| Artificial neural network is a mathematical model of biological neural network abstracted from the perspective of information processing.With the development of artificial neural networks,they have successfully solved many practical problems that are difficult to be solved by modern computers in the fields of pattern segmentation,intelligent robot,automatic control,predictive estimation,fault diagnosis and system identification,etc.,showing good intelligent characteristics,which mainly depends on the dynamic behavior of neural network.Multistability is a concept to describe the coexistence of multiple stable equilibrium states or periodic solutions.Such dynamical behavior is essential in some applications of neural networks,including image processing,pattern recognition,and associative memory storage.Hopfield neural networks have become the main model attracting a lot of interest in multistability research.In real life,periodic function can well describe the development process of systems,such as ecosystem,mechanical vibration,market supply and demand,transportation system,heartbeat and memory in biological activities,etc.,and these practical problems can be summarized as discussing the stability of periodic solutions of differential equations.Based on these,we study the multistability and control strategy for generating globally stable periodic solutions of Hopfield neural networks.In theoretical study of neural networks,the dynamic behavior of neural networks is closely related to time delay,uncertainty,random noise and diffusion.In the past two decades,many scholars have been devoted to the research on how to guarantee the global or local stability of Hopfield neural network under these factors.However,for the neural networks with impulse,time delay,and reaction-diffusion terms,how to obtain the existence and uniqueness conditions of globally stable periodic solutions with less conservatism by using matrix-based convex combinations and linear matrix inequality methods still need to be studied in depth.When piecewise linear,unsaturating piecewise linear and discontinuous nonmonotonic activation functions appear in discrete-time,continuous-time,fractional-order,Takagi-Sugeno fuzzy neural networks,how to analyze their monostability and multistability is a difficult problem.For unstable neural networks with time-varying delays,how to design impulsive controllers to generate globally stable periodic solutions need to be discussed.Aiming at these problems,we take discrete-time,continuous-time,fractional-order,TakagiSugeno fuzzy,time-dependent switching,inertial reaction-diffusion neural networks as research objects.From the geometric properties of the piecewise linear,unsaturating piecewise linear and discontinuous nonmonotonic activation function perspective,using strictly diagonally dominant matrix,contraction mapping,fixed point theorem,AscoliArzela theorem and convex combination method fully,constructing appropriate Lyapunov-Krasovskii functional,the main works accomplished in this paper are presented as follows:(1)Multistability results are developed for discrete-time neural networks and quaternion-valued neural networks.A class of piecewise linear activation function make the storage capacity for neural networks increase greatly.According to the geometric properties of piecewise linear activation functions,the n-dimensional Euclidean space is divided into many hyperrectangular regions.By using Schauder's fixed point theorem and strictly diagonally dominant matrix,some sufficient conditions for the existence and uniqueness of equilibrium points in hyperrectangular regions of neural networks are given.Sufficient conditions are derived to guarantee the local asymptotic stability of the equilibrium points of the neural network and the instability of the other equilibrium points.The estimated attractive basins for discrete-time neural networks are hyperspherical regions,which can be larger than the original rectangular regions.Without any other condition,the attraction basins for quaternion-valued neural networks,given as rectangular regions,are larger than originally rectangular regions.(2)The unsaturating piecewise linear activation function has the advantages of simple fast calculation and avoiding gradient disappearance,and it is an important part of many successful feedforward neural networks.For fractional-order neural networks with unsaturating piecewise linear activation functions,we study the monostability and multistability results of almost-periodic solutions,some globally Mittag-Leffler attractive sets are given,and the existence and uniqueness of globally Mittag-Leffler stable almost-periodic solutions are demonstrated by using Ascoli-Arzela theorem.Sufficient criteria are demonstrated to guarantee the local Mittag-Leffler stability of almost-periodic solutions by using local positive invariant sets,it is proved that there exists a locally Mittag-Leffler stable almost-periodic solution in each positive invariant set,and all trajectories converge to the periodic trajectories in this positive invariant set.(3)The multistability problem of almost-periodic solutions of Takagi-Sugeno fuzzy neural networks with nonmonotonic discontinuous activation functions and time-varying delays is discussed.Based on the geometrical properties of the nonmonotonic discontinuous activation functions,by using Ascoli-Arzela theorem and the inequality techniques,it is demonstrated that under some reasonable conditions,the addressed networks have unique locally exponentially stable almost-periodic solution in some hyperrectangular regions,and we also estimate the attraction basins of the locally stable almost-periodic solutions.These results here,which include boundedness,globally attractivity,multiple stability,attraction basin,can be extended to monostability and multistability for almost-period solutions of Takagi-Sugeno fuzzy neural networks with nonmonotonic discontinuous activation functions,which fill the gap of multistability in fuzzy neural networks.(4)A new periodic impulsive control strategy is designed for inertial reaction-diffusion neural networks and time-dependent switching neural networks with discrete and finite distributed time-varying delays to generate globally exponentially stable periodic solutions.In order to reduce the conservatism of the global uniform exponential convergence criterion,a method based on adjustable parameters and matrix-based quadratic,cubic convex combination are proposed to study the boundedness and Lagrange stability of the addressed neural networks.Sufficient conditions for the existence,uniqueness,and globally exponential stability of periodic solutions are developed by utilizing contraction mapping theorem and impulse-delaydependent Lyapunov-Krasovskii functional method.It should be pointed out that the addressed Lyapunov-Krasovskii functionals include triple integral terms and new quadruple integral terms,which will reduce the conservatism of the stability conditions of the neural networks.Even if the original neural network model is unstable or even divergent,the two types of neural networks can generate globally exponentially stable periodic solutions through impulsive control. |
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