On Synchronization Control For Fractional-Order Neural Networks

In recent years,fractional differential equations and their applications have been one of the hot research topics.In view of the hereditary and memory properties of fractional-order calculus,compared with the traditional integer-order neural net-works,fractional-order neural networks can be used to more accurately describe the complicated relations between the input and output of signals and the dynamical behaviors of neural networks such as adaptation,cognition and decision-making.At present,the dynamic behavior analysis and synchronization control of fractional-order neural networks are widely concerned by lots of domestic and foreign scholars.In this thesis,some basic theories,including fractional differential equations,the theories of neural networks and complex-variable functions,differential inclusion theory,system control theory and Laplace transform,are comprehensively utilized to investigate the exponential stability of fractional-order impulsive control systems in sense of Caputo derivative and the synchronization control of three types of frac-tional neural networks,namely,the exponential synchronization of fractional-order Cohen-Grossberg neural networks under impulsive control,projective synchroniza-tion of fractional complex-valued recurrent neural networks based on linear feedback control and adaptive control,finite-time synchronization of fractional-order memris-tive neural networks by means of discontinuous control.In the first part,the exponential stability of fractional-order impulsive control systems and the exponential synchronization of fractional-order Cohen-Grossberg neural networks are studied via impulsive control scheme.First of all,based on the generalized Caputo derivative,several new fractional differential inequalities are es-tablished by applying L'Hopital rule and Laplace transform.Next,by means of the definition of Dirac function and the theory of fractional-order calculus,the impul-sive control systems are translated into the fractional impulsive systems by rigorous theoretical deduction.Furthermore,some criteria of exponential stability for impul-sive control systems are obtained by employing the established fractional differential inequalities and iterative method.Furthermore,by constructing appropriate Lya-punov functions and applying inequality technique,some conditions are derived to ensure the exponential synchronization of fractional Cohen-Grossberg neural net-works in sense of 1-norm and p-norm respectively.Finally,two numerical examples and simulations are provided to show the validity of the theoretical results.In the second part,without separating the complex-valued neural networks into two real-valued systems according to real and imaginary parts,by using the theo-ry of complex-variable functions,the projective synchronization of fractional-order complex-valued recurrent neural networks is investigated based on linear feedback control and adaptive control.Firstly,two new fractional-order inequalities are es-tablished by using the theory of complex-variable functions,Laplace transform and the properties of Mittag-Leffler functions,and several criteria are derived to en-sure quasi-projective synchronization of the complex-valued neural networks with fractional-order based on the established fractional-order inequalities and linear con-trol strategy.Moreover,the error bounds of quasi-projective synchronization are es-timated.In addition,a suitable adaptive control law is designed for fractional-order complex-valued recurrent neural networks and several criteria are derived to guar-antee the realization of the projective synchronization.Finally,numerical examples with simulations are given.In the third part,the finite-time synchronization is discussed under discon-tinuous control for a class of fractional-order memristive neural networks.Firstly,the fractional-order memristive neural networks are transformed into systems with interval parameters according to the characteristics of memristor,which solve the d-ifficulty of theoretical analysis caused by the discontinuity of memristive connection weights.Besides,a suitable discontinuous controller is designed and the finite-time synchronization is investigated by utilizing differential inclusion theory,the method of contradiction and inequality technique in sense of 1-norm and p-norm.Some synchronization criteria are obtained and the upper estimation of the settling time is derived.Lastly,numerical simulations are given to validate the correctness and effectiveness.