Dynamical Analysis And Control Of Memristive Neural Networks With Time-varying Delays

It is a common sense that there are three basic circuit elements(resistor,inductor and capacitor)for a long time.In 1971,Dr.Chua however predicted that another basic circuit element should exist and it was named as a memristor(abbreviated by memory and resistor)by him.The memristor,as the fourth basic circuit element,has been the subject of research in the world.Particularly,in 2008,a team at Hewlett-Packard Labs claimed to have found memristor real physical model based on a nanometre scale TiO2 thin film.After that,more attention has been paid on the memristor and the memristive neural networks.Many scholars have participated in this area and numerous results have been achieved.The memristor is a passive two-terminal electronic elememt and is also a nanometer element.Meanwhile,it has the features of low-energy consumption,high-storage,small-volume and non-volatile.As a new type of memory device,the memristor has similar characteristics as human brain synapses,which is expected to realize the integration of information storage and processing and breaks the current von Neumann computer architecture,and provide new design architecture for the next generation of computer.Among them,the distinct characteristic is its memory function,which is very similar to the synaptic of biological neurons.Such function exhibits its broad applications.In recent years,some researchers have replaced the synaptic connections in Hopfield neural networks by the memristor,and have established a new neural network based on memristor.In a word,the advent of the memristor makes it possible for artificial neural networks to simulate the human brain,and greatly promotes the development of artificial intelligence.In this paper,the dynamic behavior and control of neural networks based on the memristor with time-delays,i.e.delayed memristive neural networks(DMNN)are studied.Since the DMNN are special state-dependent switched systems,and they are discontinuous on their right hand side,the classical differential equation theory is no longer applicable and has to open up technical paths.For this reason,we introduce the theory of differential inclusion and set-valued maps,and transform the DMNN into the conventional delayed neural network.With the help of Lyapunov stability theory,we focus on the DMNN dynamic behavior such as stability,dissipativity and passivity.Sufficient conditions related to the dynamic behavior mentioned above are derived in terms of linear matrix inequality(LMI)or M-matrix and the design of the controllers is given.The main works are summarized as follows:Firstly,a mathematical model of the DMNN with discrete time delay ?(t)in a given range[?1,?2]is established.On this basis,the global asymptotic stability and exponential stability of the DMNN are analyzed.With aid of the non-smooth analysis and differential inclusion theory,the MNN are transformed into the conventional neural networks.It is the first time that we have analyzed the global asymptotic stability by using a form of Lyapunov functional with time-delay product.The sufficient conditions guaranteeing the global asymptotic stability and exponential stability of the DMNN in the form of LMI are obtained,and the corresponding exponential stabilization controllers are designed.Secondly,the dissipativity and passivity of a class of MNN with mixed delays(including discrete,distribute and leakage delay)are studied.Based on the theory of set-valued maps and differential inclusion,the passivity and dissipativity of this kind of MNN are discussed.By establishing appropriate Lyapunov-Krasovskii functionals(LKF)and an improved Wirtinger integral inequality together with the reciprocal convex combination inequality,the criteria with less conservatism on the strict(Q,R,S)-?-dissipativity and passivity of the proposed neural networks are acquried.Due to the fact that the delay discussed here is mixed,these criteria derived in this paper cover some results in the literature.There are always random factors in practical engineering,and they can not be ignored in many cases.Hence we turn to study a type of stochastic DMNN,and discuss its passivity and dissipativity.With aid of the theory of nonsmooth analysis and stochastic differential inclusion,the passivity and dissipativity of such DMNN are analyzed,and the sufficient conditions for the passivity and the strict(Q,R,S)-?-dissipativity in the mean square sense are gained.In addition,the finite time control is more practical than the infinite time control,and has attracted more attention.Therefore,we finally investigate the finite time stability and stabilization of a class of DMNN.With the help of Dini derivative theory,we construct an appropriate Lyapunov functional,and deduce some sufficient conditions for finite time stability in the form of M-matrix.Based on these conditions,nonlinear state-feedback controllers for finite time stabilization are designed.