Stability And Synchronization Of Impulsive Fractional-order And Quaternion-valued Neural Network

In recent decades,fractional calculus,as a generalization of integer-order differentiation and integration to arbitrary non-integer order,has gained a heated attention in both theoretical and applied aspects of numerous branches of science and engineering.This mainly due to the fact that fractional derivatives are nonlocal and have weakly singular kernels when compared with integer-order calculus,the main superiority is that they can act as an excellent instrument for the description of memory and hereditary properties of various materials and processes.As is well known,real-valued neural networks(RVNNs)perform less well in implement of geometrical transformations like two-dimensional(2D)affine transformations.In view of this,complex-valued neural networks(CVNNs)have been bought into to improve performance on 2D affine transformations,which mainly benefits that CVNNs enable the modeling of a point in 2D space as a single entity instead of a set of two data items.However,when it comes to 3D data case such as body images and color images,CVNNs have to lose their dominant because they are incapable of implementing 3D data directly although such data can be processed by employing many neurons of CVNNs.Fortunately,quaternion-valued neural networks(QVNNs),an extension of CVNNs,have begun to receive a heated research in recent years,due to QVNNs are capable of implementing direct encoding on 3D affine transformations high-efficiently and compactly,especially spatial rotation,which can be extensively applied to various areas such as image impression,attitude control of satellites,and computer graphics.In addition,it is commonly acknowledged that the states in many actual models of application systems such as frequency-modulated signal processing system,bursting rhythm models in pathology,population interactions models and flying object motions are always inevitably subject to instantaneous perturbations and characterized by abrupt changes at certain instants,which may arise from frequency change,switching phenomenon or other sudden noises,i.e.,they manifest impulsive effects.In this dissertation,we will mainly combine the aforementioned content,and investigate the stability and synchronization of nonlinear dynamical systems.This dissertation is organized as follows:Firstly,stability analyses for a class of fractional-order impulsive nonlinear systems and a class of fractional-order impulsive nonlinear systems with uncertain parameters are considered.Based on the theory of fractional calculus,impulsive differential equation and S-procedure,Lyapunov direct method and some matrix inequalities,several sufficient criteria are established to guarantee the Mittag-Leffler stability for the addressed models with appropriate impulsive controller.Secondly,stability analyses for a class of state-dependent impulsive fractional-order nonlinear systems and a class of state-dependent impulsive fractional-order neural networks are discussed.By applying the B-equivalence method and the theory of fractional calculus,the concerned state-dependent impulsive fractional-order nonlinear systems(or neural networks)can be reduced to fixed-time impulsive fractional-order nonlinear systems(or neural networks),and the latter can be regarded as the comparison system of the original ones.Meanwhile,a series of sufficient criteria demonstrating the same stability properties between both state-dependent impulsive fractional-order nonlinear systems(or neural networks)and the fixed-time alternatives are obtained.Furthermore,some sufficient conditions guaranteeing the stability of the considered models are presented.Thirdly,the global asymptotical synchronization problem of delayed fractional-order memristor-based complex-valued neural networks with uncertain parameters is addressed.Under the framework of Filippov solution and differential inclusion theory,several sufficient criteria ensuring the global asymptotical synchronization for the considered drive-response models are derived based on Lyapunov direct method and comparison theorem by employing feedback control strategy.Fourthly,the global Mittag-Leffler stability and synchronization problems for a class of fractional-order quaternion-valued neural networks(FQVNNs)with linear threshold neurons are investigated.On account of the non-commutativity of quaternion multiplication resulting from Hamilton rules,the FQVNN models are separated into four RVNNs.Consequently,the dynamic analysis of FQVNNs can be realized by investigating the realvalued ones.Based on the method of M-matrix,the existence and uniqueness of the equilibrium point of the FQVNNs are obtained without detailed proof.Afterwards,several sufficient criteria ensuring the global Mittag-Leffler stability for the unique equilibrium point of the FQVNNs are derived by applying the Lyapunov direct method,the theory of fractional differential equation,the theory of matrix eigenvalue,and some inequality techniques.In the meanwhile,global Mittag-Leffler synchronization for the drive-response models of the addressed FQVNNs are investigated explicitly.Fifthly,the impulsive effects on stability of FQVNNs with parametric uncertainties are discussed.Two cases are discussed:(i)robust Mittag-Leffler stability analysis on impulsive FQVNNs with stable continuous subsystems and stabilizing impulses;(ii)robust exponential stability analysis on impulsive FQVNNs with unstable continuous subsystems and stabilizing impulses.For case(i),by means of Lyapunov direct method,theory of impulsive differential equation,and method of Laplace transformation,some sufficient conditions ensuring the robust Mittag-Leffler stability for the zero solution of the concerned models are presented.For case(ii),by applying Lyapunov direct method,theory of impulsive differential equation,and the method of mathematical induction,several sufficient criteria guaranteeing the robust exponential stability for the zero solution of the addressed models are established.Finally,the state-dependent impulsive effects on robust exponential stability for a class of QVNNs with parametric uncertainties are discussed.The concerned QVNNs are separated into four real-valued parts due to the non-commutativity of quaternion multiplication.Then,several assumptions ensuring every solution of the separated statedependent impulsive NNs intersects each of the discontinuous surface exactly once are proposed.In the meantime,by applying the B-equivalent method,the addressed statedependent impulsive models are reduced to fixed-time ones,and the latter can be regarded as the comparative systems of the former.For the subsequent analysis,a novel norm inequality of block matrix,which can be utilized to analyze the same stability properties of the separated state-dependent impulsive models and the reduced ones efficaciously is proposed.Afterwards,several sufficient conditions are well presented to guarantee the robust exponentially stability of the origin of the considered models,it is worth mentioning that two cases of addressed models are analyzed concretely,that is,models with exponential stable continuous subsystems and destabilizing impulses,and models with unstable continuous subsystems and stabilizing impulses.