Banach空间中线性离散时间系统的一致多项式膨胀性

给出了Banach 空间中线性离散时间系统一致多项式膨胀性的概念,并讨论了其离散特征。借助Lyapunov函数给出了线性离散时间系统满足一致多项式膨胀的充要条件。所得结论将一致指数稳定性、指数膨胀性及多项式稳定性中的若干经典结论推广到了一致多项式膨胀性的情形。     

Global exponential stability of impulsive discrete-time neural networks with time-varying delays

This paper studies the problem of global exponential stability and exponential convergence rate for a class of impulsive discrete-time neural networks with time-varying delays. Firstly, by means of the Lyapunov stability theory, some inequality analysis techniques and a discrete-time Halanay-type inequality technique, sufficient conditions for ensuring global exponential stability of discrete-time neural networks are derived, and the estimated exponential convergence rate is provided as well. The obtained results are then applied to derive global exponential stability criteria and exponential convergence rate of impulsive discrete-time neural networks with time-varying delays. Finally, numerical examples are provided to illustrate the effectiveness and usefulness of the obtained criteria.     

Global exponential stability of static neural networks with delay and impulses: Discrete-time case

In this paper, we investigate the exponential stability of discrete-time static neural networks with impulses and variable time delay. The discrete-time neural networks are derived by discretizing the corresponding continuous-time counterparts with implicit-explicit-θ (IMEX-θ) method. The impulses are classified into three classes: input disturbances, stabilizing and “neutral” type— the impulses are neither helpful for stabilizing nor destabilizing the neural networks, and then by using a very excellent ideology introduced recently the connections between the impulses and the utilized Lyapunov function are fully explored with respect to each type of impulse. New analysis techniques that used to realize the ideology in discrete-time situation are proposed and it is shown that they are essentially different from the ones used in continuous-time case. Several criteria for global exponential stability of the static neural networks in discrete-time case are established in terms of linear matrix inequalities (LMIs) and numerical simulations are given to validate the obtained theoretical results.     

Exponential stability of discrete-time neural networks with delay and impulses

In this paper, we investigate the exponential stability of discrete-time neural networks with impulses and time-varying delay. The discrete-time neural networks are derived by discretizing the corresponding continuous-time counterparts with different discretization methods. The impulses are classified into three classes: input disturbances, stabilizing and “neutral” type - the impulses are neither helpful for stabilizing nor destabilizing the neural networks, and then by using the excellent ideology introduced recently by Chen and Zheng [W.H. Chen, W.X. Zheng, Global exponential stability of impulsive neural networks with variable delay: an LMI approach, IEEE Trans. Circuits Syst. I 56 (6) (2009) 1248-1259], the connections between the impulses and the utilized Lyapunov function are fully explored with respect to each type of impulse. Novel techniques that used to realize the ideology in discrete-time situation are proposed and it is shown that they are essentially different from the continuous-time case. Several criteria for global exponential stability of the discrete-time neural networks are established in terms of matrix inequalities and based on these theoretical results numerical simulations are given to compare the capability of different discretization methods.     

Global input-to-state stability and stabilization of discrete-time piecewise affine systems

This article presents conditions for global input-to-state stability (ISS) and stabilization of discrete-time, possibly discontinuous, piecewise affine (PWA) systems. Piecewise quadratic, possibly discontinuous candidate ISS Lyapunov functions are employed for both analysis and synthesis purposes. This enables us to obtain sufficient conditions based on linear matrix inequalities, which can be solved efficiently. One of the advantages of using the ISS framework is that additive disturbance inputs are explicitly taken into account in the analysis and synthesis procedures. Furthermore, the results apply to PWA systems in their full generality, i.e. non-zero affine terms are allowed in the regions in the partition whose closure contains the origin.     

Robust exponential stability analysis of discrete-time switched Hopfield neural networks with time delay

The robust exponential stability problem in this paper for discrete-time switched Hopfield neural networks with time delay and uncertainty is considered. Firstly, the mathematical model of the system is established. Then by constructing a new Lyapunov–Krasovskii functional, some new delay-dependent criteria are developed, which guarantee the robust exponential stability of discrete-time switched Hopfield neural networks. A numerical example is provided to demonstrate the potential and effectiveness of the results obtained.     

Input-to-state stability of impulsive systems and their networks

This paper considers input-to-state stability (ISS) characterization for a class of impulsive systems which jump map depends on time. We provide sufficient conditions in terms of exponential ISS-Lyapunov functions equipped with an appropriate dwell-time condition for establishing ISS property. Some modifications of dwell times which are more conservative, but easy to be verified are being introduced. We also show that impulsive system with multiple jump maps can naturally represent an interconnection of several impulsive systems with different impulse time sequences. Then we present a procedure to verify ISS of such networks.     

