GLOBAL DYNAMICS OF DELAYED BIDIRECTIONAL ASSOCIATIVE MEMORY （BAM） NEURAL NETWORKS

IntroductionConsiderthebidirectionalassociativememory (BAM )neuralnetworkswithconstanttransmissiondelaysdescribedbyasystemofdelaydifferentialequationsoftheform[1,2 ]:dxi(t)dt =-aixi(t) nj=1bijfj(yj(t-σij) ) Ii,　　i=1 ,2 ,… ,m ,dyj(t)dt =-cjyj(t) mi=1djigi(xi(t-τji) ) Jj,　　j=1 ,2 ,… ,n ,fort >0 .Thesystem ( 1 )consistsoftwosetsofneurons (orunits)arrangedontwolayers,namely ,I_layerandJ_layer.Inthesystem ( 1 ) ,xi( ·)andyj( ·)denotemembranepotentialoftheithneuronsfromtheI_laye…     

Global Attractivity and Global Exponential Stability for Delayed Hopfield Neural Network Models

ＩｎｔｒｏｄｕｃｔｉｏｎＨｏｐｆｉｅｌｄｎｅｕｒａｌｎｅｔｗｏｒｋｍｏｄｅｌｉｓｏｎｅｏｆｔｈｅｍｏｓｔｐｏｐｕｌａｒｍｏｄｅｌｓｉｎｔｈｅｌｉｔｅｒｒａｔｕｒｅｏｆａｒｔｉｆｉｃｉａｌｎｅｕｒａｌｎｅｔｗｏｒｋｓ,ｗｈｉｃｈｉｓｄｅｓｃｒｉｂｅｄｂｙｔｈｅｆｏｌｌｏｗｉｎｇｎｏｎｌｉｎｅａｒｄｙｎａｍｉｃｓｅｑｕａｔｉｏｎｓ[1,2 ]:Ｃｉｄｕｉ(ｔ)ｄｔ =-ｕｉ(ｔ)Ｒｉ ∑ｎｊ=1Ｔｉｊｇｊ(ｕｊ(ｔ) ) Ｉｉ　　 (ｉ=1 ,2 ,… ,ｎ) ,( 1 )ｗｈｅｒｅｎ≥ 2ｉｓｔｈｅｎｕｍｂｅｒｏｆｎｅｕｒｏｎｓｉｎｔｈｅ…     

Delay-independent exponential stability of stochastic Cohen–Grossberg neural networks with time-varying delays and reaction–diffusion terms

Different from the approaches used in the earlier papers, in this paper, the Halanay inequality technique, in combination with the Lyapunov method, is exploited to establish a delay-independent sufficient condition for the exponential stability of stochastic Cohen–Grossberg neural networks with time-varying delays and reaction–diffusion terms. Moreover, for the deterministic delayed Cohen–Grossberg neural networks, with or without reaction–diffusion terms, sufficient criteria for their global exponential stability are also obtained. The proposed results improve and extend those in the earlier literature and are easier to verify. An example is also given to illustrate the correctness of our results.     

Global exponential estimates for uncertain Markovian jump neural networks with reaction-diffusion terms

In this paper, the robust global exponential estimating problem is investigated for Markovian jumping reaction-diffusion delayed neural networks with polytopic uncertainties under Dirichlet boundary conditions. The information on transition rates of the Markov process is assumed to be partially known. By introducing a new inequality, some diffusion-dependent exponential stability criteria are derived in terms of relaxed linear matrix inequalities. Those criteria depend on decay rate, which may be freely selected in a range according to practical situations, rather than required to satisfy a transcendental equation. Estimates of the decay rate and the decay coefficient are presented by solving these established linear matrix inequalities. Numerical examples are provided to demonstrate the advantage and effectiveness of the proposed method.     

Control of friction oscillator by Lyapunov redesign based on delayed state feedback

Abstract The stability and boundedness of mechanical system have been one of important research topics. In this paper ultimate boundedness of a dry friction oscillator, belonging to nonsmooth mechanical system, is investigated by proposing a controller design method. Firstly a sufficient condition of the stability for the nominal system with delayed state feedback is derived by constructing a Lyapunov-Krasovskii function. The delayed feedback gain matrix is calculated by applying linear matrix inequality method. Secondly on the basis of the delayed state feedback, a continuous function is designed by Lyapunov redesign to ensure that the solutions of the friction oscillator system are ultimately bounded under the overall control. Moreover, the ultimate bound can be adjusted in practice by choosing appropriate parameter. Accordingly friction-induced vibration or instability can be controlled effectively. Numerical results show that the pro- posed method is valid.     

