Dynamics Analysis And Control Of Fractional-order Neural Networks

Fractional calculus is the theory on derivative and integral with arbitrary order,which is essentially a generalization of classical integer-order calculus.It accumulates the weighted global information of the function,also known as memory,which is more coincident with the biological neural network.As a result,fractional calculus is introduced into artificial neural network for more accurate mathematical modelling,especially for those processes and materials in heredity and memorability,so as to further improve the design,characterization and control capabilities for such systems.Therefore,fractional-order neural networks have great application prospects and research value.This thesis addresses the dynamic behaviors and the control of Caputo fractional-order Hopfield neural networks.The main work can be summarized as follows:In the second chapter,a non-Lyapunov stability,named short-time stability is considered for a class of fractional-order complex valued neural networks with delays.Unlike the asymptotic stability of the system trajectory in the sense of Lyapunov,the finite-time stability sustains the trajectories do not exceed a certain threshold during a fixed short time under a given bound near the initial conditions.Firstly,the considered network is equivalently transformed into a new real-valued network.Then,by means of the properties of fractional calculus and some inequality scaling strategies,the short-time stability of the system is derived respectively under two cases with order 1/2 ? ? < 1 and 0 < ? < 1/2.In the third chapter,the synchronization problem for two neural networks with discontinuous activation functions are investigated on the framework of Filippov solution.Firstly,the complex-valued neural network in presence of discontinuous activation functions and parameter uncertainties is concerned.A novel feedback control procedure is designed to realize the fixed-time synchronization of this network.Besides,criteria of modified controller for assurance of fixed-time anti-synchronization are also derived for the same system,while an upper bound of the settling time is acquired as well,which allows to be modulated to predefined values independently on initial conditions.Secondly,a class of fractional-order Bidirectional associative memory(BAM)neural networks with discontinuous activation functions is considered.The global existence of solution under the framework of Filippov for such networks is firstly obtained based on the fixed point theorem for condensing map.Then state feedback and impulsive controller are respectively designed to ensure the Mittag-Leffler synchronization of such networks and two new synchronization criteria are obtained,which are voiced in terms of a fractional comparison principle and Razumikhin techniques.In the fourth chapter,some dynamics of two fractional-order coupled neural networks are studied.Firstly,the static coupled fractional-order complex network is focused for its dissipativity,which is a more general characteristic than stability.Applying the stability result for linear fractional delayed differential equations,together with Laplace transformation,some sufficient criteria are obtained to guarantee the global dissipativity of the concerned nonlinear networks.Then,considering the dynamic coupling,an adaptive coupling matrix is adopted in an array of fractional-order neural network.For the synchronization,a pinning control strategy with the free selection of pinning nodes is designed.Then,by absorbing the information of eigenvectors and adaptive laws for the coupling matrix,the sufficient condition for MittagLeffler synchronization of the fractional-order network is established.Accordingly,an easier verifiable algebraic criterion is further derived by means of some matrix inequalities.In the fifth chapter,the passivity analysis for an array model of coupled inertial delayed neural networks with impulses is investigated respectively under different network structures,namely directed and undirected topologies.Since the second-order inertia term in the system could hinder the dynamic analysis,the reduction of order for the networks is firstly implemented by the introduction of new variable.For the transformed networks,utilizing the information of eigenvectors for the directed coupling matrix,and benefited from the symmetry of undirected coupling matrix,different Lyapunov functions are constructed,and sufficient criteria of passivity,including input-strictly passivity and output-strictly passivity are derived for these two network topologies.It is seen that the passivity conditions for undirected coupled networks are more concise and easier to be verified.