Stability analysis and stabilization of linear symmetric matrix-valued continuous,discrete, and impulsive dynamical systems — A unified approach for the stability analysis and the stabilization of linear systems

Matrix-valued dynamical systems are an important class of systems that can describe important processes such as covariance/second-order moment processes, or processes on manifolds and Lie Groups. We address here the case of processes that leave the cone of positive semidefinite matrices invariant, thereby including covariance and second-order moment processes. Both the continuous-time and the discrete-time cases are first considered. In the LTV case, the obtained stability and stabilization conditions are expressed as differential and difference Lyapunov conditions which are equivalent, in the LTI case, to some spectral conditions for the generators of the processes. Convex stabilization conditions are also obtained in both the continuous-time and the discrete-time setting. It is proven that systems with constant delays are stable provided that the systems with zero-delays are stable—which mirrors existing results for linear positive systems. The results are then extended and unified into an impulsive formulation for which similar results are obtained. The proposed framework is very general and can recover and/or extend many of the existing results in the literature on linear systems related to (mean-square) exponential (uniform) stability. Several examples are discussed to illustrate this claim by deriving stability conditions for stochastic systems driven by Brownian motion and Poissonian jumps, Markov jump systems, (stochastic) switched systems, (stochastic) impulsive systems, (stochastic) sampled-data systems, and all their possible combinations.     

A new stability criterion for discrete-time neural networks: Nonlinear spectral radius

In this paper, the exponential stability of nonlinear discrete-time systems is studied. A novel notion of nonlinear spectral radius is defined. Under the assumption of Lipschitz continuity for the activation function, the developed approach is applied to stability analysis of discrete-time neural networks. A series of sufficient conditions for global exponential stability of the neural networks are established and an estimate of the exponential decay rate is also derived for each case.     

Exponential stability preservation in discrete-time analogues of artificial neural networks with distributed delays

This paper demonstrates that there is a discrete-time analogue which does not require any restriction on the size of the time-step in order to preserve the exponential stability of an artificial neural network with distributed delays. The analysis exploits an appropriate Lyapunov sequence and a discrete-time system of Halanay inequalities, and also either a Young inequality or a geometric-arithmetic mean inequality, to derive several sufficient conditions on the network parameters for the exponential stability of the analogue. The sufficiency conditions are independent of the time-step, and they correspond to those that establish the exponential stability of the continuous-time network.     

Stability of interconnected impulsive systems with and without time delays,using Lyapunov methods

In this paper, we consider the input-to-state stability (ISS) of impulsive control systems with and without time delays. We prove that, if the time-delay system possesses an exponential Lyapunov–Razumikhin function or an exponential Lyapunov–Krasovskii functional, then the system is uniformly ISS provided that the average dwell-time condition is satisfied. Then, we consider large-scale networks of impulsive systems with and without time delays and prove that the whole network is uniformly ISS under the small-gain and the average dwell-time condition. Moreover, these theorems provide us with tools to construct a Lyapunov function (for time-delay systems, a Lyapunov–Krasovskii functional or a Lyapunov–Razumikhin function) and the corresponding gains of the whole system, using the Lyapunov functions of the subsystems and the internal gains, which are linear and satisfy the small-gain condition. We illustrate the application of the main results on examples.     

STABILITY OF LINEAR MEASURE LARGE SCALE SYSTEMS WITH IMPULSIVE EFFECT

ＳＴＡＢＩＬＩＴＹＯＦＬＩＮＥＡＲＭＥＡＳＵＲＥＬＡＲＧＥＳＣＡＬＥＳＹＳＴＥＭＳＷＩＴＨＩＭＰＵＬＳＩＶＥＥＦＦＥＣＴＧｕａｎＺｈｉｈｏｎｇ（关治洪）（ＪｉａｎｇｈａｎＰｅｔｒｏｌｅｕｍＩｎｓｔｉｔｕｔｅ，江汉石油学院，邮编：４３４１０２）ＷｅｎＸ...     

Exponential stability criteria for discrete-time recurrent neural networks with time-varying delay

In this paper, the robust global exponential stability is investigated for the discrete-time recurrent neural networks (RNNs) with time-varying interval delay. By choosing an augmented Lyapunov–Krasovskii functional, delay-dependent results guaranteeing the global exponential stability and the robust exponential stability of the concerned neural network are obtained. The results are shown to be a generalization of some previous results, and less conservative than the existing works. Two numerical examples are given to demonstrate the applicability of the proposed method.     

Stochastic stabilization of hybrid neural networks by periodically intermittent control based on discrete-time state observations

This paper is concerned with stabilization of hybrid neural networks by intermittent control based on continuous or discrete-time state observations. By means of exponential martingale inequality and the ergodic property of the Markov chain, we establish a sufficient stability criterion on hybrid neural networks by intermittent control based on continuous-time state observations. Meantime, by M-matrix theory and comparison method, we show that hybrid neural networks can be stabilized by intermittent control based on discrete-time state observations. Finally, two examples are presented to illustrate our theory.     

