Research On Stability And Control Of Fractional-order Memristor-based Neural Networks

Neural network is a new interdisciplinary subject,which began in 1940 s.It is an important part of artificial intelligence,which has become the focus of brain science,neuroscience,cognitive science,psychology,computer science,mathematics and physics.Artificial neural network is an artificial system which can simulate human brain and has the functions of learning,association,memory,pattern recognition and other intelligent information processing.The fractional-order differentiation provides neurons with a fundamental and general computational ability that contributes to efficient information processing,stimulus anticipation and frequency-independent phase shifts in oscillatory neuronal firings.Therefore,fractional calculus can be well applied in the study of neural networks.In addition,as the fourth basic circuit component,memristor has the characteristics of time memory,which is similar to the synapse in human brain.Because of this feature,more and more researchers construct the memristor-based neural networks to emulate the human brain through replacing resistor by memristor and study the dynamics of fractional-order memristor-based neural networks for the purpose of successful applications in neural learning,pattern recognition and associative memories.On the other hand,chaos synchronization is widely used in physics,mechanics,optics,electronics,chemistry,information science,biology and power system protection.In particular,the chaotic synchronization of fractional-order neural network has a bright future in the fields of secure communication,image processing,pattern recognition and so on.At present,most models of fractional-order neural networks are based on the definition of Caputo fractional derivative,and a few models are based on the definition of Riemann-Liouville fractional derivative.These two definitions have their own advantages,and the research methods of the two kinds of fractional systems are not the same.This paper studies the drive-response synchronization of Caputo fractional-order neural network and Riemann-Liouville fractional-order neural network,respectively.The memristor-based neural network,competitive neural network and inertia neural network are investigated.The different kinds of effective controllers are designed and sufficient conditions for synchronization are obtained,respectively.The validity of the theoretical results are verified by numerical simulation.The detailed work is as follows:1.Most of the stability results of Caputo fractional-order neural network are obtained by Lyapunov method.However,this method requires that Lyapunov function is continuous and derivable,which is only suitable for continuous fractional system.The important inequality for Caputo fractional derivative is given for discontinuous Caputo fractional-order neural network.Furthermore,using this inequality,combining the comparison theorem of fractional delay system and the stability theorem of linear fractional system,the synchronization condition of fractional-order memristor-based neural network is obtained.2.There are a few results and methods about Riemann-Liouville fractional-order neural network.This paper studies the synchronization of Riemann-Liouville fractionalorder memristor-based neural network.The important inequality for Riemann-Liouville fractional derivative is given,which plays an important role in the study of the discontinuous fractional system.By this inequality,we construct Lyapunov function including Riemann-Liouville fractional integral terms.According to the properties of Riemann-Liouville fractional calculus,the synchronization condition of the fractionalorder memristor-based neural network is obtained by Lyapunov direct method.3.A lot of research results on fractional neural network assume that the parameters of neural network are known.In fact,it is impossible to know the values of parameters exactly.These uncertain factors will affect and even destroy the synchronization of the system.For Caputo fractional-order neural network with unknown parameters and Riemann-Liouville fractional-order neural network with unknown parameters,different adaptive controllers and parameter updating laws are designed,respectively.The unknown parameters can be identified simultaneously,while the synchronization between the drive system and response system is achieved.4.The fractional competitive neural network with different time scales is studied.Considering the different characteristics of Short-Term-Memory variable and LongTerm-Memory variable,different orders are selected for fractional differential equation.Therefore,the incommensurate fractional competitive neural network is more suitable for the actual situation.The synchronization conditions of Riemann-Liouville fractional competitive neural networks with known and unknown parameters are given.Based on the chaotic masking method,the example of chaos synchronization application in the field of secure communication is provided.5.The inertia term is introduced into the fractional-order neural network,and the model of fractional-order inertia neural network is proposed.Evidently different from the existing fractional-order neural networks,fractional-order inertial neural network is described by the fractional differential equation involving two different fractional derivatives of the state.According to the properties of Riemann-Liouville fractional calculus,the fractional inertial system is transformed into a conventional fractional system through a proper variable substitution.Lastly,several novel feedback controllers are proposed for different cases of fractional-order time-delayed inertial neural networks.The stability condition and synchronization condition of the fractional-order time-delayed inertial neural network are obtained.