), based on which, some sufficient conditions ensuring the global asymptotic and exponential stability of such systems are obtained in terms of a linear matrix inequality (LMI). The simulation result of a numerical example is also provided to substantiate the effectiveness of the proposed results. Compared to the chapter three, the Lipschitz continuity condition for this chapter is less conservative.