Observer design for discrete-time nonlinear systems

This paper is a geometric study of the observer design for discrete-time nonlinear systems. First, we obtain necessary and sufficient conditions for local exponential observers for Lyaupnov stable discrete-time nonlinear systems. We also show that the definition of local exponential observers can be considerably weakened for neutrally stable discrete-time nonlinear systems. As an application of our local observer design, we consider a class of discrete-time nonlinear systems with an input generator (exosystem) and show that for this class of nonlinear systems, under some stability assumptions, the existence of local exponential observers in the presence of inputs implies and is implied by the existence of local exponential observers in the absence of inputs.     

Banach空间中线性斜积半流指数稳定性的Holder型刻画

为了在Banach空间X的对偶空间上刻画线性斜演化半流的一致指数稳定性,借助泛函分析与算子理论得到了其一致指数稳定的一些H?lder型充要条件,所得结果推广了稳定性理论中的一些已有结论.     

Exponential Stability in Mean Square for a General Class of Discrete-Time Linear Stochastic Systems

Abstract

The problem of the mean square exponential stability for a class of discrete-time linear stochastic systems subject to independent random perturbations and Markovian switching is investigated. The case of the linear systems whose coefficients depend both to present state and the previous state of the Markov chain is considered. Three different definitions of the concept of exponential stability in mean square are introduced and it is shown that they are not always equivalent. One definition of the concept of mean square exponential stability is done in terms of the exponential stability of the evolution defined by a sequence of linear positive operators on an ordered Hilbert space. The other two definitions are given in terms of different types of exponential behavior of the trajectories of the considered system. In our approach the Markov chain is not prefixed. The only available information about the Markov chain is the sequence of probability transition matrices and the set of its states. In this way one obtains that if the system is affected by Markovian jumping the property of exponential stability is independent of the initial distribution of the Markov chain.

The definition expressed in terms of exponential stability of the evolution generated by a sequence of linear positive operators, allows us to characterize the mean square exponential stability based on the existence of some quadratic Lyapunov functions.

The results developed in this article may be used to derive some procedures for designing stabilizing controllers for the considered class of discrete-time linear stochastic systems in the presence of a delay in the transmission of the data.     

Discrete-time analogues of integrodifferential equations modelling bidirectional neural networks

We formulate discrete-time analogues of integrodifferential equations modelling bidirectional neural networks studied by Gopalsamy and He. The discrete-time analogues are considered to be numerical discretizations of the continuous-time networks and we study their dynamical characteristics. It is shown that the discrete-time analogues preserve the equilibria of the continuous-time networks. By constructing a Lyapunov-type sequence, we obtain easily verifiable sufficient conditions under which every solution of the discrete-time analogue converges exponentially to the unique equilibrium. The sufficient conditions are identical to those obtained by Gopalsamy and He for the uniqueness and global asymptotic stability of the equilibrium of the continuous-time network. By constructing discrete-time versions of Halanay-type inequalities, we obtain another set of easily verifiable sufficient conditions for the global exponential stability of the unique equilibrium of the discrete-time analogue. The latter sufficient conditions have not been obtained in the literature of continuous-time bidirectional neural networks. Several computer simulations are provided to illustrate the advantages of our discrete-time analogue in numerically simulating the continuous-time network with distributed delays over finite intervals.     

EXPONENTIAL STABILITY OF LINEAR TIME－VARYING IMPULSIVE DIFFERENTIAL SYSTEMS WITHDELAYS

ＥＸＰＯＮＥＮＴＩＡＬＳＴＡＢＩＬＩＴＹＯＦＬＩＮＥＡＲＴＩＭＥ－ＶＡＲＹＩＮＧＩＭＰＵＬＳＩＶＥＤＩＦＦＥＲＥＮＴＩＡＬＳＹＳＴＥＭＳＷＩＴＨＤＥＬＡＹＳ￥ＧｕａｎＺｈｉｈｏｎｇ（关治洪）；ＬｉｕＹｏｎｇｑｉｎｇ（刘永清）（ＳｏｕｔｈＣｈｉｎａＵｎ...     

Global exponential stability of BAM neural networks with time-varying delays: The discrete-time case

This paper deals with the problem of stability analysis for a class of discrete-time bidirectional associative memory (BAM) neural networks with time-varying delays. By employing the Lyapunov functional and linear matrix inequality (LMI) approach, a new sufficient conditions is proposed for the global exponential stability of discrete-time BAM neural networks. The proposed LMI based results can be easily checked by LMI control toolbox. Moreover, an example is also provided to demonstrate the effectiveness of the proposed method.