Synchronization And Control For Several Kinds Of Caputo-type Fractional-Order Neural Networks

Since its historical reliance,non-locality and hereditary,the features of neurons’ memorability,cognition and decision-making can be accurately described by introducing fractional calculus into neural networks.Moreover,fractional-order system can exhibit more complicated dynamic behaviors under the influence of commensurate order.Therefore,it is a significant,attractive and challenging issue for exploring the dynamics of fractional-order neural networks.Combining fractional-order Lyapunov stability theory,neural network with the properties of special functions,the synchronization of complexvalued,quaternion-valued as well as reaction-diffusion neural networks is investigated in the sense of Caputo fractional-order derivative of this thesis.The primary contents are concluded as follows:In view of its outstanding performance in two-dimensional date processing,the synchronization of fractional-order complex-valued networks is investigated by employing non-separation theory in the third and forth parts.(i)The projective synchronization of fractional-order complex-valued neural networks is studied by designing two different hybrid controllers.Firstly,an essential fractional-order inequality is established which is associated with complex-valued functions and p-norm.Secondly,by using M-matrix theory,several sufficient conditions are obtained to ensure the Mittag-Leffler projective synchronization.Finally,some novel analysis tools,including the method of contradiction,the properties of fractional calculus as well as the divergence of harmonic series,are developed to demonstrate the adaptive synchronization.(ii)The finite-time cluster synchronization problem for fractional-order complex-valued networks with nonlinear coupling is analyzed.Firstly,a new finite-time fractional-order inequality is established by using L’Hospital’s rule,Laplace transform and contradiction method.Furthermore,two different control schemes are proposed on the basis of complex-valued sign functions.In addition,by employing fractional-order stability theory and complex function theory,several criteria are deduced to ensure finite-time cluster synchronization.Lastly,the setting time is effectively estimated by utilizing finite-time fractional-order differential inequality.Actually,the processing of the high-dimensional data and chromatic image are inevitable in many practical fields.To provide an effective and convenient method for resolving these problems,fractional-order quaternion-valued memristive networks with discontinuous activation functions are introduced and the finite-time projective synchronization is discussed on the basis of non-separation method in the fifth part.Firstly,the addressed networks are converted into systems with parametric uncertainty in the framework of differential inclusion and measurable selection.Subsequently,the sign function and a new norm in the sense of absolute value are extended into the quaternion field,and some important inequalities associate with quaternion-valued function are established.Finally,by designing two quaternion-valued controllers,some sufficient conditions are derived to guarantee the projective synchronization between systems in a finite time.To accurately depict reaction-diffusion phenomena,the synchronization of fractionalorder reaction-diffusion neural network is explored under the Dirichlet-type boundary condition in the sixth and seventh parts of this thesis.(i)The synchronization of fractionalorder competitive neural networks with reaction-diffusion terms is investigated.Firstly,the proposed network model encompasses both discrete delay and leakage delay.Secondly,by designing two different controllers,some sufficient criteria are derived to guarantee global synchronization through M-matrix theory,comparison principle as well as the method of contradiction.(ii)The synchronization of reaction-diffusion coupled neural networks with fractional-order and impulses is concerned.Firstly,an extended Halanaytype inequality is established to cope with the hybrid delay-dependent impulsive problem by utilizing the mathematical induction.Furthermore,a direct error method is introduced by constructing Lyapunov function for the addressed networks to investigate the exponential synchronization under impulsive effects.Finally,by utilizing the concept of average impulsive interval and average impulsive strength,some sufficient synchronization criteria are derived,which are closely associated with time delay and commensurate order for fractional-order systems.In addition,the above mentioned theoretical results are demonstrated by several numerical examples.