Robustness of stability of time-varying index-1 DAEs

We study exponential stability and its robustness for time-varying linear index-1 differential-algebraic equations. The effect of perturbations on the leading coefficient matrix is investigated. An appropriate class of allowable perturbations is introduced. Robustness of exponential stability with respect to a certain class of perturbations is proved in terms of the Bohl exponent and perturbation operator. Finally, a stability radius involving these perturbations is introduced and investigated. In particular, a lower bound for the stability radius is derived. The results are presented by means of illustrative examples.     

Switched nonlinear differential algebraic equations: Solution theory,Lyapunov functions,and stability

We study switched nonlinear differential algebraic equations (DAEs) with respect to existence and nature of solutions as well as stability. We utilize piecewise-smooth distributions introduced in earlier work for linear switched DAEs to establish a solution framework for switched nonlinear DAEs. In particular, we allow induced jumps in the solutions. To study stability, we first generalize Lyapunov’s direct method to non-switched DAEs and afterwards obtain Lyapunov criteria for asymptotic stability of switched DAEs. Developing appropriate generalizations of the concepts of a common Lyapunov function and multiple Lyapunov functions for DAEs, we derive sufficient conditions for asymptotic stability under arbitrary switching and under sufficiently slow average dwell-time switching, respectively.     

Preservation of exponential stability for linear non-autonomous functional differential systems

We consider preservation of exponential stability for a system of linear equations with a distributed delay under the addition of new terms and a delay perturbation. As particular cases, the system includes models with concentrated delays and systems of integrodifferential equations. Our method is based on Bohl–Perron type theorems.     

Stability of linear switched differential algebraic equations with stable and unstable subsystems

This article studies linear switched differential algebraic equations (DAEs), which contains stable and unstable subsystems. We prove sufficient conditions for stability of switched DAEs based on the existence of suitable Lyapunov functions. The result shows that stability is preserved under switching with an average dwell time and an additional condition involving consistency projectors holds. Furthermore, we also give an example to illustrate the result.     

Two-sided bounds for the largest Lyapunov exponent and exponential stability criteria for nonlinear systems with arbitrary delays

We consider a system of nonlinear differential equations with a given linear part, a nonlinear term bounded in the norm, and variable concentrated and distributed delays. We find two-sided bounds on the maximal Lyapunov exponent expressed via the norm of the nonlinear term and maxima of the delay functions. For some systems, we find the exact value of this exponent. These results give sufficient (and, in some cases, necessary) conditions for a system’s exponential stability which are invariant with respect to the delay. We give examples that illustrate our method.     

二阶神经网络的全局指数稳定性分析

当神经网络应用于最优化计算时,理想的情形是只有一个全局渐近稳定的平衡点,并且以指数速度趋近于平衡点,从而减少神经网络所需计算时间,二阶神经网络较一般神经网络具有更快的收敛速度,对于二阶连续型Hopfield神经网络,用Lyapunov方法讨论平衡点的全局指数稳定性,给出了平衡点全局指数稳定的几个判别准则,作为特例,获得了连续型Hopfield神经网络全局指数稳定的新判据。     

Implicit-system approach to the robust stability for a class of singularly perturbed linear systems

Asymptotic stability and the complex stability radius of a class of singularly perturbed systems of linear differential-algebraic equations (DAEs) are studied. The asymptotic behavior of the stability radius for a singularly perturbed implicit system is characterized as the parameter in the leading term tends to zero. The main results are obtained in direct and short ways which involve some basic results in linear algebra and classical analysis, only. Our results can be extended to other singular perturbation problems for DAEs of more general form.     

Stabilization of linear time varying systems over uncertain channels

In this paper, we study the problem of control of discrete‐time linear time varying systems over uncertain channels. The uncertainty in the channels is modeled as a stochastic random variable. We use exponential mean square stability of the closed‐loop system as a stability criterion. We show that fundamental limitations arise for the mean square exponential stabilization for the closed‐loop system expressed in terms of statistics of channel uncertainty and the positive Lyapunov exponent of the open‐loop uncontrolled system. Our results generalize the existing results known in the case of linear time invariant systems, where Lyapunov exponents are shown to emerge as the generalization of eigenvalues from linear time invariant systems to linear time varying systems. Simulation results are presented to verify the main results of this paper. Copyright © 2012 John Wiley & Sons, Ltd.     