Global asymptotic stability of cellular neural networks with proportional delays

Proportional delay, which is different from distributed delay, is a kind of unbounded delay. The proportional delay system as an important mathematical model often rises in some fields such as physics, biology systems, and control theory. In this paper, the uniqueness and the global asymptotic stability of equilibrium point of cellular neural networks with proportional delays are analyzed. By using matrix theory and constructing suitable Lyapunov functional, delay-dependent and delay-independent sufficient conditions are obtained for the global asymptotic stability of cellular neural networks with proportional delays. These results extend previous works on these issues for the delayed cellular neural networks. Two numerical examples and their simulation are given to illustrate the effectiveness of obtained results.     

Exponential synchronization of stochastic delayed discrete-time complex networks

This paper is concerned with the problem of synchronization for stochastic discrete-time drive-response networks with time-varying delay. By employing the Lyapunov functional method combined with the stochastic analysis as well as the feedback control technique, several sufficient conditions are established that guarantee the exponentially mean-square synchronization of two identical delayed networks with stochastic disturbances. These sufficient conditions, which are expressed in terms of linear matrix inequalities (LMIs), can be solved efficiently by the LMI toolbox in Matlab. A particular feature of the LMI-based synchronization criteria is that they are dependent not only on the connection matrices in the drive networks and the feedback gains in the response networks, but also on the lower and upper bounds of the time-varying delay, and are therefore less conservative than the delay-independent ones. Two numerical examples are exploited to demonstrate the feasibility and applicability of the proposed synchronization approaches.     

Adaptive-impulsive projective synchronization of drive-response delayed complex dynamical networks with time-varying coupling

This paper investigates the adaptive-impulsive projective synchronization of drive-response delayed complex dynamical networks with time-varying coupling, in which the weights of links between two connected nodes are time varying. By the stability analysis of the impulsive functional differential equation, the sufficient conditions for achieving projective synchronization are obtained, and a hybrid controller, that is, an adaptive feedback controller with impulsive control effects is designed. The numerical examples are presented to illustrate the effectiveness and advantage of the proposed synchronization criteria.     

A delay partitioning approach to output feedback control for uncertain discrete time-delay systems with actuator saturation

This paper is concerned with the problems of output feedback control for uncertain discrete time-delay systems with input saturation. The delay partitioning approach is proposed to obtain new stability criteria. The dynamic output feedback controller is designed based on a linear matrix inequality framework. A sufficient condition is developed, which guarantees the existence of dynamic output feedback controllers such that all trajectories of the closed-loop system starting from an admissible initial condition domain converge to a smaller ellipsoid. Simulation examples are provided to show the potential of the proposed techniques.     

Periodically intermittent control strategies for $$\varvec{\alpha }$$-exponential stabilization of fractional-order complex-valued delayed neural networks

This paper studies the global \(\alpha \)-exponential stabilization of a kind of fractional-order neural networks with time delay in complex-valued domain. To end this, several useful fractional-order differential inequalities are set up, which generalize and improve the existing results. Then, a suitable periodically intermittent control scheme with time delay is put forward for the global \(\alpha \)-exponential stabilization of the addressed networks, which include feedback control as a special case. Utilizing these useful fractional-order differential inequalities and combining with the Lyapunov approach and other inequality techniques, some novel delay-independent criteria in terms of real-valued algebraic inequalities are obtained to ensure global \(\alpha \)-exponential stabilization of the discussed networks, which are very simple to implement in practice and avert to calculate the complex matrix inequalities. Finally, the availability of the theoretical criteria is verified by an illustrative example with simulations.     

Stability and almost periodicity for delayed high-order Hopfield neural networks with discontinuous activations

In this paper, the stability and almost periodicity for delayed high-order Hopfield neural networks with discontinuous activations are investigated. Some new criteria ensuring the existence and global exponential stability of almost periodic solution for the considered neural network model are established by employing the differential inclusion theory, differential inequality technique, and Lyapunov functional approach, the results of this paper improve and complement previously known results. Finally, examples with numerical simulations are presented to demonstrate the effectiveness of theoretical results.     

A LMI-based approach to global asymptotic stability of neural networks with time varying delays

In this paper, the asymptotic stability of neural networks with time varying delay is studied by using the nonsmooth analysis, Lyapunov functional method and linear matrix inequality (LMI) technique. It is noted that the proposed results do not require smoothness of the behaved function and activation function as well as boundedness of the activation function. Several sufficient conditions are presented to show the uniqueness and the global asymptotical stability of the equilibrium point. Also, a high-dimensional matrix condition to ensure the uniqueness and the global asymptotical stability of equilibrium point can be reduced to a low-dimensional condition. The obtained results are easy to apply and improve some earlier works. Finally, we give two simulations to justify the theoretical analysis in this paper. This work was supported by the Natural Science Foundation of Jiangsu Province of China under Grant No. BK2006093.     

