Study On The Stability Of Quaternion-valued Neural Networks With Time Delays

Since 1980 s the mathematical model of neural network has been established by Hopfield who is the biophysicist at California Institute of Technology,a variety of neural networks have been extensively studied.Quaternion-valued neural networks(QVNNs),as an extension of real-valued neural networks and complex-valued neural(CVNNs)networks,have attracted many researchers and lots of interesting results have been obtained in recent years.Compared with CVNNs,QVNNs are more perplexed because that their connection weights,states and activation functions are all quaternion-valued.It is because that the quaternion neural network can directly process quaternion data and can be widely used in various fields,it has aroused the research interest of many scholars.When solving the practical problems through the QVNNs,it mainly discusses the stability of QVNNs,and needs to correctly select the network parameters and activation functions to ensure the normal operation of the network.Therefore,it is of great significance to conduct in-depth research on the stability of quaternionic neural networks.The thesis mainly studies the following five aspects:(1)Global exponential stability of quaternion-valued neural networks with mixed time-varying delaysThe global exponential stability of Clifford-valued recurrent neural networks with both asynchronous time-varying and continuously distributed delays is investigated.Firstly,exploring the existence and uniqueness for the equilibrium of delayed Clifford-valued neural networks,by using inequality technique and M matrix properties.Then,using mathematical analysis method,can get some determinant conditions ensuring the global exponential stability of such systems.The numerical simulation examples show that the model tends to be stable at t =20s,indicating the validity and feasibility of the results.(2)Global ?-stability of quaternion-valued neural networks with undifferentiable time-varying delaysThe quaternion-valued neural networks with non-differentiable time-varying delays is studied.Firstly,by using the method of plural decomposition,QVNNs can be decomposed into two complex-valued neural networks.Some sufficient criteria in linear matrix inequality form are derived to guarantee the existence and uniqueness of the equilibrium point for considered QVNNs by using the homeomorphism mapping principle of complex domain.Secondly,based on applying the free weighting matrix method and constructing Lyapunov-Krasovskii functional,several conditions are established in complex-valued linear matrix inequality to ensure the the global ?-stability of QVNNs.Finally,by employing the predictor-corrector approach,two numerical examples show that the model tends to be stable at t =2s,indicating the validity and feasibility of the results.(3)Lagrange stability analysis for complex-valued neural networks with leakage delay and mixed time-varying delaysThe stability in Lagrange sense for complex-valued neural networks with timevarying discrete delays and distributed delays as well as leakage delay is discussed.By constructing an appropriate Lyapunov-Krasovskii functional,and employing free weighting matrix approach and inequality techniques in matrix form,a sufficient criterion to guarantee global exponential stability in Lagrange sense is obtained for the investigated neural networks.The given criterion is delay-dependent and is shown as linear matrix inequalities in complex domain,which can be calculated numerically applying valid YALMIP toolbox in MATLAB.The numerical simulation examples show that the model tends to be stable at t =20s,indicating the validity and feasibility of the results.(4)Global exponential stability in Lagrabge sense for quaternion-valued neural networks with leakage delay and mixed time-varying delays.The global exponential stability in Lagrange sense for quaternion-valued neural networks with leakage delay,discrete time-varying delays and distributed delays is studied.By structuring an advisable Lyapunov-Krasovskii functional in quaternion field,and adopting free weighting matrix method and inequality technique,a sufficient condition in quaternion-valued linear matrix inequality to guarantee the global exponential stability in Lagrange sense is acquired,and the domain of attraction is estimated.The numerical simulation examples show that the model tends to be stable at t =3s,indicating the validity and feasibility of the results.(5)Stabilization of time-varying financial systemsThe stabilization of financial systems with time-varying delays is studied.The considered financial system only needs to satisfy the boundedness of state variables.As far as we know,many scholars have studied financial systems without time delay.Almost no scholars have studied financial systems with time-varying delays.In this paper,the method of studying neural networks is used in the financial systems.By constructing the appropriate Lyapunov-Krasovskii functional and using the inequality technique,it is proved that the system is stable under intermittent control.