Stability And Dissipativity Of Complex Systems:Continuous-Time And Discrete-Time Cases

A complex system is a system composed of many components which may interact with each other,and is also the main research tasks of complexity science.Neural network(NN),as one of the main manifestations of complex systems,is not only an interdisci-plinary subject involving biology,mathematics,computer science and other fields,but also shows good intelligent characteristics in practical applications.In addition,as a new circuit element,memristor has attracted wide attention due to its natural memory func-tion and nano-scale physical size.Therefore,the study on the memristive neural networks(MNNs)has greater theoretical and application value undoubtedly.Up to now,many scholars have studied MNNs,but mostly concentrated on continuous-time MNNs.However,discrete-time systems have irreplaceable advantages over continuous-time systems in practice.Hence,we will analyze and design continuous-time and discrete-time MNNs respectively,while improve the previous results and explore new methods in this paper.On this basis,some other problems also should be further considered here,such as,saturation nonlinearity,discontinuity,disturbance input,performance index and so on.The main contents of this paper are listed as follows:In the first part,the stability of discrete-time high-order neural networks(HONNs)with delays and impulses is considered.Firstly,a basic lemma is proposed to illustrate the stability relationship between the system with impulses and the corresponding system without impulse.Secondly,instead of Lyapunov stability theory,both the asymptotic stability and exponential stability of discrete-time HONNs are analyzed via using the fixed point theory.Finally,the numerical simulation further shows that the conservativeness of the criteria is reduced in this paper.In the second part,both the stability and synchronization of discrete-time delayed NNs with discontinuous activations are studied.Firstly,based on differential inclusion theory and Kakutani fixed point theorem,the existence and uniqueness of solutions are ensured.Secondly,the attractivity and stability of the system are investigated through the Halanay-type inequality and comparison principle.Furthermore,under weaker as-sumptions than Lipschitz conditions,two different controllers are separately designed and some sufficient conditions for the synchronization of discrete-time NNs with discontinuous right-hand sides are obtained.Finally,the corresponding numerical simulations are given to verify the validity of the conclusions.In the third part,the delay-dependent stability of continuous-time and discrete-time memristive complex-valued neural networks(MCVNNs)is investigated.Firstly,the complex-valued systems are transformed into real-valued systems by constructing linear mappings.Secondly,in view of an extended matrix inequality,the stability of continuous-time systems is discussed when activation functions are continuous but not Lipschitz con-tinuous.Thirdly,when activation functions are discontinuous,a discontinuous adaptive controller is designed to guarantee its stability.Meanwhile,two different methods are proposed to explore the dynamical behaviors of the corresponding discrete-time systems.Finally,a detailed numerical analysis is provided to demonstrate the correctness of the results.In the fourth part,the H_?control problem for memristive neural networks(MNNs)with aperiodic sampling and actuator saturation is addressed.Firstly,combined with discrete-time Lyapunov theory and aperiodic sampling system theory,stability condi-tions of the estimation error system with input saturation are given and its ellipsoidal region of stability is determined by choosing a polyhedral set and designing a saturating sampled-data control.Secondly,we analyze the H_?performance of the system with ex-ternal disturbances.Finally,to further illustrate the effectiveness of the conclusions,both the comparative analysis on the the existing ellipsoidal regions of the stability and the corresponding numerical simulations are given here.In the last part,the exponential dissipativity of discrete-time switched MNNs with actuator saturation is considered.Under the mode-dependent average dwell time(M-DADT)switching,firstly,a lemma is improved to deal with the problem of saturation nonlinearity.Secondly,based on the quasi-time-dependent(QTD)method,we analyze the exponential stability of closed-loop systems,and acquire the(Q,S,R)-exponentially dissipative criteria.Similarly,in the case of external disturbances,the H_?performance of the system is discussed,and its observer and controller gains are also obtained.Finally,a numerical example is provided to show the feasibility and superiority of the conclusions.