A less conservative stability test for second-order linear time-varying vector differential equations

System stability and stability bounds play an essential role in control theory. This note is concerned with the exponential stability of a class of second-order linear time-varying vector differential equations with real piecewise continuous coefficient matrices. A less conservative explicit condition for stability of such a system is derived using the matrix measure theory and a more accurate upper bound for the decay exponent of its stable solution is established. Examples are included for illustration.     

Impulses-induced exponential stability in recurrent delayed neural networks

The present paper formulates and studies a model of recurrent neural networks with time-varying delays in the presence of impulsive connectivity among the neurons. This model can well describe practical architectures of more realistic neural networks. Some novel yet generic criteria for global exponential stability of such neural networks are derived by establishing an extended Halanay differential inequality on impulsive delayed dynamical systems. The distinctive feature of this work is to address exponential stability issues without a priori stability assumption for the corresponding delayed neural networks without impulses. It is shown that the impulses in neuronal connectivity play an important role in inducing global exponential stability of recurrent delayed neural networks even if it may be unstable or chaotic itself. Furthermore, example and simulation are given to illustrate the practical nature of the novel results.     

Stability of a Damped Hyperbolic Timoshenko System Coupled with a Heat Equation

We consider a one‐dimensional damped hyperbolic Timoshenko beam that is coupled with a heat equation. When its wave speeds are different, it is known that the Timoshenko beam that is coupled with a heat equation, under Cattaneo's law, does not have exponential stability. With two internal dampings being introduced, in this paper, we show that the system under Cattaneo's law is exponentially stable. We also show the exponential stability when the system is considered under Fourier's law.     

Improved exponential estimates for neutral systems

Improved exponential estimates and a new sufficient condition for the exponential stability of neutral type time‐delay systems are established in terms of linear matrix inequalities (LMIs). A new Lyapunov‐Krasovskii functional candidate with appropriately constructed exponential terms is introduced to prove the exponential stability and to reduce the conservatism. For the uncertain case, a corresponding condition for the robust exponential stability is also given. It is shown by numerical examples that the proposed conditions are less conservative than existing results. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

具分布参数的随机Hopfield神经网络的指数稳定

基于随机Fubini定理,将随机偏微分方程描述的Hopfield神经网络系统转化为用相应的随机常微分方程来描述.利用关于空间变量平均的Lyapunov函数与Ito^公式,通过对所构造的Lyapunov函数在Ito^微分规则下对相应系统求导的方法,获得了系统指数稳定的代数判据及其Lyapunov指数估计.实现了运用Lyapunov直接法对分布参数系统稳定性的研究.     

Quantum Computing and Information Extraction for Dynamical Quantum Systems

We discuss the simulation of complex dynamical systems on a quantum computer. We show that a quantum computer can be used to efficiently extract relevant physical information. It is possible to simulate the dynamical localization of classical chaos and extract the localization length with quadratic speed up with respect to any known classical computation. We can also compute with algebraic speed up the diffusion coefficient and the diffusion exponent, both in the regimes of Brownian and anomalous diffusion. Finally, we show that it is possible to extract the fidelity of the quantum motion, which measures the stability of the system under perturbations, with exponential speed up. The so-called quantum sawtooth map model is used as a test bench to illustrate these results. PACS: 03.67.Lx, 05.45.Mt     

Size distribution and waiting times for the avalanches of the Cell Network Model of Fracture

The Cell Network Model is a fracture model recently introduced that resembles the microscopical structure and drying process of the parenchymatous tissue of the Bamboo Guadua angustifolia. The model exhibits a power-law distribution of avalanche sizes, with exponent −3.0 when the breaking thresholds are randomly distributed with uniform probability density. Hereby we show that the same exponent also holds when the breaking thresholds obey a broad set of Weibull distributions, and that the humidity decrements between successive avalanches (the equivalent to waiting times for this model) follow in all cases an exponential distribution. Moreover, the fraction of remaining junctures shows an exponential decay in time. In addition, introducing partial breakings and cumulative damages induces a crossover behavior between two power-laws in the histogram of avalanche sizes. This results support the idea that the Cell Network Model may be in the same universality class as the Random Fuse Model.     

