The Stability Analysis Of Quaternion-valued Recurrent Neural Networks With Time-varying Delays

Recently,due to the extensive applications of neural networks to such as phys-ical sciences,biological sciences,mathematics and communications,neural networks have been a hot topic.Hitherto,researchers have done extensive works for the sta-bility problem of the equilibrium point of(recurrent)neural networks.In addition,due to the extensive applications of complex-valued neural networks which can be seen as an extension of real-valued neural networks,more and more researchers are considering complex-valued neural networks.Then naturally,complex-valued neural networks can be extended to the quaternion-valued neural networks,which can be ap-plied into the polarized wave process,the simulation of wind profile image processing as well as aviation and satellite tracking.Therefore,the investigation for quaternion-valued neural networks is more and more important,especially the stability problem of quaternion-valued neural networks.In fact,quaternion-valued neural networks have quaternion-valued matrices,quaternion-valued activation functions and quaternion-valued inputs.However,the non-communication of quaternion multiplication makes the investigation more complicated than complicated-valued neural networks.For this difficulty,the plural decomposition method,real-valued decomposition and direct methods are applied in this paper.In addition,combined with the Halanay inequal-ity method,{ζ,∞}-norm,Lyapunov functional method and linear matrix inequality technique,this paper respectively considers the global μ-stability,exponential stability and power stability of quaternion-valued neural networks.The main contributions of this paper are presented as follows.In Chapter 2,we first analyze the dynamics of delayed quaternion-valued neural networks and give the definition of stability,and then decompose the quaternion-valued neural network into two complex-valued systems which are equivalent to the quaternion-valued neural network,and then some sufficient conditions are derived by means of the Lyapunov-Krasovskii functional method and linear matrix inequality technique.Chapter 3 investigates the global μ-stability,exponential stability and power sta-bility of delayed quaternion-valued neural networks,where the quaternion-valued neu-ral network is decomposed into four real-valued systems with the real-valued decom-position method.Then based on Hermite matrices,Lyapunov function and linear matrix inequality technique,some sufficient conditions are presented.Next,based on quaternion-valued self-conjugate matrices and V function,some sufficient conditions are established with the direct method where the quaternion-valued neural network is considered as a whole without being decomposed.Based on the Halanay inequality and the real-valued decomposition method,Chapter 4 considers the global exponential stability and give some sufficient condi-tions.Combined with {ζ,∞} and the real-valued decomposition method,Chapter 5 considers the global exponential stability and establishes some sufficient conditions.