Asymptotic stability for time-variant systems and observability: Uniform and nonuniform criteria

This paper presents some new criteria for uniform and nonuniform asymptotic stability of equilibria for time-variant differential equations and this within a Lyapunov approach. The stability criteria are formulated in terms of certain observability conditions with the output derived from the Lyapunov function. For some classes of systems, this system theoretic interpretation proves to be fruitful since—after establishing the invariance of observability under output injection—this enables us to check the stability criteria on a simpler system. This procedure is illustrated for some classical examples.     

Packing Random Intervals On-Line

Starting at time 0, unit-length intervals arrive and are placed on the positive real line by a unit-intensity Poisson process in two dimensions; the probability of an interval arriving in the time interval [t,t+ t] with its left endpoint in [y,y+ y] is t y + o( t y ). Fix x 0. An arriving interval is accepted if and only if it is contained in [0,x] and overlaps no interval already accepted. We study the number N _x (t) of intervals accepted during [0,t] . By Laplace-transform methods, we derive large-x estimates of EN _x (t) and VarN _x (t) with error terms exponentially small in x uniformly in t (0,T) , where T is any fixed positive constant. We prove that, as , EN _x (t) , VarN _x (t) , uniformly in t (0,T) , where and are given by explicit, albeit complicated formulas. Using these asymptotic estimates we show that N _x (t) satisfies a central limit theorem, i.e., for any fixed t where (0,1) is a standard normal random variable, and denotes convergence in distribution. This stochastic, on-line interval packing problem generalizes the classical parking problem, the latter corresponding only to the absorbing states of the interval packing process, where successive packed intervals are separated by gaps less than 1 in length. We verify that, as , (t) and (t) converge to _* = 0.748 . . . and _* = 0.03815 . . ., the constants of Rényi and Mackenzie for the parking problem. Thus, by comparison with the parking analysis in a single space variable, ours is a transient analysis involving both a time and a space variable. Our interval packing problem has applications similar to those of the parking problem in the physical sciences, but the primary source of our interest is the modeling of reservation systems, especially those designed for multimedia communication systems to handle high-bandwidth, real-time demands. Received December 4, 1997; revised February 15, 1998.     

关于Volterra积分微分方程的渐近稳定性

本文利用V-泛函方法讨论了Volterra积分微分方程x′=A(t)x+integral from n=0 to t c(t,s)x(s)ds零解的渐近稳定性,特别是引入了向量V-泛函方法,得到若干判别法,有的十分简便。     

一类非线性常微分方程解的渐近性态

本文研究了一类柯西问题（１）含参量α的解ｙ（ｘ，α）的渐近性态，证明了在［０，＋∞）上ｙ（ｘ，α）是振幅衰减且趋于零的振荡解。     

删失数据下部分线性模型的估计及渐近正态性

研究了删失数据下的部分线性回归模型,通过数据变换,利用局部多项式方法和两阶段估计,给出了函数系数和参数部分的估计。并给出了估计的渐近正态性。     

The Periodogram of fractional processes1

Abstract. We analyse asymptotic properties of the discrete Fourier transform and the periodogram of time series obtained through (truncated) linear filtering of stationary processes. The class of filters contains the fractional differencing operator and its coefficients decay at an algebraic rate, implying long‐range‐dependent properties for the filtered processes when the degree of integration α is positive. These include fractional time series which are nonstationary for any value of the memory parameter (α ≠ 0) and possibly nonstationary trending (α ≥ 0.5). We consider both fractional differencing or integration of weakly dependent and long‐memory stationary time series. The results obtained for the moments of the Fourier transform and the periodogram at Fourier frequencies in a degenerating band around the origin are weaker compared with the stationary nontruncated case for α > 0, but sufficient for the analysis of parametric and semiparametric memory estimates. They are applied to the study of the properties of the log‐periodogram regression estimate of the memory parameter α for Gaussian processes, for which asymptotic normality could not be showed using previous results. However, only consistency can be showed for the trending cases, 0.5 ≤ α < 1. Several detrending and initialization mechanisms are studied and only local conditions on spectral densities of stationary input series and transfer functions of filters are assumed.     

Online maintenance policy for a deteriorating system with random change of mode

Most of maintenance policies proposed in the literature for gradually deteriorating systems, consider a stationary deterioration process. This paper is an attempt to take into account stochastically deteriorating systems which are subject to a sudden change in their degradation process. A technical device subject to gradual degradation is considered. It is assumed that the level of degradation can be resumed by a single scalar variable. An online maintenance decision rule is proposed, which makes it possible to take into account in real time the online information available on the operating mode of the system as well as its actual deterioration level. We show the efficiency of considering online decision rules for maintenance with respect to traditional maintenance policies based on a static alarm threshold. Numerical simulations are given, to assess and optimize the performance of the maintained system from its asymptotic unavailability point of view. It is compared to the results obtained with classical control-limit maintenance policies.     

On a convective condition in the diffusion of a solvent into a polymer with non-constant conductivity coefficient

We studied a one-dimensional free boundary problem arising in the polymer industry. Imposing a convective boundary condition, the solution has an interesting asymptotic behavior. It is found that the free boundary is bounded by a constant which does not depend on the conductivity coefficient, which holds even if the diffusion process is non-linear. Numerical methods are presented to compute the solutions and to compare with the theoretical results.     

Trimming and Tapering Semi‐Parametric Estimates in Asymmetric Long Memory Time Series

Abstract. This paper considers semi‐parametric frequency domain inference for seasonal or cyclical time series with asymmetric long memory properties. It is shown that tapering the data reduces the bias caused by the asymmetry of the spectral density at the cyclical frequency. We provide a joint treatment of different tapering schemes and of the log‐periodogram regression and Gaussian semi‐parametric estimates of the memory parameters. Tapering allows for a less restrictive trimming of frequencies for the analysis of the asymptotic properties of both estimates when allowing for asymmetries. Simple rules for inference are feasible thanks to tapering and their validity in finite samples is investigated in a simulation exercise and for an empirical example.     

Asymptotic buckling analysis of imperfect shallow spherical shells on non-linear elastic foundation

An asymptotic solution is formulated for non-linear buckling of elastically restrained imperfect shallow spherical shells continuously supported on a non-linear elastic foundation. The asymptotic iteration method is introduced to result in a cubic non-linear analytical relation between the external load and central transverse displacement (deflection) of the structures incorporating the effects of geometrical imperfection, edge-restraint coefficients, moduli of foundation and characteristic geometrical parameter. The resulting expression can be used easily to evaluate the effects of these factors on buckling behaviors. Numerical examples are given, and comparisons of the available results show validity of the method suggested in the present work.