Synchronization of unknown chaotic delayed competitive neural networks with different time scales based on adaptive control and parameter identification

In this paper, we investigate the synchronization problems of delayed competitive neural networks with different time scales and unknown parameters. A simple and robust adaptive controller is designed such that the response system can be synchronized with a drive system with unknown parameters by utilizing Lyapunov stability theory and parameter identification. Our synchronization criteria are easily verified and do not need to solve any linear matrix inequality. This research also demonstrates the effectiveness of application in secure communication. Numerical simulations are carried out to illustrate the main results.     

Integral-based event-triggered synchronization criteria for chaotic Lur’e systems with networked PD control

Integral-based event-triggered synchronization criteria are firstly presented for networked chaotic systems with proportional-derivative (PD) control. The event-triggered scheme effectively utilizes network resources; however, the PD-type control subject to the conventional triggering inequality may cause excessive triggering and have difficulty in obtaining a feasible solution. To solve these problems, the integrated event-triggering inequality is employed and the modified integral inequality with free-weighting matrix is proposed to fill the empty diagonal terms, which overcomes the difficulties of the integration of delayed signal vectors upon integral event-triggering condition. Based on Lyapunov stability, the synchronization criteria are derived as linear matrix inequalities. Finally, the effectiveness of the integral-based event-triggered synchronization method is demonstrated by numerical examples.     

Stability analysis of delayed cellular neural networks with and without noise perturbation

The stability of a class of delayed cellular neural networks (DCNN) with or without noise perturbation is studied.After presenting a simple and easily checkable condition for the global exponential stability of a deterministic system,we further investigate the case with noise perturbation.When DCNN is perturbed by external noise,the system is globally stable.An important fact is that,when the system is perturbed by internal noise,it is globally exponentially stable only if the total noise strength is within a certain bound.This is significant since the stochastic resonance phenomena have been found to exist in many nonlinear systems.     

Globally exponential stability of stochastic neutral-type delayed neural networks with impulsive perturbations and Markovian switching

The problem of globally exponential stability of stochastic neutral-type delayed neural networks with impulsive perturbations and Markovian switching is studied in this paper. By using the Lyapunov?CKrasovskii method and the stochastic analysis approach, a sufficient condition to ensure globally exponential stability for the stochastic neutral-type delayed neural networks with impulsive perturbations and Markovian switching is derived. Finally, a numerical example is given to illustrate the effectiveness of the result proposed in this paper.     

New augmented Lyapunov–Krasovskii functional approach to stability analysis of neural networks with time-varying delays

This paper is concerned with the problem of stability analysis for neural networks with time-varying delays. By constructing a newly augmented Lyapunov functional and some novel techniques, delay-dependent criteria to guarantee the asymptotic stability of the concerned networks are derived in terms of linear matrix inequalities (LMIs). The improvement of feasible region of the proposed criteria comparing with the previous works is shown by two numerical examples.     

Adaptive synchronization of different Cohen–Grossberg chaotic neural networks with unknown parameters and time-varying delays

In this paper, an adaptive linear feedback controller is presented to study the synchronization problem of different Cohen–Grossberg neural networks with unknown parameters and time-varying delays. Lyapunov stability theory and Barbalat’s lemma are used to guarantee the response system can be synchronized with the drive system. The synchronization criteria of this paper which do not solve any linear matrix inequality are easily verified. These results remove some restrictions on amplification functions and activation functions. Finally, numerical simulations are carried out to illustrate the effectiveness of the obtained results.     

Exponential synchronization of chaotic Lur’e systems with delayed feedback control

The exponential synchronization problem is studied in this paper for a class of chaotic Lur’e systems by using delayed feedback control. An augmented Lyapunov functional based approach is proposed to deal with this issue. A delay-dependent condition is established such that the controlled slave system can exponentially synchronize with the master system. It is shown that the delayed feedback gain matrix and the exponential decay rate can be obtained by solving a set of linear matrix inequalities. The decay coefficient can be also easily calculated. Finally, as an example, the Chua’s circuit is used to illustrate the effectiveness of the developed approach and the improvement over some existing results.     

Exponential stability in the mean square for stochastic neural networks with mixed time-delays and Markovian jumping parameters

In this paper, the stability analysis problem is considered for a class of stochastic neural networks with mixed time-delays and Markovian jumping parameters. The mixed delays include discrete and distributed time-delays, and the jumping parameters are generated from a continuous-time discrete-state homogeneous Markov process. The aim of this paper is to establish some criteria under which the delayed stochastic neural networks are exponentially stable in the mean square. By constructing suitable Lyapunov functionals, several stability conditions are derived on the basis of inequality techniques and the stochastic analysis. An example is also provided in the end of this paper to demonstrate the usefulness of the proposed criteria.