Robust Stability Of Quaternion-valued Neural Networks With Time Delays

In this thesis,based on the classical Hopfield real-valued and complex-valued neural networks models,considering the communication delay between neurons,a quaternion neural network model with transmission delay and leakage delay is established.Via the homeomorphic mapping principle,the contraction mapping theorem,the Lyapunov stability theorem and matrix inequality techniques,the robust stability of quaternion-valued neural networks is researched systematically.The existence,uniqueness and robust stability of the solution are obtained,which enrich the theoretical achievements of neural networks in pattern recognition,signal processing,associative memory,optimal control and other applications.In the first part of the thesis,the continuous quaternion-valued neural network model with time delays is considered.By the characters of the homeomorphic mapping,the existence and uniqueness of the equilibrium point are obtained.Through constructing an appropriate Lyapunov functional,the sufficient conditions for global robust stability of the system are derived.The relevant results consider the sign of the connection weight of the network,and the given elements in the judging matrix depend not only on the lower bounds,but also on the upper bounds of the interval parameters.The relevant work of delayed complex-valued neural networks model are generalized to quaternion-valued neural networks,moreover,the robust stability conditions of delayed quaternion-valued neural network model are extended.In the second part of the thesis,the robust stability of the discrete quaternionvalued neural network model with time delays is studied.Firstly,utilizing contraction mapping theorem,the existence and uniqueness of the equilibrium point for the model are proved.Then,according to the appropriate discrete Lyapunov functional,the global robust exponential stability of the system is obtained.It is the first time to discuss the robust stability of discrete quaternion-valued neural network,the obtained judging matrix is decided by the upper and lower bounds of connection weights,the sign of connection weights are not ignored,which reduces the conservatism of the system.In the third part of the thesis,the quaternion-valued neural network model with time delays on time scales is analyzed.On the basis of the calculus theory on time scales,both the continuous-time and discrete-time quaternion-valued neural networks can be described in the same framework.Using the similar methods in the new framework,the existence and uniqueness of the equilibrium point are derived.For the selected Lyapunov functional,the judging criteria for the global robust stability is established by utilizing the quaternion-valued inequality technique and the derivative rules on time scales.Under the framework of time scales,there is no need to decompose the quaternion-valued neural network model into two complex-valued neural network models,the previous results are improved substantially.