On solving hybrid optimal control problems with higher index DAEs

The paper deals with hybrid optimal control problems described by higher index differential–algebraic equations (DAEs). We introduce a numerical procedure for solving these problems. The procedure has the following features: it is based on the appropriately defined adjoint equations formulated for the discretized equations being the result of the numerical integration of systems equations by an implicit Runge–Kutta method; the consistent initialization procedure is applied whenever control functions jumps, or state variables transition occurs. The procedure can cope with hybrid optimal control problems which are defined by DAEs with the index not exceeding three. Our approach does not require differentiation of some system equations in order to transform higher index DAEs to the underlying ordinary differential equations (ODEs). The presented numerical examples show that the proposed approach can be used to solve efficiently hybrid optimal control problems with higher index DAEs.     

On the reliability exponent of the additive exponential noise channel

We study the reliability exponent of the additive exponential noise channel (AENC) in the absence as well as in the presence of noiseless feedback. For rates above the critical rate, the fixed-transmission-time reliability exponent of the AENC is completely determined, while below the critical rate an expurgated exponent is obtained. Using a fixed-block-length code ensemble (with block length denoting the number of recorded departures), we obtain a lower bound on the random-transmission-time reliability exponent of the AENC. Finally, with a variable-block-length code ensemble, a lower bound on the random-transmission-time zero-error capacity of the AENC with noiseless feedback is obtained.     

Regularity of distributional differential algebraic equations

Time-varying differential algebraic equations (DAEs) of the form E[(x)\dot]=Ax+f{E\dot{x}=Ax+f} are considered. The solutions x and the inhomogeneities f are assumed to be distributions (generalized functions). As a new approach, distributional entries in the time-varying coefficient matrices E and A are allowed as well. Since a multiplication for general distributions is not possible, the smaller space of piecewise-smooth distributions is introduced. This space consists of distributions which could be written as the sum of a piecewise-smooth function and locally finite Dirac impulses and derivatives of Dirac impulses. A restriction can be defined for the space of piecewise-smooth distributions, this restriction is used to study DAEs with inconsistent initial values; basically, it is assumed that some past trajectory for x is given and the DAE is activated at some initial time. If this initial trajectory problem has a unique solution for all initial trajectories and all inhomogeneities, then the DAE is called regular. This generalizes the regularity for classical DAEs (i.e. a DAE with constant coefficients). Sufficient and necessary conditions for the regularity of distributional DAEs are given.     

Minimax state estimation for linear discrete-time differential-algebraic equations

This paper presents a state estimation approach for an uncertain linear equation with a non-invertible operator in Hilbert space. The approach addresses linear equations with uncertain deterministic input and noise in the measurements, which belong to a given convex closed bounded set. A new notion of a minimax observable subspace is introduced. By means of the presented approach, new equations describing the dynamics of a minimax recursive estimator for discrete-time non-causal differential-algebraic equations (DAEs) are presented. For the case of regular DAEs it is proved that the estimator’s equation coincides with the equation describing the seminal Kalman filter. The properties of the estimator are illustrated by a numerical example.     

Nonlinear measures: a new approach to exponential stabilityanalysis for Hopfield-type neural networks

In this paper, a new concept called nonlinear measure is introduced to quantify stability of nonlinear systems in the way similar to the matrix measure for stability of linear systems. Based on the new concept, a novel approach for stability analysis of neural networks is developed. With this approach, a series of new sufficient conditions for global and local exponential stability of Hopfield type neural networks is presented, which generalizes those existing results. By means of the introduced nonlinear measure, the exponential convergence rate of the neural networks to stable equilibrium point is estimated, and, for local stability, the attraction region of the stable equilibrium point is characterized. The developed approach can be generalized to stability analysis of other general nonlinear systems.